Bernoulli Articles (Project Euclid)
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A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables
http://projecteuclid.org/euclid.bj/1274821072
<strong>Michael V. Boutsikas</strong>, <strong>Eutichia Vaggelatou</strong><p><strong>Source: </strong>Bernoulli, Volume 16, Number 2, 301--330.</p><p><strong>Abstract:</strong><br/>
Let X 1 , X 2 , …, X n be a sequence of independent or locally dependent random variables taking values in ℤ + . In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑ i =1 n X i and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This “smoothness factor” is of order O( σ −2 ), according to a heuristic argument, where σ 2 denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.
</p>projecteuclid.org/euclid.bj/1274821072_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTRecovering the Brownian coalescent point process from the Kingman coalescent by conditional samplinghttps://projecteuclid.org/euclid.bj/1544605242<strong>Amaury Lambert</strong>, <strong>Emmanuel Schertzer</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 148--173.</p><p><strong>Abstract:</strong><br/>
We consider a continuous population whose dynamics is described by the standard stationary Fleming–Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert and Uribe Bravo ( P-Adic Numbers Ultrametric Anal. Appl. 9 (2017) 22–38)), we show that the genealogy of the descendance of the MRCA of the sample on the timescale $\varepsilon$ converges as $\varepsilon\to0$. The limit is the so-called Brownian coalescent point process (CPP) stopped at an independent Gamma random variable with parameter $n$, which can be seen as the genealogy at a large time of the total population of a rescaled critical birth–death process, biased by the $n$th power of its size. Second, we show that in this limit the coalescence times of the $n$ sampled individuals are i.i.d. uniform random variables in $(0,1)$. These results provide a coupling between two standard models for the genealogy of a random exchangeable population: the Kingman coalescent and the Brownian CPP.
</p>projecteuclid.org/euclid.bj/1544605242_20181212040117Wed, 12 Dec 2018 04:01 ESTSubexponential decay in kinetic Fokker–Planck equation: Weak hypocoercivityhttps://projecteuclid.org/euclid.bj/1544605243<strong>Shulan Hu</strong>, <strong>Xinyu Wang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 174--188.</p><p><strong>Abstract:</strong><br/>
We consider here quantitative convergence to equilibrium for the kinetic Fokker–Planck equation. We present a weak hypocoercivity approach à la Villani, using weak Poincaré inequality, ensuring subexponential convergence to equilibrium in $\mathcal{H}^{1}$ sense or in $L^{2}$ sense.
</p>projecteuclid.org/euclid.bj/1544605243_20181212040117Wed, 12 Dec 2018 04:01 ESTPólya urns with immigration at random timeshttps://projecteuclid.org/euclid.bj/1544605244<strong>Erol Peköz</strong>, <strong>Adrian Röllin</strong>, <strong>Nathan Ross</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 189--220.</p><p><strong>Abstract:</strong><br/>
We study the number of white balls in a classical Pólya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the inter-arrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending on the inter-arrival distribution and are difficult to describe explicitly. However, we show that the limits are fixed points of certain probabilistic distributional transformations, and this fact provides a proof of convergence and leads to properties of the limits. The model can alternatively be viewed as a preferential attachment random graph model where added vertices initially have a random number of edges, and from this perspective, our results describe the limit of the degree of a fixed vertex.
</p>projecteuclid.org/euclid.bj/1544605244_20181212040117Wed, 12 Dec 2018 04:01 ESTFeller property of the multiplicative coalescent with linear deletionhttps://projecteuclid.org/euclid.bj/1544605245<strong>Balázs Ráth</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 221--240.</p><p><strong>Abstract:</strong><br/>
We modify the definition of Aldous’ multiplicative coalescent process ( Ann. Probab. 25 (1997) 812–854) and introduce the multiplicative coalescent with linear deletion (MCLD). A state of this process is a square-summable decreasing sequence of cluster sizes. Pairs of clusters merge with a rate equal to the product of their sizes and clusters are deleted with a rate linearly proportional to their size. We prove that the MCLD is a Feller process. This result is a key ingredient in the description of scaling limits of the evolution of component sizes of the mean field frozen percolation model ( J. Stat. Phys. 137 (2009) 459–499) and the so-called rigid representation of such scaling limits ( Electron. J. Probab. To appear).
</p>projecteuclid.org/euclid.bj/1544605245_20181212040117Wed, 12 Dec 2018 04:01 ESTAsymptotic power of Rao’s score test for independence in high dimensionshttps://projecteuclid.org/euclid.bj/1544605246<strong>Dennis Leung</strong>, <strong>Qiman Shao</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 241--263.</p><p><strong>Abstract:</strong><br/>
Let $\mathbf{R}$ be the Pearson correlation matrix of $m$ normal random variables. The Rao’s score test for the independence hypothesis $H_{0}:\mathbf{R}=\mathbf{I}_{m}$, where $\mathbf{I}_{m}$ is the identity matrix of dimension $m$, was first considered by Schott ( Biometrika 92 (2005) 951–956) in the high dimensional setting. In this paper, we study the exact power function of this test, under an asymptotic regime in which both $m$ and the sample size $n$ tend to infinity with the ratio $m/n$ upper bounded by a constant. In particular, our result implies that the Rao’s score test is minimax rate-optimal for detecting the dependency signal $\Vert\mathbf{R}-\mathbf{I}_{m}\Vert_{F}$ of order $\sqrt{m/n}$, where $\Vert\cdot\Vert_{F}$ is the matrix Frobenius norm.
</p>projecteuclid.org/euclid.bj/1544605246_20181212040117Wed, 12 Dec 2018 04:01 ESTExtreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimizationhttps://projecteuclid.org/euclid.bj/1544605247<strong>Abdelaati Daouia</strong>, <strong>Stéphane Girard</strong>, <strong>Gilles Stupfler</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 264--309.</p><p><strong>Abstract:</strong><br/>
The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of $L^{p}$ optimization. These generalized $L^{p}$-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for $1<p<2$. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate $L^{p}$-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme $L^{p}$-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme $L^{p}$-quantiles and elaborate the choice of $p$ through applications to some simulated and financial real data.
</p>projecteuclid.org/euclid.bj/1544605247_20181212040117Wed, 12 Dec 2018 04:01 ESTError bounds for sequential Monte Carlo samplers for multimodal distributionshttps://projecteuclid.org/euclid.bj/1544605248<strong>Daniel Paulin</strong>, <strong>Ajay Jasra</strong>, <strong>Alexandre Thiery</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 310--340.</p><p><strong>Abstract:</strong><br/>
In this paper, we provide bounds on the asymptotic variance for a class of sequential Monte Carlo (SMC) samplers designed for approximating multimodal distributions. Such methods combine standard SMC methods and Markov chain Monte Carlo (MCMC) kernels. Our bounds improve upon previous results, and unlike some earlier work, they also apply in the case when the MCMC kernels can move between the modes. We apply our results to the Potts model from statistical physics. In this case, the problem of sharp peaks is encountered. Earlier methods, such as parallel tempering, are only able to sample from it at an exponential (in an important parameter of the model) cost. We propose a sequence of interpolating distributions called interpolation to independence , and show that the SMC sampler based on it is able to sample from this target distribution at a polynomial cost. We believe that our method is generally applicable to many other distributions as well.
</p>projecteuclid.org/euclid.bj/1544605248_20181212040117Wed, 12 Dec 2018 04:01 ESTOn the convex Poincaré inequality and weak transportation inequalitieshttps://projecteuclid.org/euclid.bj/1544605249<strong>Radosław Adamczak</strong>, <strong>Michał Strzelecki</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 341--374.</p><p><strong>Abstract:</strong><br/>
We prove that for a probability measure on $\mathbb{R}^{n}$, the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line.
The proof relies on modified logarithmic Sobolev inequalities of Bobkov–Ledoux type for convex and concave functions, which are of independent interest.
We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar, and Tetali.
</p>projecteuclid.org/euclid.bj/1544605249_20181212040117Wed, 12 Dec 2018 04:01 ESTOn the longest gap between power-rate arrivalshttps://projecteuclid.org/euclid.bj/1544605250<strong>Søren Asmussen</strong>, <strong>Jevgenijs Ivanovs</strong>, <strong>Johan Segers</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 375--394.</p><p><strong>Abstract:</strong><br/>
Let $L_{t}$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $\lambda_{t}$ proportional to $t^{\alpha-1}$ for some $\alpha\in(0,1)$. It is shown that $\lambda_{t}L_{t}-b_{t}$ has a limiting Gumbel distribution for suitable constants $b_{t}$ and that the distance of this longest gap from $t$ is asymptotically of the form $(t/\log t)E$ for an exponential random variable $E$. The analysis is performed via weak convergence of related point processes. Subject to a weak technical condition, the results are extended to include a slowly varying term in $\lambda_{t}$.
</p>projecteuclid.org/euclid.bj/1544605250_20181212040117Wed, 12 Dec 2018 04:01 ESTNonparametric depth and quantile regression for functional datahttps://projecteuclid.org/euclid.bj/1544605251<strong>Joydeep Chowdhury</strong>, <strong>Probal Chaudhuri</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 395--423.</p><p><strong>Abstract:</strong><br/>
We investigate nonparametric regression methods based on spatial depth and quantiles when the response and the covariate are both functions. As in classical quantile regression for finite dimensional data, regression techniques developed here provide insight into the influence of the functional covariate on different parts, like the center as well as the tails, of the conditional distribution of the functional response. Depth and quantile based nonparametric regression methods are useful to detect heteroscedasticity in functional regression. We derive the asymptotic behavior of the nonparametric depth and quantile regression estimates, which depend on the small ball probabilities in the covariate space. Our nonparametric regression procedures are used to analyze a dataset about the influence of per capita GDP on saving rates for 125 countries, and another dataset on the effects of per capita net disposable income on the sale of cigarettes in some states in the US.
</p>projecteuclid.org/euclid.bj/1544605251_20181212040117Wed, 12 Dec 2018 04:01 ESTEstimation and hypotheses testing in boundary regression modelshttps://projecteuclid.org/euclid.bj/1544605252<strong>Holger Drees</strong>, <strong>Natalie Neumeyer</strong>, <strong>Leonie Selk</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 424--463.</p><p><strong>Abstract:</strong><br/>
Consider a nonparametric regression model with one-sided errors and regression function in a general Hölder class. We estimate the regression function via minimization of the local integral of a polynomial approximation. We show uniform rates of convergence for the simple regression estimator as well as for a smooth version. These rates carry over to mean regression models with a symmetric and bounded error distribution. In such a setting, one obtains faster rates for irregular error distributions concentrating sufficient mass near the endpoints than for the usual regular distributions. The results are applied to prove asymptotic $\sqrt{n}$-equivalence of a residual-based (sequential) empirical distribution function to the (sequential) empirical distribution function of unobserved errors in the case of irregular error distributions. This result is remarkably different from corresponding results in mean regression with regular errors. It can readily be applied to develop goodness-of-fit tests for the error distribution. We present some examples and investigate the small sample performance in a simulation study. We further discuss asymptotically distribution-free hypotheses tests for independence of the error distribution from the points of measurement and for monotonicity of the boundary function as well.
</p>projecteuclid.org/euclid.bj/1544605252_20181212040117Wed, 12 Dec 2018 04:01 ESTConsistent order estimation for nonparametric hidden Markov modelshttps://projecteuclid.org/euclid.bj/1544605253<strong>Luc Lehéricy</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 464--498.</p><p><strong>Abstract:</strong><br/>
We consider the problem of estimating the number of hidden states (the order ) of a nonparametric hidden Markov model (HMM). We propose two different methods and prove their almost sure consistency without any prior assumption, be it on the order or on the emission distributions. This is the first time a consistency result is proved in such a general setting without using restrictive assumptions such as a priori upper bounds on the order or parametric restrictions on the emission distributions. Our main method relies on the minimization of a penalized least squares criterion. In addition to the consistency of the order estimation, we also prove that this method yields rate minimax adaptive estimators of the parameters of the HMM – up to a logarithmic factor. Our second method relies on estimating the rank of a matrix obtained from the distribution of two consecutive observations. Finally, numerical experiments are used to compare both methods and study their ability to select the right order in several situations.
</p>projecteuclid.org/euclid.bj/1544605253_20181212040117Wed, 12 Dec 2018 04:01 ESTCentral limit theorem for Fourier transform and periodogram of random fieldshttps://projecteuclid.org/euclid.bj/1544605254<strong>Magda Peligrad</strong>, <strong>Na Zhang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 499--520.</p><p><strong>Abstract:</strong><br/>
In this paper, we show that the limiting distribution of the real and the imaginary part of the Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field’s spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense. The results go beyond the Bernoulli fields and apply to both short range and long range dependence. They can be easily applied to derive the asymptotic behavior of the periodogram associated to the random field. The method of proof is based on new probabilistic methods involving martingale approximations and also on borrowed and new tools from harmonic analysis. Several examples to linear, Volterra and Gaussian random fields will be presented.
</p>projecteuclid.org/euclid.bj/1544605254_20181212040117Wed, 12 Dec 2018 04:01 ESTA multidimensional analogue of the arcsine law for the number of positive terms in a random walkhttps://projecteuclid.org/euclid.bj/1544605255<strong>Zakhar Kabluchko</strong>, <strong>Vladislav Vysotsky</strong>, <strong>Dmitry Zaporozhets</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 521--548.</p><p><strong>Abstract:</strong><br/>
Consider a random walk $S_{i}=\xi_{1}+\cdots+\xi_{i}$, $i\in\mathbb{N}$, whose increments $\xi_{1},\xi_{2},\ldots$ are independent identically distributed random vectors in $\mathbb{R}^{d}$ such that $\xi_{1}$ has the same law as $-\xi_{1}$ and $\mathbb{P}[\xi_{1}\in H]=0$ for every affine hyperplane $H\subset\mathbb{R}^{d}$. Our main result is the distribution-free formula
\[\mathbb{E}\bigg[\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\mathbb{1}_{\{0\notin\operatorname{Conv}(S_{i_{1}},\ldots,S_{i_{k}})\}}\bigg]=2\binom{n}{k}\frac{B(k,d-1)+B(k,d-3)+\cdots}{2^{k}k!},\] where the $B(k,j)$’s are defined by their generating function $(t+1)(t+3)\ldots(t+2k-1)=\sum_{j=0}^{k}B(k,j)t^{j}$. The expected number of $k$-tuples above admits the following geometric interpretation: it is the expected number of $k$-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type $B_{n}$ that are not intersected by a generic linear subspace $L\subset\mathbb{R}^{n}$ of codimension $d$. The case $d=1$ turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required.
</p>projecteuclid.org/euclid.bj/1544605255_20181212040117Wed, 12 Dec 2018 04:01 ESTLimit properties of the monotone rearrangement for density and regression function estimationhttps://projecteuclid.org/euclid.bj/1544605256<strong>Dragi Anevski</strong>, <strong>Anne-Laure Fougères</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 549--583.</p><p><strong>Abstract:</strong><br/>
The monotone rearrrangement algorithm was introduced by Hardy, Littlewood and Pólya as a sorting device for functions. Assuming that $x$ is a monotone function and that an estimate $x_{n}$ of $x$ is given, consider the monotone rearrangement $\hat{x}_{n}$ of $x_{n}$. This new estimator is shown to be uniformly consistent as soon as $x_{n}$ is. Under suitable assumptions, pointwise limit distribution results for $\hat{x}_{n}$ are obtained. The framework is general and allows for weakly dependent and long range dependent stationary data. Applications in monotone density and regression function estimation are detailed. Asymptotics for rearrangement estimators with vanishing derivatives are also obtained in these two contexts.
</p>projecteuclid.org/euclid.bj/1544605256_20181212040117Wed, 12 Dec 2018 04:01 ESTSequential Monte Carlo as approximate sampling: bounds, adaptive resampling via $\infty$-ESS, and an application to particle Gibbshttps://projecteuclid.org/euclid.bj/1544605257<strong>Jonathan H. Huggins</strong>, <strong>Daniel M. Roy</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 584--622.</p><p><strong>Abstract:</strong><br/>
Sequential Monte Carlo (SMC) algorithms were originally designed for estimating intractable conditional expectations within state-space models, but are now routinely used to generate approximate samples in the context of general-purpose Bayesian inference. In particular, SMC algorithms are often used as subroutines within larger Monte Carlo schemes, and in this context, the demands placed on SMC are different: control of mean-squared error is insufficient—one needs to control the divergence from the target distribution directly. Towards this goal, we introduce the conditional adaptive resampling particle filter, building on the work of Gordon, Salmond, and Smith (1993), Andrieu, Doucet, and Holenstein (2010), and Whiteley, Lee, and Heine (2016). By controlling a novel notion of effective sample size, the $\infty$-ESS, we establish the efficiency of the resulting SMC sampling algorithm, providing an adaptive resampling extension of the work of Andrieu, Lee, and Vihola (2018). We apply our results to arrive at new divergence bounds for SMC samplers with adaptive resampling as well as an adaptive resampling version of the Particle Gibbs algorithm with the same geometric-ergodicity guarantees as its nonadaptive counterpart.
</p>projecteuclid.org/euclid.bj/1544605257_20181212040117Wed, 12 Dec 2018 04:01 ESTOptimal rates of statistical seriationhttps://projecteuclid.org/euclid.bj/1544605258<strong>Nicolas Flammarion</strong>, <strong>Cheng Mao</strong>, <strong>Philippe Rigollet</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 623--653.</p><p><strong>Abstract:</strong><br/>
Given a matrix, the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimation of the matrices. Specifically, when the columns are either unimodal or monotone, we show that the least squares estimator is optimal up to logarithmic factors and adapts to matrices with a certain natural structure. Finally, we propose a computationally efficient estimator in the monotonic case and study its performance both theoretically and experimentally. Our work is at the intersection of shape constrained estimation and recent work that involves permutation learning, such as graph denoising and ranking.
</p>projecteuclid.org/euclid.bj/1544605258_20181212040117Wed, 12 Dec 2018 04:01 ESTSecond order correctness of perturbation bootstrap M-estimator of multiple linear regression parameterhttps://projecteuclid.org/euclid.bj/1544605259<strong>Debraj Das</strong>, <strong>S.N. Lahiri</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 654--682.</p><p><strong>Abstract:</strong><br/>
Consider the multiple linear regression model $y_{i}=\mathbf{x}'_{i}\boldsymbol{\beta}+\varepsilon_{i}$, where $\varepsilon_{i}$’s are independent and identically distributed random variables, $\mathbf{x}_{i}$’s are known design vectors and $\boldsymbol{\beta}$ is the $p\times1$ vector of parameters. An effective way of approximating the distribution of the M-estimator $\bar{\boldsymbol{\beta}}_{n}$, after proper centering and scaling, is the Perturbation Bootstrap Method. In this current work, second order results of this non-naive bootstrap method have been investigated. Second order correctness is important for reducing the approximation error uniformly to $o(n^{-1/2})$ to get better inferences. We show that the classical studentized version of the bootstrapped estimator fails to be second order correct. We introduce an innovative modification in the studentized version of the bootstrapped statistic and show that the modified bootstrapped pivot is second order correct (S.O.C.) for approximating the distribution of the studentized M-estimator. Additionally, we show that the Perturbation Bootstrap continues to be S.O.C. when the errors $\varepsilon_{i}$’s are independent, but may not be identically distributed. These findings establish perturbation Bootstrap approximation as a significant improvement over asymptotic normality in the regression M-estimation.
</p>projecteuclid.org/euclid.bj/1544605259_20181212040117Wed, 12 Dec 2018 04:01 ESTRandom polymers on the complete graphhttps://projecteuclid.org/euclid.bj/1544605260<strong>Francis Comets</strong>, <strong>Gregorio Moreno</strong>, <strong>Alejandro F. Ramí rez</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 683--711.</p><p><strong>Abstract:</strong><br/>
Consider directed polymers in a random environment on the complete graph of size $N$. This model can be formulated as a product of i.i.d. $N\times N$ random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path.
Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index $\alpha\in(0,1)$. Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large $N$ asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws.
</p>projecteuclid.org/euclid.bj/1544605260_20181212040117Wed, 12 Dec 2018 04:01 ESTSum rules and large deviations for spectral matrix measureshttps://projecteuclid.org/euclid.bj/1544605261<strong>Fabrice Gamboa</strong>, <strong>Jan Nagel</strong>, <strong>Alain Rouault</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 712--741.</p><p><strong>Abstract:</strong><br/>
In the paradigm of random matrices, one of the most classical object under study is the empirical spectral distribution. This random measure is the uniform distribution supported by the eigenvalues of the random matrix. In this paper, we give large deviation theorems for another popular object built on Hermitian random matrices: the spectral measure. This last probability measure is a random weighted version of the empirical spectral distribution. The weights involve the eigenvectors of the random matrix. We have previously studied the large deviations of the spectral measure in the case of scalar weights. Here, we will focus on matrix valued weights. Our probabilistic results lead to deterministic ones called “sum rules” in spectral theory. A sum rule relative to a reference measure on $\mathbb{R}$ is a relationship between the reversed Kullback–Leibler divergence of a positive measure on $\mathbb{R}$ and some non-linear functional built on spectral elements related to this measure. By using only probabilistic tools of large deviations, we extend the sum rules to the case of Hermitian matrix-valued measures.
</p>projecteuclid.org/euclid.bj/1544605261_20181212040117Wed, 12 Dec 2018 04:01 ESTWeak subordination of multivariate Lévy processes and variance generalised gamma convolutionshttps://projecteuclid.org/euclid.bj/1544605262<strong>Boris Buchmann</strong>, <strong>Kevin W. Lu</strong>, <strong>Dilip B. Madan</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 742--770.</p><p><strong>Abstract:</strong><br/>
Subordinating a multivariate Lévy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Lévy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create Lévy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs $(T,X)$ of Lévy processes which creates a Lévy process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary Lévy process with possibly dependent components. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. We illustrate our methods giving a weak formulation of the variance-$\boldsymbol{\alpha}$-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of Lévy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.
</p>projecteuclid.org/euclid.bj/1544605262_20181212040117Wed, 12 Dec 2018 04:01 ESTEstimating the interaction graph of stochastic neural dynamicshttps://projecteuclid.org/euclid.bj/1544605263<strong>Aline Duarte</strong>, <strong>Antonio Galves</strong>, <strong>Eva Löcherbach</strong>, <strong>Guilherme Ost</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 1, 771--792.</p><p><strong>Abstract:</strong><br/>
In this paper, we address the question of statistical model selection for a class of stochastic models of biological neural nets. Models in this class are systems of interacting chains with memory of variable length. Each chain describes the activity of a single neuron, indicating whether it spikes or not at a given time. The spiking probability of a given neuron depends on the time evolution of its presynaptic neurons since its last spike time. When a neuron spikes, its potential is reset to a resting level and postsynaptic current pulses are generated, modifying the membrane potential of all its postsynaptic neurons . The relationship between a neuron and its pre- and postsynaptic neurons defines an oriented graph, the interaction graph of the model. The goal of this paper is to estimate this graph based on the observation of the spike activity of a finite set of neurons over a finite time. We provide explicit exponential upper bounds for the probabilities of under- and overestimating the interaction graph restricted to the observed set and obtain the strong consistency of the estimator. Our result does not require stationarity nor uniqueness of the invariant measure of the process.
</p>projecteuclid.org/euclid.bj/1544605263_20181212040117Wed, 12 Dec 2018 04:01 ESTExpansion for moments of regression quantiles with applications to nonparametric testinghttps://projecteuclid.org/euclid.bj/1551862835<strong>Enno Mammen</strong>, <strong>Ingrid Van Keilegom</strong>, <strong>Kyusang Yu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 793--827.</p><p><strong>Abstract:</strong><br/>
We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya–Watson quantile smoothing estimator.
An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth.
Our alternative mathematical approach requires the calculation of moments of Nadaraya–Watson quantile regression estimators. This calculation is done by application of higher order Edgeworth expansions.
</p>projecteuclid.org/euclid.bj/1551862835_20190306040120Wed, 06 Mar 2019 04:01 ESTOn squared Bessel particle systemshttps://projecteuclid.org/euclid.bj/1551862836<strong>Piotr Graczyk</strong>, <strong>Jacek Małecki</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 828--847.</p><p><strong>Abstract:</strong><br/>
We study the existence and uniqueness of solutions of SDEs describing squared Bessel particle systems in full generality. We define nonnegative and non-colliding squared Bessel particle systems and we study their properties. Particle systems dissatisfying non-colliding and unicity properties are pointed out. The structure of squared Bessel particle systems is described.
</p>projecteuclid.org/euclid.bj/1551862836_20190306040120Wed, 06 Mar 2019 04:01 ESTSmooth, identifiable supermodels of discrete DAG models with latent variableshttps://projecteuclid.org/euclid.bj/1551862837<strong>Robin J. Evans</strong>, <strong>Thomas S. Richardson</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 848--876.</p><p><strong>Abstract:</strong><br/>
We provide a parameterization of the discrete nested Markov model, which is a supermodel that approximates DAG models (Bayesian network models) with latent variables. Such models are widely used in causal inference and machine learning. We explicitly evaluate their dimension, show that they are curved exponential families of distributions, and fit them to data. The parameterization avoids the irregularities and unidentifiability of latent variable models. The parameters used are all fully identifiable and causally-interpretable quantities.
</p>projecteuclid.org/euclid.bj/1551862837_20190306040120Wed, 06 Mar 2019 04:01 ESTBayesian consistency for a nonparametric stationary Markov modelhttps://projecteuclid.org/euclid.bj/1551862838<strong>Minwoo Chae</strong>, <strong>Stephen G. Walker</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 877--901.</p><p><strong>Abstract:</strong><br/>
We consider posterior consistency for a Markov model with a novel class of nonparametric prior. In this model, the transition density is parameterized via a mixing distribution function. Therefore, the Wasserstein distance between mixing measures can be used to construct neighborhoods of a transition density. The Wasserstein distance is sufficiently strong, for example, if the mixing distributions are compactly supported, it dominates the sup-$L_{1}$ metric. We provide sufficient conditions for posterior consistency with respect to the Wasserstein metric provided that the true transition density is also parametrized via a mixing distribution. In general, when it is not be parameterized by a mixing distribution, we show the posterior distribution is consistent with respect to the average $L_{1}$ metric. Also, we provide a prior whose support is sufficiently large to contain most smooth transition densities.
</p>projecteuclid.org/euclid.bj/1551862838_20190306040120Wed, 06 Mar 2019 04:01 ESTLow-frequency estimation of continuous-time moving average Lévy processeshttps://projecteuclid.org/euclid.bj/1551862839<strong>Denis Belomestny</strong>, <strong>Vladimir Panov</strong>, <strong>Jeannette H.C. Woerner</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 902--931.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the problem of statistical inference for a continuous-time moving average Lévy process of the form
\[Z_{t}=\int_{\mathbb{R}}\mathcal{K}(t-s)\,dL_{s},\qquad t\in\mathbb{R},\] with a deterministic kernel $\mathcal{K}$ and a Lévy process $L$. Especially the estimation of the Lévy measure $\nu$ of $L$ from low-frequency observations of the process $Z$ is considered. We construct a consistent estimator, derive its convergence rates and illustrate its performance by a numerical example. On the mathematical level, we establish some new results on exponential mixing for continuous-time moving average Lévy processes.
</p>projecteuclid.org/euclid.bj/1551862839_20190306040120Wed, 06 Mar 2019 04:01 ESTFréchet means and Procrustes analysis in Wasserstein spacehttps://projecteuclid.org/euclid.bj/1551862840<strong>Yoav Zemel</strong>, <strong>Victor M. Panaretos</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 932--976.</p><p><strong>Abstract:</strong><br/>
We consider two statistical problems at the intersection of functional and non-Euclidean data analysis: the determination of a Fréchet mean in the Wasserstein space of multivariate distributions; and the optimal registration of deformed random measures and point processes. We elucidate how the two problems are linked, each being in a sense dual to the other. We first study the finite sample version of the problem in the continuum. Exploiting the tangent bundle structure of Wasserstein space, we deduce the Fréchet mean via gradient descent. We show that this is equivalent to a Procrustes analysis for the registration maps, thus only requiring successive solutions to pairwise optimal coupling problems. We then study the population version of the problem, focussing on inference and stability: in practice, the data are i.i.d. realisations from a law on Wasserstein space, and indeed their observation is discrete, where one observes a proxy finite sample or point process. We construct regularised nonparametric estimators, and prove their consistency for the population mean, and uniform consistency for the population Procrustes registration maps.
</p>projecteuclid.org/euclid.bj/1551862840_20190306040120Wed, 06 Mar 2019 04:01 ESTAre there needles in a moving haystack? Adaptive sensing for detection of dynamically evolving signalshttps://projecteuclid.org/euclid.bj/1551862841<strong>Rui M. Castro</strong>, <strong>Ervin Tánczos</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 977--1012.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the problem of detecting dynamically evolving signals. We model the signal as an $n$ dimensional vector that is either zero or has $s$ non-zero components. At each time step $t\in\mathbb{N}$ the nonzero components change their location independently with probability $p$. The statistical problem is to decide whether the signal is a zero vector or in fact it has non-zero components. This decision is based on $m$ noisy observations of individual signal components collected at times $t=1,\ldots,m$. We consider two different sensing paradigms, namely adaptive and non-adaptive sensing. For non-adaptive sensing, the choice of components to measure has to be decided before the data collection process started, while for adaptive sensing one can adjust the sensing process based on observations collected earlier. We characterize the difficulty of this detection problem in both sensing paradigms in terms of the aforementioned parameters, with special interest to the speed of change of the active components. In addition, we provide an adaptive sensing algorithm for this problem and contrast its performance to that of non-adaptive detection algorithms.
</p>projecteuclid.org/euclid.bj/1551862841_20190306040120Wed, 06 Mar 2019 04:01 ESTTowards a general theory for nonlinear locally stationary processeshttps://projecteuclid.org/euclid.bj/1551862842<strong>Rainer Dahlhaus</strong>, <strong>Stefan Richter</strong>, <strong>Wei Biao Wu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1013--1044.</p><p><strong>Abstract:</strong><br/>
In this paper, some general theory is presented for locally stationary processes based on the stationary approximation and the stationary derivative. Laws of large numbers, central limit theorems as well as deterministic and stochastic bias expansions are proved for processes obeying an expansion in terms of the stationary approximation and derivative. In addition it is shown that this applies to some general nonlinear non-stationary Markov-models. In addition the results are applied to derive the asymptotic properties of maximum likelihood estimates of parameter curves in such models.
</p>projecteuclid.org/euclid.bj/1551862842_20190306040120Wed, 06 Mar 2019 04:01 ESTProperties of switching jump diffusions: Maximum principles and Harnack inequalitieshttps://projecteuclid.org/euclid.bj/1551862843<strong>Xiaoshan Chen</strong>, <strong>Zhen-Qing Chen</strong>, <strong>Ky Tran</strong>, <strong>George Yin</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1045--1075.</p><p><strong>Abstract:</strong><br/>
This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.
</p>projecteuclid.org/euclid.bj/1551862843_20190306040120Wed, 06 Mar 2019 04:01 ESTError bounds in local limit theorems using Stein’s methodhttps://projecteuclid.org/euclid.bj/1551862844<strong>A.D. Barbour</strong>, <strong>Adrian Röllin</strong>, <strong>Nathan Ross</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1076--1104.</p><p><strong>Abstract:</strong><br/>
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős–Rényi random graph, and of the Curie–Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein’s method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.
</p>projecteuclid.org/euclid.bj/1551862844_20190306040120Wed, 06 Mar 2019 04:01 ESTStability for gains from large investors’ strategies in $M_{1}$/$J_{1}$ topologieshttps://projecteuclid.org/euclid.bj/1551862845<strong>Dirk Becherer</strong>, <strong>Todor Bilarev</strong>, <strong>Peter Frentrup</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1105--1140.</p><p><strong>Abstract:</strong><br/>
We prove continuity of a controlled SDE solution in Skorokhod’s $M_{1}$ and $J_{1}$ topologies and also uniformly, in probability, as a nonlinear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that $M_{1}$-continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.
</p>projecteuclid.org/euclid.bj/1551862845_20190306040120Wed, 06 Mar 2019 04:01 ESTConvergence rates for a class of estimators based on Stein’s methodhttps://projecteuclid.org/euclid.bj/1551862846<strong>Chris J. Oates</strong>, <strong>Jon Cockayne</strong>, <strong>François-Xavier Briol</strong>, <strong>Mark Girolami</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1141--1159.</p><p><strong>Abstract:</strong><br/>
Gradient information on the sampling distribution can be used to reduce the variance of Monte Carlo estimators via Stein’s method. An important application is that of estimating an expectation of a test function along the sample path of a Markov chain, where gradient information enables convergence rate improvement at the cost of a linear system which must be solved. The contribution of this paper is to establish theoretical bounds on convergence rates for a class of estimators based on Stein’s method. Our analysis accounts for (i) the degree of smoothness of the sampling distribution and test function, (ii) the dimension of the state space, and (iii) the case of non-independent samples arising from a Markov chain. These results provide insight into the rapid convergence of gradient-based estimators observed for low-dimensional problems, as well as clarifying a curse-of-dimension that appears inherent to such methods.
</p>projecteuclid.org/euclid.bj/1551862846_20190306040120Wed, 06 Mar 2019 04:01 ESTMallows and generalized Mallows model for matchingshttps://projecteuclid.org/euclid.bj/1551862847<strong>Ekhine Irurozki</strong>, <strong>Borja Calvo</strong>, <strong>Jose A. Lozano</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1160--1188.</p><p><strong>Abstract:</strong><br/>
The Mallows and Generalized Mallows Models are two of the most popular probability models for distributions on permutations. In this paper, we consider both models under the Hamming distance. This models can be seen as models for matchings instead of models for rankings. These models cannot be factorized, which contrasts with the popular MM and GMM under Kendall’s-$\tau$ and Cayley distances. In order to overcome the computational issues that the models involve, we introduce a novel method for computing the partition function. By adapting this method we can compute the expectation, joint and conditional probabilities. All these methods are the basis for three sampling algorithms, which we propose and analyze. Moreover, we also propose a learning algorithm. All the algorithms are analyzed both theoretically and empirically, using synthetic and real data from the context of e-learning and Massive Open Online Courses (MOOC).
</p>projecteuclid.org/euclid.bj/1551862847_20190306040120Wed, 06 Mar 2019 04:01 ESTStable limit theorems for empirical processes under conditional neighborhood dependencehttps://projecteuclid.org/euclid.bj/1551862848<strong>Ji Hyung Lee</strong>, <strong>Kyungchul Song</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1189--1224.</p><p><strong>Abstract:</strong><br/>
This paper introduces a new concept of stochastic dependence among many random variables which we call conditional neighborhood dependence (CND). Suppose that there are a set of random variables and a set of sigma algebras where both sets are indexed by the same set endowed with a neighborhood system. When the set of random variables satisfies CND, any two non-adjacent sets of random variables are conditionally independent given sigma algebras having indices in one of the two sets’ neighborhood. Random variables with CND include those with conditional dependency graphs and a class of Markov random fields with a global Markov property. The CND property is useful for modeling cross-sectional dependence governed by a complex, large network. This paper provides two main results. The first result is a stable central limit theorem for a sum of random variables with CND. The second result is a Donsker-type result of stable convergence of empirical processes indexed by a class of functions satisfying a certain bracketing entropy condition when the random variables satisfy CND.
</p>projecteuclid.org/euclid.bj/1551862848_20190306040120Wed, 06 Mar 2019 04:01 ESTOracle inequalities for high-dimensional predictionhttps://projecteuclid.org/euclid.bj/1551862849<strong>Johannes Lederer</strong>, <strong>Lu Yu</strong>, <strong>Irina Gaynanova</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1225--1255.</p><p><strong>Abstract:</strong><br/>
The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bound can help to identify the potential of specific estimators, and they can help to get a sense of the prediction accuracy in a given application.
</p>projecteuclid.org/euclid.bj/1551862849_20190306040120Wed, 06 Mar 2019 04:01 ESTTruncated random measureshttps://projecteuclid.org/euclid.bj/1551862850<strong>Trevor Campbell</strong>, <strong>Jonathan H. Huggins</strong>, <strong>Jonathan P. How</strong>, <strong>Tamara Broderick</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1256--1288.</p><p><strong>Abstract:</strong><br/>
Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper, we detail two major classes of sequential CRM representations— series representations and superposition representations —within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context.
</p>projecteuclid.org/euclid.bj/1551862850_20190306040120Wed, 06 Mar 2019 04:01 ESTMinimax optimal estimation in partially linear additive models under high dimensionhttps://projecteuclid.org/euclid.bj/1551862851<strong>Zhuqing Yu</strong>, <strong>Michael Levine</strong>, <strong>Guang Cheng</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1289--1325.</p><p><strong>Abstract:</strong><br/>
In this paper, we derive minimax rates for estimating both parametric and nonparametric components in partially linear additive models with high dimensional sparse vectors and smooth functional components. The minimax lower bound for Euclidean components is the typical sparse estimation rate that is independent of nonparametric smoothness indices. However, the minimax lower bound for each component function exhibits an interplay between the dimensionality and sparsity of the parametric component and the smoothness of the relevant nonparametric component. Indeed, the minimax risk for smooth nonparametric estimation can be slowed down to the sparse estimation rate whenever the smoothness of the nonparametric component or dimensionality of the parametric component is sufficiently large. In the above setting, we demonstrate that penalized least square estimators can nearly achieve minimax lower bounds.
</p>projecteuclid.org/euclid.bj/1551862851_20190306040120Wed, 06 Mar 2019 04:01 ESTStrong Gaussian approximation of the mixture Rasch modelhttps://projecteuclid.org/euclid.bj/1551862852<strong>Friedrich Liese</strong>, <strong>Alexander Meister</strong>, <strong>Johanna Kappus</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1326--1354.</p><p><strong>Abstract:</strong><br/>
We consider the famous Rasch model, which is applied to psychometric surveys when $n$ persons under test answer $m$ questions. The score is given by a realization of a random binary $n\times m$-matrix. Its $(j,k)$th component indicates whether or not the answer of the $j$th person to the $k$th question is correct. In the mixture, Rasch model one assumes that the persons are chosen randomly from a population. We prove that the mixture Rasch model is asymptotically equivalent to a Gaussian observation scheme in Le Cam’s sense as $n$ tends to infinity and $m$ is allowed to increase slowly in $n$. For that purpose, we show a general result on strong Gaussian approximation of the sum of independent high-dimensional binary random vectors. As a first application, we construct an asymptotic confidence region for the difficulty parameters of the questions.
</p>projecteuclid.org/euclid.bj/1551862852_20190306040120Wed, 06 Mar 2019 04:01 ESTTime-frequency analysis of locally stationary Hawkes processeshttps://projecteuclid.org/euclid.bj/1551862853<strong>François Roueff</strong>, <strong>Rainer von Sachs</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1355--1385.</p><p><strong>Abstract:</strong><br/>
Locally stationary Hawkes processes have been introduced in order to generalise classical Hawkes processes away from stationarity by allowing for a time-varying second-order structure. This class of self-exciting point processes has recently attracted a lot of interest in applications in the life sciences (seismology, genomics, neuro-science, …), but also in the modeling of high-frequency financial data. In this contribution, we provide a fully developed nonparametric estimation theory of both local mean density and local Bartlett spectra of a locally stationary Hawkes process. In particular, we apply our kernel estimation of the spectrum localised both in time and frequency to two data sets of transaction times revealing pertinent features in the data that had not been made visible by classical non-localised approaches based on models with constant fertility functions over time.
</p>projecteuclid.org/euclid.bj/1551862853_20190306040120Wed, 06 Mar 2019 04:01 ESTQuenched central limit theorem rates of convergence for one-dimensional random walks in random environmentshttps://projecteuclid.org/euclid.bj/1551862854<strong>Sung Won Ahn</strong>, <strong>Jonathon Peterson</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1386--1411.</p><p><strong>Abstract:</strong><br/>
Unlike classical simple random walks, one-dimensional random walks in random environments (RWRE) are known to have a wide array of potential limiting distributions. Under certain assumptions, however, it is known that CLT-like limiting distributions hold for the walk under both the quenched and averaged measures. We give upper bounds on the rates of convergence for the quenched central limit theorems for both the hitting time and position of the RWRE with polynomial rates of convergence that depend on the distribution on environments.
</p>projecteuclid.org/euclid.bj/1551862854_20190306040120Wed, 06 Mar 2019 04:01 ESTFrom random partitions to fractional Brownian sheetshttps://projecteuclid.org/euclid.bj/1551862855<strong>Olivier Durieu</strong>, <strong>Yizao Wang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1412--1450.</p><p><strong>Abstract:</strong><br/>
We propose discrete random-field models that are based on random partitions of $\mathbb{N}^{2}$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.
</p>projecteuclid.org/euclid.bj/1551862855_20190306040120Wed, 06 Mar 2019 04:01 ESTA Bernstein-type inequality for functions of bounded interactionhttps://projecteuclid.org/euclid.bj/1551862856<strong>Andreas Maurer</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1451--1471.</p><p><strong>Abstract:</strong><br/>
We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein’s inequality from sums to more general functions, whose variation in any argument does not depend too much on the other arguments. Applications sharpen existing bounds for U-statistics and the generalization error of regularized least squares.
</p>projecteuclid.org/euclid.bj/1551862856_20190306040120Wed, 06 Mar 2019 04:01 ESTAn extreme-value approach for testing the equality of large U-statistic based correlation matriceshttps://projecteuclid.org/euclid.bj/1551862857<strong>Cheng Zhou</strong>, <strong>Fang Han</strong>, <strong>Xin-Sheng Zhang</strong>, <strong>Han Liu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1472--1503.</p><p><strong>Abstract:</strong><br/>
There has been an increasing interest in testing the equality of large Pearson’s correlation matrices. However, in many applications it is more important to test the equality of large rank-based correlation matrices since they are more robust to outliers and nonlinearity. Unlike the Pearson’s case, testing the equality of large rank-based statistics has not been well explored and requires us to develop new methods and theory. In this paper, we provide a framework for testing the equality of two large U-statistic based correlation matrices, which include the rank-based correlation matrices as special cases. Our approach exploits extreme value statistics and the Jackknife estimator for uncertainty assessment and is valid under a fully nonparametric model. Theoretically, we develop a theory for testing the equality of U-statistic based correlation matrices. We then apply this theory to study the problem of testing large Kendall’s tau correlation matrices and demonstrate its optimality. For proving this optimality, a novel construction of least favorable distributions is developed for the correlation matrix comparison.
</p>projecteuclid.org/euclid.bj/1551862857_20190306040120Wed, 06 Mar 2019 04:01 ESTNumerically stable online estimation of variance in particle filtershttps://projecteuclid.org/euclid.bj/1551862858<strong>Jimmy Olsson</strong>, <strong>Randal Douc</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1504--1535.</p><p><strong>Abstract:</strong><br/>
This paper discusses variance estimation in sequential Monte Carlo methods, alternatively termed particle filters. The variance estimator that we propose is a natural modification of that suggested by H.P. Chan and T.L. Lai [ Ann. Statist. 41 (2013) 2877–2904], which allows the variance to be estimated in a single run of the particle filter by tracing the genealogical history of the particles. However, due particle lineage degeneracy, the estimator of the mentioned work becomes numerically unstable as the number of sequential particle updates increases. Thus, by tracing only a part of the particles’ genealogy rather than the full one, our estimator gains long-term numerical stability at the cost of a bias. The scope of the genealogical tracing is regulated by a lag, and under mild, easily checked model assumptions, we prove that the bias tends to zero geometrically fast as the lag increases. As confirmed by our numerical results, this allows the bias to be tightly controlled also for moderate particle sample sizes.
</p>projecteuclid.org/euclid.bj/1551862858_20190306040120Wed, 06 Mar 2019 04:01 ESTNew tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chainshttps://projecteuclid.org/euclid.bj/1551862859<strong>Amir Sepehri</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1536--1567.</p><p><strong>Abstract:</strong><br/>
This paper introduces two new families of non-parametric tests of goodness-of-fit on the compact classical groups. One of them is a family of tests for the eigenvalue distribution induced by the uniform distribution, which is consistent against all fixed alternatives. The other is a family of tests for the uniform distribution on the entire group, which is again consistent against all fixed alternatives. The construction of these tests heavily employs facts and techniques from the representation theory of compact groups. In particular, new Cauchy identities are derived and proved for the characters of compact classical groups, in order to accommodate the computation of the test statistic. We find the asymptotic distribution under the null and general alternatives. The tests are proved to be asymptotically admissible. Local power is derived and the global properties of the power function against local alternatives are explored.
The new tests are validated on two random walks for which the mixing-time is studied in the literature. The new tests, and several others, are applied to the Markov chain sampler proposed by Jones, Osipov and Rokhlin [ Proc. Natl. Acad. Sci. 108 (2011) 15679–15686], providing strong evidence supporting the claim that the sampler mixes quickly.
</p>projecteuclid.org/euclid.bj/1551862859_20190306040120Wed, 06 Mar 2019 04:01 ESTMacroscopic analysis of determinantal random ballshttps://projecteuclid.org/euclid.bj/1551862860<strong>Jean-Christophe Breton</strong>, <strong>Adrien Clarenne</strong>, <strong>Renan Gobard</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 2, 1568--1601.</p><p><strong>Abstract:</strong><br/>
We consider a collection of Euclidean random balls in $\mathbb{R}^{d}$ generated by a determinantal point process inducing inhibitory interaction into the balls. We study this model at a macroscopic level obtained by a zooming-out and three different regimes – Gaussian, Poissonian and stable – are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.
</p>projecteuclid.org/euclid.bj/1551862860_20190306040120Wed, 06 Mar 2019 04:01 ESTSparse Hanson–Wright inequalities for subgaussian quadratic formshttps://projecteuclid.org/euclid.bj/1560326421<strong>Shuheng Zhou</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1603--1639.</p><p><strong>Abstract:</strong><br/>
In this paper, we provide a proof for the Hanson–Wright inequalities for sparse quadratic forms in subgaussian random variables. This provides useful concentration inequalities for sparse subgaussian random vectors in two ways. Let $X=(X_{1},\ldots,X_{m})\in\mathbf{R}^{m}$ be a random vector with independent subgaussian components, and $\xi=(\xi_{1},\ldots,\xi_{m})\in\{0,1\}^{m}$ be independent Bernoulli random variables. We prove the large deviation bound for a sparse quadratic form of $(X\circ\xi)^{T}A(X\circ\xi)$, where $A\in\mathbf{R}^{m\times m}$ is an $m\times m$ matrix, and random vector $X\circ\xi$ denotes the Hadamard product of an isotropic subgaussian random vector $X\in\mathbf{R}^{m}$ and a random vector $\xi\in\{0,1\}^{m}$ such that $(X\circ\xi)_{i}=X_{i}\xi_{i}$, where $\xi_{1},\ldots,\xi_{m}$ are independent Bernoulli random variables. The second type of sparsity in a quadratic form comes from the setting where we randomly sample the elements of an anisotropic subgaussian vector $Y=HX$ where $H\in\mathbf{R}^{m\times m}$ is an $m\times m$ symmetric matrix; we study the large deviation bound on the $\ell_{2}$-norm $\lVert D_{\xi}Y\rVert_{2}^{2}$ from its expected value, where for a given vector $x\in\mathbf{R}^{m}$, $D_{x}=\operatorname{diag}(x)$ denotes the diagonal matrix whose main diagonal entries are the entries of $x$. This form arises naturally from the context of covariance estimation.
</p>projecteuclid.org/euclid.bj/1560326421_20190612040036Wed, 12 Jun 2019 04:00 EDTDual attainment for the martingale transport problemhttps://projecteuclid.org/euclid.bj/1560326422<strong>Mathias Beiglböck</strong>, <strong>Tongseok Lim</strong>, <strong>Jan Obłój</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1640--1658.</p><p><strong>Abstract:</strong><br/>
We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [ Ann. Probab. 45 (2017) 3038–3074] established such existence for weak (quasi-sure) duality, [ Finance Stoch. 17 (2013) 477–501] showed existence for the natural stronger (pointwise) duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when $y\mapsto c(x,y)$ is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost $c(x,y)$ is twice continuously differentiable in $y$. Further, this may not be improved as we give examples with $c(x,\cdot)\in C^{2-\varepsilon}$, $\varepsilon>0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.
</p>projecteuclid.org/euclid.bj/1560326422_20190612040036Wed, 12 Jun 2019 04:00 EDTMartingale decompositions and weak differential subordination in UMD Banach spaceshttps://projecteuclid.org/euclid.bj/1560326423<strong>Ivan S. Yaroslavtsev</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1659--1689.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider Meyer–Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in(1,\infty)$, any $X$-valued $L^{p}$-martingale $M$ has a unique decomposition $M=M^{d}+M^{c}$ such that $M^{d}$ is a purely discontinuous martingale, $M^{c}$ is a continuous martingale, $M^{c}_{0}=0$ and \[\mathbb{E}\big\|M^{d}_{\infty}\big\|^{p}+\mathbb{E}\big\|M^{c}_{\infty}\big\|^{p}\leq c_{p,X}\mathbb{E}\|M_{\infty}\|^{p}.\] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps.
As an application, we show that $X$ is a UMD Banach space if and only if for any fixed $p\in(1,\infty)$ and for all $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb{E}\|N_{\infty}\|^{p}\leq C_{p,X}\mathbb{E}\|M_{\infty}\|^{p}$.
</p>projecteuclid.org/euclid.bj/1560326423_20190612040036Wed, 12 Jun 2019 04:00 EDTMaximum likelihood estimators based on the block maxima methodhttps://projecteuclid.org/euclid.bj/1560326424<strong>Clément Dombry</strong>, <strong>Ana Ferreira</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1690--1723.</p><p><strong>Abstract:</strong><br/>
The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observations. Among other semi-parametric methods (such as the popular Hill estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method for independent and identically distributed observations in the max-domain of attraction of some extreme value distribution. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second-order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.
</p>projecteuclid.org/euclid.bj/1560326424_20190612040036Wed, 12 Jun 2019 04:00 EDTMixing properties and central limit theorem for associated point processeshttps://projecteuclid.org/euclid.bj/1560326425<strong>Arnaud Poinas</strong>, <strong>Bernard Delyon</strong>, <strong>Frédéric Lavancier</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1724--1754.</p><p><strong>Abstract:</strong><br/>
Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove $\alpha $-mixing properties for associated spatial point processes by controlling their $\alpha $-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the $\alpha $-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.
</p>projecteuclid.org/euclid.bj/1560326425_20190612040036Wed, 12 Jun 2019 04:00 EDTPerpetual integrals via random time changeshttps://projecteuclid.org/euclid.bj/1560326426<strong>Franziska Kühn</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1755--1769.</p><p><strong>Abstract:</strong><br/>
Let $(X_{t})_{t\geq0}$ be a $d$-dimensional Feller process with symbol $q$, and let $f:\mathbb{R}^{d}\to(0,\infty)$ be a continuous function. In this paper, we establish a growth condition in terms of $q$ and $f$ such that the perpetual integral \begin{equation*}\int_{0}^{\infty}f(X_{s})\,ds\end{equation*} is infinite almost surely. The result applies, in particular, if $(X_{t})_{t\geq0}$ is a Lévy process. The key idea is to approach perpetuals integrals via random time changes. As a by-product of the proof, a sufficient condition for the non-explosion of solutions to martingale problems is obtained. Moreover, we establish a condition which ensures that the random time change of a Feller process is a conservative $C_{b}$-Feller process.
</p>projecteuclid.org/euclid.bj/1560326426_20190612040036Wed, 12 Jun 2019 04:00 EDTUniform behaviors of random polytopes under the Hausdorff metrichttps://projecteuclid.org/euclid.bj/1560326427<strong>Victor-Emmanuel Brunel</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1770--1793.</p><p><strong>Abstract:</strong><br/>
We study the Hausdorff distance between a random polytope, defined as the convex hull of i.i.d. random points, and the convex hull of the support of their distribution. As particular examples, we consider uniform distributions on convex bodies, densities that decay at a certain rate when approaching the boundary of a convex body, projections of uniform distributions on higher dimensional convex bodies and uniform distributions on the boundary of convex bodies. We essentially distinguish two types of convex bodies: those with a smooth boundary and polytopes. In the case of uniform distributions, we prove that, in some sense, the random polytope achieves its best statistical accuracy under the Hausdorff metric when the support has a smooth boundary and its worst statistical accuracy when the support is a polytope. This is somewhat surprising, since the exact opposite is true under the Nikodym metric. We prove rate optimality of most our results in a minimax sense. In the case of uniform distributions, we extend our results to a rescaled version of the Hausdorff metric. We also tackle the estimation of functionals of the support of a distribution such as its mean width and its diameter. Finally, we show that high dimensional random polytopes can be approximated with simple polyhedral representations that significantly decrease their computational complexity without affecting their statistical accuracy.
</p>projecteuclid.org/euclid.bj/1560326427_20190612040036Wed, 12 Jun 2019 04:00 EDTOn the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension onehttps://projecteuclid.org/euclid.bj/1560326428<strong>Adrien Saumard</strong>, <strong>Jon A. Wellner</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1794--1815.</p><p><strong>Abstract:</strong><br/>
First, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Second, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger’s inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincaré inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger’s inequality, which is a $L_{p}$-Poincaré inequality for $p=2$, to any real $p\geq1$.
</p>projecteuclid.org/euclid.bj/1560326428_20190612040036Wed, 12 Jun 2019 04:00 EDTA one-sample test for normality with kernel methodshttps://projecteuclid.org/euclid.bj/1560326429<strong>Jérémie Kellner</strong>, <strong>Alain Celisse</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1816--1837.</p><p><strong>Abstract:</strong><br/>
We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence, our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure.
</p>projecteuclid.org/euclid.bj/1560326429_20190612040036Wed, 12 Jun 2019 04:00 EDTCentral limit theorem for linear spectral statistics of large dimensional separable sample covariance matriceshttps://projecteuclid.org/euclid.bj/1560326430<strong>Zhidong Bai</strong>, <strong>Huiqin Li</strong>, <strong>Guangming Pan</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1838--1869.</p><p><strong>Abstract:</strong><br/>
Suppose that $\mathbf{X}_{n}=(x_{jk})$ is $N\times n$ whose elements are independent complex variables with mean zero, variance 1. The separable sample covariance matrix is defined as $\mathbf{B}_{n}=\frac{1}{N}\mathbf{T}_{2n}^{1/2}\mathbf{X}_{n}\mathbf{T}_{1n}\mathbf{X}_{n}^{*}\mathbf{T}_{2n}^{1/2}$ where $\mathbf{T}_{1n}$ is a Hermitian matrix and $\mathbf{T}_{2n}^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $\mathbf{T}_{2n}$. Its linear spectral statistics (LSS) are shown to have Gaussian limits when $n/N$ approaches a positive constant under some conditions.
</p>projecteuclid.org/euclid.bj/1560326430_20190612040036Wed, 12 Jun 2019 04:00 EDTAsymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observationshttps://projecteuclid.org/euclid.bj/1560326431<strong>Masaaki Fukasawa</strong>, <strong>Tetsuya Takabatake</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1870--1900.</p><p><strong>Abstract:</strong><br/>
This paper proposes feasible asymptotically efficient estimators for a certain class of Gaussian noises with self-similarity and stationarity properties, which includes the fractional Gaussian noises, under high frequency observations. In this setting, the optimal rate of estimation depends on whether either the Hurst or diffusion parameters is known or not. This is due to the singularity of the asymptotic Fisher information matrix for simultaneous estimation of the above two parameters. One of our key ideas is to extend the Whittle estimation method to the situation of high frequency observations. We show that our estimators are asymptotically efficient in Fisher’s sense. Further by Monte-Carlo experiments, we examine finite sample performances of our estimators. Finite sample modifications of the asymptotic variances of the estimators are also given, which exhibit almost perfect fits to the numerical results.
</p>projecteuclid.org/euclid.bj/1560326431_20190612040036Wed, 12 Jun 2019 04:00 EDTSparse covariance matrix estimation in high-dimensional deconvolutionhttps://projecteuclid.org/euclid.bj/1560326432<strong>Denis Belomestny</strong>, <strong>Mathias Trabs</strong>, <strong>Alexandre B. Tsybakov</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1901--1938.</p><p><strong>Abstract:</strong><br/>
We study the estimation of the covariance matrix $\Sigma$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of $\Sigma$. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in $n/\log p$. We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.
</p>projecteuclid.org/euclid.bj/1560326432_20190612040036Wed, 12 Jun 2019 04:00 EDTHybrid regularisation and the (in)admissibility of ridge regression in infinite dimensional Hilbert spaceshttps://projecteuclid.org/euclid.bj/1560326433<strong>Anirvan Chakraborty</strong>, <strong>Victor M. Panaretos</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1939--1976.</p><p><strong>Abstract:</strong><br/>
We consider the problem of estimating the slope function in a functional regression with a scalar response and a functional covariate. This central problem of functional data analysis is well known to be ill-posed, thus requiring a regularised estimation procedure. The two most commonly used approaches are based on spectral truncation or Tikhonov regularisation of the empirical covariance operator. In principle, Tikhonov regularisation is the more canonical choice. Compared to spectral truncation, it is robust to eigenvalue ties, while it attains the optimal minimax rate of convergence in the mean squared sense, and not just in a concentration probability sense. In this paper, we show that, surprisingly, one can strictly improve upon the performance of the Tikhonov estimator in finite samples by means of a linear estimator, while retaining its stability and asymptotic properties by combining it with a form of spectral truncation. Specifically, we construct an estimator that additively decomposes the functional covariate by projecting it onto two orthogonal subspaces defined via functional PCA; it then applies Tikhonov regularisation to the one component, while leaving the other component unregularised. We prove that when the covariate is Gaussian, this hybrid estimator uniformly improves upon the MSE of the Tikhonov estimator in a non-asymptotic sense, effectively rendering it inadmissible. This domination is shown to also persist under discrete observation of the covariate function. The hybrid estimator is linear, straightforward to construct in practice, and with no computational overhead relative to the standard regularisation methods. By means of simulation, it is shown to furnish sizeable gains even for modest sample sizes.
</p>projecteuclid.org/euclid.bj/1560326433_20190612040036Wed, 12 Jun 2019 04:00 EDTConsistency of adaptive importance sampling and recycling schemeshttps://projecteuclid.org/euclid.bj/1560326434<strong>Jean-Michel Marin</strong>, <strong>Pierre Pudlo</strong>, <strong>Mohammed Sedki</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1977--1998.</p><p><strong>Abstract:</strong><br/>
Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. Sequential adaptive algorithms have been proposed to calibrate the sampling distribution. Cornuet et al. [ Scand. J. Stat. 39 (2012) 798–812] provides a significant improvement in stability and effective sample size by the introduction of a recycling procedure. However, the consistency of such algorithms have been rarely tackled because of their complexity. Moreover, the recycling strategy of the AMIS estimator adds another difficulty and its consistency remains largely open. In this work, we prove the convergence of sequential adaptive sampling, with finite Monte Carlo sample size at each iteration, and consistency of recycling procedures. Contrary to Douc et al. [ Ann. Statist. 35 (2007) 420–448], results are obtained here in the asymptotic regime where the number of iterations is going to infinity while the number of drawings per iteration is a fixed, but growing sequence of integers. Hence, some of the results shed new light on adaptive population Monte Carlo algorithms in that last regime and give advices on how the sample sizes should be fixed.
</p>projecteuclid.org/euclid.bj/1560326434_20190612040036Wed, 12 Jun 2019 04:00 EDTOn posterior consistency of tail index for Bayesian kernel mixture modelshttps://projecteuclid.org/euclid.bj/1560326435<strong>Cheng Li</strong>, <strong>Lizhen Lin</strong>, <strong>David B. Dunson</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 1999--2028.</p><p><strong>Abstract:</strong><br/>
Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. We show that posterior inconsistency in tail index is surprisingly common for both parametric and nonparametric mixture models. We then present a set of sufficient conditions under which posterior consistency in tail index can be achieved, and verify these conditions for Pareto mixture models under general mixing priors.
</p>projecteuclid.org/euclid.bj/1560326435_20190612040036Wed, 12 Jun 2019 04:00 EDTThe unusual properties of aggregated superpositions of Ornstein–Uhlenbeck type processeshttps://projecteuclid.org/euclid.bj/1560326436<strong>Danijel Grahovac</strong>, <strong>Nikolai N. Leonenko</strong>, <strong>Alla Sikorskii</strong>, <strong>Murad S. Taqqu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2029--2050.</p><p><strong>Abstract:</strong><br/>
Superpositions of Ornstein–Uhlenbeck type (supOU) processes form a rich class of stationary processes with a flexible dependence structure. The asymptotic behavior of the integrated and partial sum supOU processes can be, however, unusual. Their cumulants and moments turn out to have an unexpected rate of growth. We identify the property of fast growth of moments or cumulants as intermittency . Many proofs are given in a supplemental article (Grahovac, Leonenko, Sikorskii and Taqqu (2018)).
</p>projecteuclid.org/euclid.bj/1560326436_20190612040036Wed, 12 Jun 2019 04:00 EDTGibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: Closing the Ising gaphttps://projecteuclid.org/euclid.bj/1560326437<strong>Florian Henning</strong>, <strong>Richard C. Kraaij</strong>, <strong>Christof Külske</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2051--2074.</p><p><strong>Abstract:</strong><br/>
We complete the investigation of the Gibbs properties of the fuzzy Potts model on the $d$-dimensional torus with Kac interaction which was started by Jahnel and one of the authors in (Sharp thresholds for Gibbs-non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction (2017)). As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes (containing precisely two spin values). The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.
</p>projecteuclid.org/euclid.bj/1560326437_20190612040036Wed, 12 Jun 2019 04:00 EDTRegularization, sparse recovery, and median-of-means tournamentshttps://projecteuclid.org/euclid.bj/1560326438<strong>Gábor Lugosi</strong>, <strong>Shahar Mendelson</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2075--2106.</p><p><strong>Abstract:</strong><br/>
We introduce a regularized risk minimization procedure for regression function estimation. The procedure is based on median-of-means tournaments, introduced by the authors in Lugosi and Mendelson (2018) and achieves near optimal accuracy and confidence under general conditions, including heavy-tailed predictor and response variables. It outperforms standard regularized empirical risk minimization procedures such as LASSO or SLOPE in heavy-tailed problems.
</p>projecteuclid.org/euclid.bj/1560326438_20190612040036Wed, 12 Jun 2019 04:00 EDTRoot-$n$ consistent estimation of the marginal density in semiparametric autoregressive time series modelshttps://projecteuclid.org/euclid.bj/1560326439<strong>Lionel Truquet</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2107--2136.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the problem of estimating the marginal density in some autoregressive time series models for which the conditional mean and variance have a parametric specification. Under some regularity conditions, we show that a kernel type estimate based on the residuals can be root-$n$ consistent even if the noise density is unknown. Our results substantially extend those existing in the literature. Our assumptions are carefully checked for some standard time series models such as ARMA or GARCH processes. Asymptotic expansion of our estimator is obtained by combining some martingale type arguments and a coupling method for time series which is of independent interest. We also study the uniform convergence of our estimator on compact intervals.
</p>projecteuclid.org/euclid.bj/1560326439_20190612040036Wed, 12 Jun 2019 04:00 EDTIntegration with respect to the non-commutative fractional Brownian motionhttps://projecteuclid.org/euclid.bj/1560326440<strong>Aurélien Deya</strong>, <strong>René Schott</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2137--2162.</p><p><strong>Abstract:</strong><br/>
We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian motion in a non-commutative probability setting.
When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an Itô-type approach is known to be possible.
When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined Lévy area process. We show that such an object can indeed be “canonically” constructed for any $H\in(\frac{1}{4},\frac{1}{2})$. Finally, when $H\leq1/4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) Lévy area above the standard fractional Brownian motion.
</p>projecteuclid.org/euclid.bj/1560326440_20190612040036Wed, 12 Jun 2019 04:00 EDTConstruction of marginally coupled designs by subspace theoryhttps://projecteuclid.org/euclid.bj/1560326441<strong>Yuanzhen He</strong>, <strong>C. Devon Lin</strong>, <strong>Fasheng Sun</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2163--2182.</p><p><strong>Abstract:</strong><br/>
Recent researches on designs for computer experiments with both qualitative and quantitative factors have advocated the use of marginally coupled designs. This paper proposes a general method of constructing such designs for which the designs for qualitative factors are multi-level orthogonal arrays and the designs for quantitative factors are Latin hypercubes with desirable space-filling properties. Two cases are introduced for which we can obtain the guaranteed low-dimensional space-filling property for quantitative factors. Theoretical results on the proposed constructions are derived. For practical use, some constructed designs for three-level qualitative factors are tabulated.
</p>projecteuclid.org/euclid.bj/1560326441_20190612040036Wed, 12 Jun 2019 04:00 EDTConsistency of Bayesian nonparametric inference for discretely observed jump diffusionshttps://projecteuclid.org/euclid.bj/1560326442<strong>Jere Koskela</strong>, <strong>Dario Spanò</strong>, <strong>Paul A. Jenkins</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2183--2205.</p><p><strong>Abstract:</strong><br/>
We introduce verifiable criteria for weak posterior consistency of Bayesian nonparametric inference for jump diffusions with unit diffusion coefficient and uniformly Lipschitz drift and jump coefficients in arbitrary dimension. The criteria are expressed in terms of coefficients of the SDEs describing the process, and do not depend on intractable quantities such as transition densities. We also show that priors built from discrete nets, wavelet expansions, and Dirichlet mixture models satisfy our conditions. This generalises known results by incorporating jumps into previous work on unit diffusions with uniformly Lipschitz drift coefficients.
</p>projecteuclid.org/euclid.bj/1560326442_20190612040036Wed, 12 Jun 2019 04:00 EDTOn the risk of convex-constrained least squares estimators under misspecificationhttps://projecteuclid.org/euclid.bj/1560326443<strong>Billy Fang</strong>, <strong>Adityanand Guntuboyina</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2206--2244.</p><p><strong>Abstract:</strong><br/>
We consider the problem of estimating the mean of a noisy vector. When the mean lies in a convex constraint set, the least squares projection of the random vector onto the set is a natural estimator. Properties of the risk of this estimator, such as its asymptotic behavior as the noise tends to zero, have been well studied. We instead study the behavior of this estimator under misspecification, that is, without the assumption that the mean lies in the constraint set. For appropriately defined notions of risk in the misspecified setting, we prove a generalization of a low noise characterization of the risk due to [ Found. Comput. Math. 16 (2016) 965–1029] in the case of a polyhedral constraint set. An interesting consequence of our results is that the risk can be much smaller in the misspecified setting than in the well-specified setting. We also discuss consequences of our result for isotonic regression.
</p>projecteuclid.org/euclid.bj/1560326443_20190612040036Wed, 12 Jun 2019 04:00 EDTA central limit theorem for the realised covariation of a bivariate Brownian semistationary processhttps://projecteuclid.org/euclid.bj/1560326444<strong>Andrea Granelli</strong>, <strong>Almut E.D. Veraart</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2245--2278.</p><p><strong>Abstract:</strong><br/>
This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.
</p>projecteuclid.org/euclid.bj/1560326444_20190612040036Wed, 12 Jun 2019 04:00 EDTThe first order correction to harmonic measure for random walks of rotationally invariant step distributionhttps://projecteuclid.org/euclid.bj/1560326445<strong>Longmin Wang</strong>, <strong>KaiNan Xiang</strong>, <strong>Lang Zou</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2279--2300.</p><p><strong>Abstract:</strong><br/>
Let $D\subset\mathbb{R}^{d}\ (d\geq2)$ be an open simply-connected bounded domain with smooth boundary $\partial D$ and $\mathbf{0}=(0,\ldots,0)\in D$. Fix any rotationally invariant probability $\mu$ on closed unit ball $\{z\in\mathbb{R}^{d}:\vert z\vert\leq1\}$ with $\mu(\{\mathbf{0}\})<1$. Let $\{S_{n}^{\mu}\}_{n=0}^{\infty}$ be the random walk with step-distribution $\mu$ starting at $\mathbf{0}$. Denote by $\omega_{\delta}(\mathbf{0},\mathrm{d}z;D)$ the discrete harmonic measure for $\{\delta S_{n}^{\mu}\}_{n=0}^{\infty}\ (\delta>0)$ exiting from $D$, which is viewed as a probability on $\partial D$ by projecting suitably the first exiting point to $\partial D$. Denote by $\omega(\mathbf{0},\mathrm{d}z;D)$ the harmonic measure for the $d$-dimensional standard Brownian motion exiting from $D$. Then in the weak convergence topology, \begin{equation*}\lim_{\delta\rightarrow0}\frac{1}{\delta}\bigl[\omega_{\delta}(\mathbf{0} ,\mathrm{d}z;D)-\omega(\mathbf{0},\mathrm{d}z;D)\bigr]=c_{\mu}\rho_{D}(z)\,\vert \mathrm{d}z\vert ,\end{equation*} where $\rho_{D}(\cdot)$ is a smooth function depending on $D$ but not on $\mu$, $c_{\mu}$ is a constant depending only on $\mu$, and $|\mathrm{d}z|$ is the Lebesgue measure with respect to $\partial D$. Additionally, $\rho_{D}(z)$ is determined by the following equation: For any smooth function $g$ on $\partial D$, \begin{equation*}\int_{\partial D}g(z)\rho_{D}(z)\,\vert \mathrm{d}z\vert =\int_{\partial D}\frac{\partial f}{\partial\mathbf{n}_{z}}(z)H_{D}(\mathbf{0},z)\,\vert \mathrm{d}z\vert ,\end{equation*} where $f$ is the harmonic function in $D$ with boundary values given by $g$, $H_{D}(\mathbf{0},z)$ is the Poisson kernel and derivative $\frac{\partial f}{\partial\mathbf{n}_{z}}$ is with respect to the inward unit normal $\mathbf{n}_{z}$ at $z\in\partial D$.
</p>projecteuclid.org/euclid.bj/1560326445_20190612040036Wed, 12 Jun 2019 04:00 EDTGromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson treeshttps://projecteuclid.org/euclid.bj/1560326446<strong>Hui He</strong>, <strong>Matthias Winkel</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2301--2329.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the vertex cut-trees of Galton–Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut’s vertex cut-tree of Galton–Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov–Hausdorff–Prokhorov convergence in the finite variance case of the Galton–Watson tree and its vertex cut-tree to Bertoin and Miermont’s joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut’s and Bertoin and Miermont’s Gromov–Prokhorov convergence to Gromov–Hausdorff–Prokhorov remains open for their models conditioned to have $n$ vertices.
</p>projecteuclid.org/euclid.bj/1560326446_20190612040036Wed, 12 Jun 2019 04:00 EDTBayesian mode and maximum estimation and accelerated rates of contractionhttps://projecteuclid.org/euclid.bj/1560326447<strong>William Weimin Yoo</strong>, <strong>Subhashis Ghosal</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2330--2358.</p><p><strong>Abstract:</strong><br/>
We study the problem of estimating the mode and maximum of an unknown regression function in the presence of noise. We adopt the Bayesian approach by using tensor-product B-splines and endowing the coefficients with Gaussian priors. In the usual fixed-in-advanced sampling plan, we establish posterior contraction rates for mode and maximum and show that they coincide with the minimax rates for this problem. To quantify estimation uncertainty, we construct credible sets for these two quantities that have high coverage probabilities with optimal sizes. If one is allowed to collect data sequentially, we further propose a Bayesian two-stage estimation procedure, where a second stage posterior is built based on samples collected within a credible set constructed from a first stage posterior. Under appropriate conditions on the radius of this credible set, we can accelerate optimal contraction rates from the fixed-in-advanced setting to the minimax sequential rates. A simulation experiment shows that our Bayesian two-stage procedure outperforms single-stage procedure and also slightly improves upon a non-Bayesian two-stage procedure.
</p>projecteuclid.org/euclid.bj/1560326447_20190612040036Wed, 12 Jun 2019 04:00 EDTBootstrapping INAR modelshttps://projecteuclid.org/euclid.bj/1560326448<strong>Carsten Jentsch</strong>, <strong>Christian H. Weiß</strong>. <p><strong>Source: </strong>Bernoulli, Volume 25, Number 3, 2359--2408.</p><p><strong>Abstract:</strong><br/>
Integer-valued autoregressive (INAR) models form a very useful class of processes to deal with time series of counts. Statistical inference in these models is commonly based on asymptotic theory, which is available only under additional parametric conditions and further restrictions on the model order. For general INAR models, such results are not available and might be cumbersome to derive. Hence, we investigate how the INAR model structure and, in particular, its similarity to classical autoregressive (AR) processes can be exploited to develop an asymptotically valid bootstrap procedure for INAR models. Although, in a common formulation, INAR models share the autocorrelation structure with AR models, it turns out that (a) consistent estimation of the INAR coefficients is not sufficient to compute proper ‘INAR residuals’ to formulate a valid model-based bootstrap scheme, and (b) a naïve application of an AR bootstrap will generally fail. Instead, we propose a general INAR-type bootstrap procedure and discuss parametric as well as semi-parametric implementations. The latter approach is based on a joint semi-parametric estimator of the INAR coefficients and the innovations’ distribution. Under mild regularity conditions, we prove bootstrap consistency of our procedure for statistics belonging to the class of functions of generalized means. In an extensive simulation study, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed INAR bootstrap over some obvious competitors. We illustrate our method by an application to a real data set about iceberg orders for the Lufthansa stock.
</p>projecteuclid.org/euclid.bj/1560326448_20190612040036Wed, 12 Jun 2019 04:00 EDT