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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.