The random coefficients model is an extension of the linear regression model that allows for unobserved heterogeneity in the population by modeling the regression coefficients as random variables. Given data from this model, the statistical challenge is to recover information about the joint density of the random coefficients which is a multivariate and ill-posed problem. Because of the curse of dimensionality and the ill-posedness, nonparametric estimation of the joint density is difficult and suffers from slow convergence rates. Larger features, such as an increase of the density along some direction or a well-accentuated mode can, however, be much easier detected from data by means of statistical tests. In this article, we follow this strategy and construct tests and confidence statements for qualitative features of the joint density, such as increases, decreases and modes. We propose a multiple testing approach based on aggregating single tests which are designed to extract shape information on fixed scales and directions. Using recent tools for Gaussian approximations of multivariate empirical processes, we derive expressions for the critical value. We apply our method to simulated and real data.

We study uniqueness in the generalized lasso problem, where the penalty is the $\ell _{1}$ norm of a matrix $D$ times the coefficient vector. We derive a broad result on uniqueness that places weak assumptions on the predictor matrix $X$ and penalty matrix $D$; the implication is that, if $D$ is fixed and its null space is not too large (the dimension of its null space is at most the number of samples), and $X$ and response vector $y$ jointly follow an absolutely continuous distribution, then the generalized lasso problem has a unique solution almost surely, regardless of the number of predictors relative to the number of samples. This effectively generalizes previous uniqueness results for the lasso problem [32] (which corresponds to the special case $D=I$). Further, we extend our study to the case in which the loss is given by the negative log-likelihood from a generalized linear model. In addition to uniqueness results, we derive results on the local stability of generalized lasso solutions that might be of interest in their own right.

The nonparametric measurement error model (NMEM) postulates that $X_{i}=\Delta +\epsilon _{i},i=1,2,\ldots ,n;\Delta \in \Re $ with $\epsilon _{i},i=1,2,\ldots ,n$, IID from $F(\cdot )\in\mathfrak{F}_{c,0}$, where $\mathfrak{F}_{c,0}$ is the class of all continuous distributions with median $0$, so $\Delta $ is the median parameter of $X$. This paper deals with the problem of constructing a confidence region (CR) for $\Delta $ under the NMEM. Aside from the NMEM, the problem setting also arises in a variety of situations, including inference about the median lifetime of a complex system arising in engineering, reliability, biomedical, and public health settings, as well as in the economic arena such as when dealing with household income. Current methods of constructing CRs for $\Delta $ are discussed, including the $T$-statistic based CR and the Wilcoxon signed-rank statistic based CR, arguably the two default methods in applied work when a confidence interval about the center of a distribution is desired. A ‘bottom-to-top’ approach for constructing CRs is implemented, which starts by imposing reasonable invariance or equivariance conditions on the desired CRs, and then optimizing with respect to their mean contents on subclasses of $\mathfrak{F}_{c,0}$. This contrasts with the usual approach of using a pivotal quantity constructed from test statistics and/or estimators and then ‘pivoting’ to obtain the CR. Applications to a real car mileage data set and to Proschan’s famous air-conditioning data set are illustrated. Simulation studies to compare performances of the different CR methods were performed. Results of these studies indicate that the sign-statistic based CR and the optimal CR focused on symmetric distributions satisfy the confidence level requirement, though they tended to have higher contents; while three of the bootstrap-based CR procedures and one of the newly-developed adaptive CR tended to be a tad more liberal, but with smaller contents. A critical recommendation for practitioners is that, under the NMEM, the $T$-statistic based and Wilcoxon signed-rank statistic based CRs should not be used since they either have very degraded coverage probabilities or inflated contents under some of the allowable error distributions under the NMEM.

We study nonparametric maximum likelihood estimation of a log-concave density function $f_{0}$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_{0}$ is known, or (b) $f_{0}$ is known to be symmetric about a fixed point $m$. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE’s), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE’s pointwise limit distribution at $m$ (either the known mode or the known center of symmetry) and at a point $x_{0}\ne m$. Software to compute the constrained estimators is available in the R package logcondens.mode.

The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of $f_{0}$. These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters.

This paper proposes a new minimum description length procedure to detect multiple changepoints in time series data when some times are a priori thought more likely to be changepoints. This scenario arises with temperature time series homogenization pursuits, our focus here. Our Bayesian procedure constructs a natural prior distribution for the situation, and is shown to estimate the changepoint locations consistently, with an optimal convergence rate. Our methods substantially improve changepoint detection power when prior information is available. The methods are also tailored to bivariate data, allowing changes to occur in one or both component series.

This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X=(X_{t})_{t\geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to construct confidence bands for the Lévy measure and propose a practical method for bandwidth selection.

With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides results for stationary covariance functions of random fields defined over $d$-dimensional spheres cross time. Specifically, we provide a bridge between the characterization in Berg and Porcu (2017) for covariance functions on spheres cross time and Gneiting’s lemma (Gneiting, 2002) that deals with planar surfaces.

We then prove that there is a valid class of covariance functions similar in form to the Gneiting class of space-time covariance functions (Gneiting, 2002) that replaces the squared Euclidean distance with the great circle distance. Notably, the provided class is shown to be positive definite on every $d$-dimensional sphere cross time, while the Gneiting class is positive definite over $\mathbb{R} ^{d}\times \mathbb{R} $ for fixed $d$ only.

In this context, we illustrate the value of our adapted Gneiting class by comparing examples from this class to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. For these datasets, we show that examples from our covariance class have better predictive performance than competing models.

We propose a simple recursive data-based partitioning scheme which produces piecewise-constant or piecewise-linear density estimates on intervals, and show how this scheme can determine the optimal $L_{1}$ minimax rate for some discrete nonparametric classes.

Estimation of a single Bernoulli parameter using pooled sampling is among the oldest problems in the group testing literature. To carry out such estimation, an array of efficient estimators have been introduced covering a wide range of situations routinely encountered in applications. More recently, there has been growing interest in using group testing to simultaneously estimate the joint probabilities of two correlated traits using a multinomial model. Unfortunately, basic estimation results, such as the maximum likelihood estimator (MLE), have not been adequately addressed in the literature for such cases. In this paper, we show that finding the MLE for this problem is equivalent to maximizing a multinomial likelihood with a restricted parameter space. A solution using the EM algorithm is presented which is guaranteed to converge to the global maximizer, even on the boundary of the parameter space. Two additional closed form estimators are presented with the goal of minimizing the bias and/or mean square error. The methods are illustrated by considering an application to the joint estimation of transmission prevalence for two strains of the Potato virus Y by the aphid Myzus persicae.

We propose a goodness-of-fit test for the distribution of errors from a multivariate indirect regression model, which we assume belongs to a location-scale family under the null hypothesis. The test statistic is based on the Khmaladze transformation of the empirical process of standardized residuals. This goodness-of-fit test is consistent at the root-$n$ rate of convergence, and the test can maintain power against local alternatives converging to the null at a root-$n$ rate.

Let $\mathcal{X}_{n}=\{X_{1},\ldots X_{n}\}\subset \mathbb{R}^{d}$ be a iid random sample of observations drawn with a probability distribution supported by $S$ a compact set satisfying that both $S$ and $\overline{S^{c}}$ are $r_{0}$-convex ($r_{0}>0$). In this paper we study some properties of an estimator of the inner medial axis of $S$ based on the $\lambda $-medial axis. The proposed estimator depends on the choices of $\mathcal{Y}\subset \mathcal{X}_{n}$ an estimator of $\partial S$ and $\hat{S}_{n}$ an estimator of $S$. In a first general theorem we prove that our medial axis estimator converges to the medial axis with a rate $O(\max _{y\in \mathcal{Y}}d(y,\partial S),(\max _{y\in \partial S}d(y,\mathcal{Y})^{2})$. A corollary being that the choice of $\mathcal{Y}$ as the intersection of the sample and its $r$-convex hull, $\mathcal{Y}=C_{r}(\mathcal{X}_{n})\cap \mathcal{X}_{n}$, allows to estimate the medial axis with a convergence rate $O((\ln n/n)^{2/(d+1)})$. In a practical point of view, computational aspects are discussed, algorithms are given and a way to tune the parameters is proposed. A small simulation study is performed to illustrate the results.

In this paper, we study a family of probability distributions, alternative to the von Mises family, called Inverse Stereographic Normal Distributions. These distributions are counterparts of the Gaussian Distribution on $\mathbb{S}^{1}$ (univariate) and $\mathbb{T}^{n}$ (multivariate). We discuss some key properties of the models, such as unimodality and closure with respect to marginalizing and conditioning. We compare this family of distributions to the von Mises’ family and the Wrapped Normal Distribution. Then, we discuss some inferential problems, introduce a notion of moments which is natural for inverse stereographic distributions and revisit a version of the CLT in this context. We construct point estimators, confidence intervals and hypothesis tests and discuss briefly sampling methods. Finally, we conclude with some applications to molecular biology and some illustrative examples. This study is motivated by the Protein Folding Problem and by the fact that a large number of proteins involved in the DNA-metabolism assume a toroidal shape with some amorphous regions.

A flexible approach for modeling both dynamic event counting and dynamic link-based networks based on counting processes is proposed, and estimation in these models is studied. We consider nonparametric likelihood based estimation of parameter functions via kernel smoothing. The asymptotic behavior of these estimators is rigorously analyzed in an asymptotic framework where the number of nodes tends to infinity. The finite sample performance of the estimators is illustrated through an empirical analysis of bike share data.

We consider two problems of estimation in high-dimensional Gaussian models. The first problem is that of estimating a linear functional of the means of $n$ independent $p$-dimensional Gaussian vectors, under the assumption that at most $s$ of the means are nonzero. We show that, up to a logarithmic factor, the minimax rate of estimation in squared Euclidean norm is between $(s^{2}\wedge n)+sp$ and $(s^{2}\wedge np)+sp$. The estimator that attains the upper bound being computationally demanding, we investigate suitable versions of group thresholding estimators that are efficiently computable even when the dimension and the sample size are very large. An interesting new phenomenon revealed by this investigation is that the group thresholding leads to a substantial improvement in the rate as compared to the element-wise thresholding. Thus, the rate of the group thresholding is $s^{2}\sqrt{p}+sp$, while the element-wise thresholding has an error of order $s^{2}p+sp$. To the best of our knowledge, this is the first known setting in which leveraging the group structure leads to a polynomial improvement in the rate.

The second problem studied in this work is the estimation of the common $p$-dimensional mean of the inliers among $n$ independent Gaussian vectors. We show that there is a strong analogy between this problem and the first one. Exploiting it, we propose new strategies of robust estimation that are computationally tractable and have better rates of convergence than the other computationally tractable robust (with respect to the presence of the outliers in the data) estimators studied in the literature. However, this tractability comes with a loss of the minimax-rate-optimality in some regimes.

We present a sequential sampling methodology for weakly structural Markov laws, arising naturally in a Bayesian structure learning context for decomposable graphical models. As a key component of our suggested approach, we show that the problem of graph estimation, which in general lacks natural sequential interpretation, can be recast into a sequential setting by proposing a recursive Feynman-Kac model that generates a flow of junction tree distributions over a space of increasing dimensions. We focus on particle McMC methods to provide samples on this space, in particular on particle Gibbs (PG), as it allows for generating McMC chains with global moves on an underlying space of decomposable graphs. To further improve the PG mixing properties, we incorporate a systematic refreshment step implemented through direct sampling from a backward kernel. The theoretical properties of the algorithm are investigated, showing that the proposed refreshment step improves the performance in terms of asymptotic variance of the estimated distribution. The suggested sampling methodology is illustrated through a collection of numerical examples demonstrating high accuracy in Bayesian graph structure learning in both discrete and continuous graphical models.

By introducing a weighted score function, we propose a new penalized method, similar to square root lasso, to study sparse Poisson regression problems. The corresponding new estimator not only has $\ell _{1}$ consistency but also enjoys the tuning free property. We further verify our theoretical results by numerical simulations and apply them to an image reconstruction problem.

We consider covariance parameter estimation for a Gaussian process under inequality constraints (boundedness, monotonicity or convexity) in fixed-domain asymptotics. We address the estimation of the variance parameter and the estimation of the microergodic parameter of the Matérn and Wendland covariance functions. First, we show that the (unconstrained) maximum likelihood estimator has the same asymptotic distribution, unconditionally and conditionally to the fact that the Gaussian process satisfies the inequality constraints. Then, we study the recently suggested constrained maximum likelihood estimator. We show that it has the same asymptotic distribution as the (unconstrained) maximum likelihood estimator. In addition, we show in simulations that the constrained maximum likelihood estimator is generally more accurate on finite samples. Finally, we provide extensions to prediction and to noisy observations.

We consider the high-dimensional linear regression model $Y=X\beta^{0}+\epsilon$ with Gaussian noise $\epsilon$ and Gaussian random design $X$. We assume that $\Sigma:=\mathrm{I\hskip-0.48emE}X^{T}X/n$ is non-singular and write its inverse as $\Theta :=\Sigma^{-1}$. The parameter of interest is the first component $\beta_{1}^{0}$ of $\beta^{0}$. We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than $\Theta_{1,1}$. For some special such cases we establish asymptotic efficiency. The conditions include $\beta^{0}$ being sparse and the first column $\Theta_{1}$ of $\Theta$ being not sparse. These sparsity conditions depend on whether $\Sigma$ is known or not.

In the multiple testing problem with independent tests, the classical linear step-up procedure controls the false discovery rate (FDR) at level $\pi_{0}\alpha$, where $\pi_{0}$ is the proportion of true null hypotheses and $\alpha$ is the target FDR level. Adaptive procedures can improve power by incorporating estimates of $\pi_{0}$, which typically rely on a tuning parameter. Fixed adaptive procedures set their tuning parameters before seeing the data and can be shown to control the FDR in finite samples. We develop theoretical results for dynamic adaptive procedures whose tuning parameters are determined by the data. We show that, if the tuning parameter is chosen according to a stopping time rule, the corresponding dynamic adaptive procedure controls the FDR in finite samples. Examples include the recently proposed right-boundary procedure and the widely used lowest-slope procedure, among others. Simulation results show that the right-boundary procedure is more powerful than other dynamic adaptive procedures under independence and mild dependence conditions. The right-boundary procedure is implemented in the Bioconductor R package calm.

We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological or geological systems. The results of the paper are classified into three parts.

In the first part, we characterize the equivalence of two Gaussian measures with GC covariance functions. Then we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn (MT) and GC covariance functions and two Gaussian measures with Generalized Wendland (GW) and GC covariance functions.

In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GC covariance model, under fixed domain asymptotics. The last part focuses on optimal prediction with GC model and specifically, we give conditions for asymptotic efficiency prediction and asymptotically correct estimation of mean square error using a misspecified GC, MT or GW model.

Our findings are illustrated through a simulation study: the first compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter of the GC model with the given asymptotic distribution. We then compare the finite-sample behavior of the prediction and its associated mean square error when the true model is GC and the prediction is performed using the true model and a misspecified GW model.

In high-dimensional problems, choosing a prior distribution such that the corresponding posterior has desirable practical and theoretical properties can be challenging. This begs the question: can the data be used to help choose a prior? In this paper, we develop a general strategy for constructing a data-driven or empirical prior and sufficient conditions for the corresponding posterior distribution to achieve a certain concentration rate. The idea is that the prior should put sufficient mass on parameter values for which the likelihood is large. An interesting byproduct of this data-driven centering is that the asymptotic properties of the posterior are less sensitive to the prior shape which, in turn, allows users to work with priors of computationally convenient forms while maintaining the desired rates. General results on both adaptive and non-adaptive rates based on empirical priors are presented, along with illustrations in density estimation, nonparametric regression, and high-dimensional normal models.

It is well known that high-dimensional procedures like the LASSO provide biased estimators of parameters in a linear model. In a 2014 paper Zhang and Zhang showed how to remove this bias by means of a two-step procedure. We show that de-biasing can also be achieved by a one-step estimator, the form of which inspires the development of a Bayesian analogue of the frequentists’ de-biasing techniques.

In this paper, we propose an estimator for $g(x)$ under the model $Y_{i}=g(Z_{i}),\ i=1,2,...,n$ where $Z_{i},\ i=1,2,...$ are random variables with known distribution but unknown observed values, $Y_{i},\ i=1,2,...$ are observed data and $g(x)$ is an unknown strictly monotonically increasing function (we call $g(x)$ transformation function). We prove the almost sure convergence of the estimator and construct confidence intervals and bands when $Z_{i},i=1,2,...$ are i.i.d data based on their asymptotic distribution. Corresponding case when $Z_{i}$ being linear process is handled by resampling method. We also design the hypothesis test regarding whether $g(x)$ equals an expected transformation function or not. The finite sample performance is evaluated by applying the method to simulated data and an urban waste water treatment plant’s dataset.

In this paper, we consider estimation of the conditional mode of an outcome variable given regressors. To this end, we propose and analyze a computationally scalable estimator derived from a linear quantile regression model and develop asymptotic distributional theory for the estimator. Specifically, we find that the pointwise limiting distribution is a scale transformation of Chernoff’s distribution despite the presence of regressors. In addition, we consider analytical and subsampling-based confidence intervals for the proposed estimator. We also conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimator together with the analytical and subsampling confidence intervals. Finally, we apply the proposed estimator to predicting the net hourly electrical energy output using Combined Cycle Power Plant Data.

In this paper, we propose a sparse generalized linear model incorporating graphical structure among predictors (sGLMg), which is an extension of [37] where they exploit the structure information among predictors to improve the performance for the linear regression. There is an explicit expression between the coefficient and the predictor graph measured by the precision matrix in the linear regression, however, this structure does not exist in generalized linear model for the explicit expression of the coefficient in generalized linear model is usually hard to be obtained. To incorporate the graphical structure among predictors for generalized linear models, we make use of the sufficient reduction techniques to reestablish the relationship between the coefficient and the precision matrix. The oracle inequalities of the estimator for sGLMg are also presented and the finite sample performance of the proposed methods is examined via numerical simulations and a breast cancer data analysis.

In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. Defining the non-negative variables $T_{1}$ (time of tumor onset), $T_{2}$ (time of death from the tumor) and $C$ (time of death from an unrelated cause), we observe $(Y,\Delta_{1},\Delta_{2})$, where $Y=\min\left\{T_{2},C\right\}$, $\Delta_{1}=1_{\left\{T_{1}\leq C\right\}}$, and $\Delta_{2}=1_{\left\{T_{2}\leq C\right\}}$. The random variables $T_{1}$ and $T_{2}$ are independent of $C$ and have a joint distribution such that $P(T_{1}\leq T_{2})=1$. Some authors call this model a “survival-sacrifice model”.

[20] (generally to be denoted by LJP (1997)) proposed a Weighted Least Squares estimator for $F_{1}$ (the marginal distribution function of $T_{1}$), using the Kaplan-Meier estimator of $F_{2}$ (the marginal distribution function of $T_{2}$). The authors claimed that their estimator is more efficient than the MLE (maximum likelihood estimator) of $F_{1}$ and that the Kaplan-Meier estimator is more efficient than the MLE of $F_{2}$. However, we show that the MLE of $F_{1}$ was not computed correctly, and that the (claimed) MLE estimate of $F_{1}$ is even undefined in the case of active constraints.

In our simulation study we used a primal-dual interior point algorithm to obtain the true MLE of $F_{1}$. The results showed a better performance of the MLE of $F_{1}$ over the weighted least squares estimator in LJP (1997) for points where $F_{1}$ is close to $F_{2}$. Moreover, application to the model, used in the simulation study of LJP (1997), showed smaller variances of the MLE estimators of the first and second moments for both $F_{1}$ and $F_{2}$, and sample sizes from 100 up to 5000, in comparison to the estimates, based on the weighted least squares estimator for $F_{1}$, proposed in LJP (1997), and the Kaplan-Meier estimator for $F_{2}$. R scripts are provided for computing the estimates either with the primal-dual interior point method or by the EM algorithm.

In spite of the long history of the model in the biometrics literature (since about 1982), basic properties of the real maximum likelihood estimator (MLE) were still unknown. We give necessary and sufficient conditions for the MLE (Theorem 3.1), as an element of a cone, where the number of generators of the cone increases quadratically with sample size. From this and a self-consistency equation, turned into a Volterra integral equation, we derive the consistency of the MLE (Theorem 4.1). We conjecture that (under some natural conditions) one can extend the methods, used to prove consistency, to proving that the MLE is $\sqrt{n}$ consistent for $F_{2}$ and cube root $n$ convergent for $F_{1}$, but this has presently not yet been proved.

It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in homoscedastic isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.

Modern multiscale type segmentation methods are known to detect multiple change-points with high statistical accuracy, while allowing for fast computation. Underpinning (minimax) estimation theory has been developed mainly for models that assume the signal as a piecewise constant function. In this paper, for a large collection of multiscale segmentation methods (including various existing procedures), such theory will be extended to certain function classes beyond step functions in a nonparametric regression setting. This extends the interpretation of such methods on the one hand and on the other hand reveals these methods as robust to deviation from piecewise constant functions. Our main finding is the adaptation over nonlinear approximation classes for a universal thresholding, which includes bounded variation functions, and (piecewise) Hölder functions of smoothness order $0<\alpha \le1$ as special cases. From this we derive statistical guarantees on feature detection in terms of jumps and modes. Another key finding is that these multiscale segmentation methods perform nearly (up to a log-factor) as well as the oracle piecewise constant segmentation estimator (with known jump locations), and the best piecewise constant approximants of the (unknown) true signal. Theoretical findings are examined by various numerical simulations.

In this paper, aliasing effects are investigated for random fields defined on the $d$-dimensional sphere $\mathbb{S}^{d}$ and reconstructed from discrete samples. First, we introduce the concept of an aliasing function on $\mathbb{S}^{d}$. The aliasing function allows one to identify explicitly the aliases of a given harmonic coefficient in the Fourier decomposition. Then, we exploit this tool to establish the aliases of the harmonic coefficients approximated by means of the quadrature procedure named spherical uniform sampling. Subsequently, we study the consequences of the aliasing errors in the approximation of the angular power spectrum of an isotropic random field, the harmonic decomposition of its covariance function. Finally, we show that band-limited random fields are aliases-free, under the assumption of a sufficiently large amount of nodes in the quadrature rule.

Weighting the $p$-values is a well-established strategy that improves the power of multiple testing procedures while dealing with heterogeneous data. However, how to achieve this task in an optimal way is rarely considered in the literature. This paper contributes to fill the gap in the case of group-structured null hypotheses, by introducing a new class of procedures named ADDOW (for Adaptive Data Driven Optimal Weighting) that adapts both to the alternative distribution and to the proportion of true null hypotheses. We prove the asymptotical FDR control and power optimality among all weighted procedures of ADDOW, which shows that it dominates all existing procedures in that framework. Some numerical experiments show that the proposed method preserves its optimal properties in the finite sample setting when the number of tests is moderately large.

We consider the problem of estimating an expected outcome from a stochastic simulation model. Our goal is to develop a theoretical framework on importance sampling for such estimation. By investigating the variance of an importance sampling estimator, we propose a two-stage procedure that involves a regression stage and a sampling stage to construct the final estimator. We introduce a parametric and a nonparametric regression estimator in the first stage and study how the allocation between the two stages affects the performance of the final estimator. We analyze the variance reduction rates and derive oracle properties of both methods. We evaluate the empirical performances of the methods using two numerical examples and a case study on wind turbine reliability evaluation.

The purpose of this paper is twofold. First, introduce a new adaptive procedure to select the optimal – up to a logarithmic factor – cutoff parameter for Fourier density estimators. Two inverse problems are considered: deconvolution and decompounding. Deconvolution is a typical inverse problem, for which our procedure is numerically simple and stable, a comparison is performed with penalized techniques. Moreover, the procedure and the proof of oracle bounds do not rely on any knowledge on the noise term. Second, for decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of $n$ increments at timestep $\Delta$, build an unified adaptive estimator which is optimal – up to a logarithmic factor – regardless the behavior of $\Delta$.

We consider tail empirical processes of long memory stochastic volatility models with heavy tails and leverage. We study the limiting behaviour of the tail empirical process with both fixed and random levels. We show a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not affected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic mean estimators. The paper can be viewed as an extension of [21] while the presence of leverage in the model creates additional theoretical problems, the limiting behaviour remains unchanged.

We systematically study the rates of contraction with respect to the integrated $L_{2}$-distance for Bayesian nonparametric regression in a generic framework, and, notably, without assuming the regression function space to be uniformly bounded. The generic framework is very flexible and can be applied to a wide class of nonparametric prior models. Three non-trivial applications of the framework are provided: The finite random series regression of an $\alpha$-Hölder function, with adaptive rates of contraction up to a logarithmic factor; The un-modified block prior regression of an $\alpha$-Sobolev function, with adaptive-and-exact rates of contraction; The Gaussian spline regression of an $\alpha$-Hölder function, with near optimal rates of contraction. These applications serve as generalization or complement of their respective results in the literature. Extensions to the fixed-design regression problem and sparse additive models in high dimensions are discussed as well.

We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form \[Y_{t}=\sum_{k=1}^{N(t)}Z_{k},~~~t\ge 0,\] where $N(t)$ is a standard Poisson process of intensity $\lambda$, and $Z_{k}$ are drawn i.i.d. from jump measure $\mu$. A high-dimensional wavelet series prior for the Lévy measure $\nu =\lambda\mu$ is devised and the posterior distribution arises from observing discrete samples $Y_{\Delta},Y_{2\Delta},\dots,Y_{n\Delta}$ at fixed observation distance $\Delta$, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior distribution around the true Lévy density that are optimal up to logarithmic factors over Hölder classes, as sample size $n$ increases. We prove a functional Bernstein–von Mises theorem for the distribution functions of both $\mu$ and $\nu$, as well as for the intensity $\lambda$, establishing the fact that the posterior distribution is approximated by an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the information lower bound for this inverse problem. As a consequence posterior based inferences, such as nonparametric credible sets, are asymptotically valid and optimal from a frequentist point of view.

When detecting a change-point in the marginal distribution of a stationary time series, bootstrap techniques are required to determine critical values for the tests when the pre-change distribution is unknown. In this paper, we propose a sequential moving block bootstrap and demonstrate its validity under a converging alternative. Furthermore, we demonstrate that power is still achieved by the bootstrap under a non-converging alternative. We follow the approach taken by Peligrad in [14], and avoid assumptions of mixing, association or near epoch dependence. These results are applied to a linear process and are shown to be valid under very mild conditions on the existence of any moment of the innovations and a corresponding condition of summability of the coefficients.

This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_{0})$ at some $x_{0}\in\mathbb{R}$ with i.i.d. contaminated observations: \[X_{1},\dots,X_{n}\sim (1-\epsilon)f+\epsilon g\] where $g$ stands for a contamination distribution. We closely track the effect of contamination by the following model index: contamination proportion $\epsilon$, smoothness of the target density $\beta_{0}$, smoothness of the contamination density $\beta_{1}$, and the local level of contamination $m$ such that $g(x_{0})\leq{m}$. The local effect of contamination is shown to depend intricately on the interplay of these parameters. In particular, under a minimax framework, the cost \[[\epsilon^{2}(1\wedge m)^{2}]\vee[n^{-\frac{2\beta_{1}}{2\beta_{1}+1}}\epsilon^{\frac{2}{2\beta_{1}+1}}]\] is shown to be the optimal cost for contamination compared with the usual minimax rate without contamination. The lower bound relies on a novel construction that involves perturbations of a density function at two different resolutions. Such a construction may be of independent interest for the study of local effect of contamination in other nonparametric estimation problems. We also study the setting without any assumption on the contamination distribution, and the minimax cost for contamination is shown to be \[\epsilon^{\frac{2\beta_{0}}{\beta_{0}+1}}.\] Finally, the minimax cost for adaptation is established both for smooth contamination and arbitrary contamination. Under arbitrary contamination, we show that while adaptation to either contamination proportion or smoothness only costs a logarithmic factor, adaptation to both numbers is impossible.

This paper introduces test and estimation procedures for abrupt and gradual changes in the entire jump behaviour of a discretely observed Itō semimartingale. In contrast to existing work we analyse jumps of arbitrary size which are not restricted to a minimum height. Our methods are based on weak convergence of a truncated sequential empirical distribution function of the jump characteristic of the underlying Itō semimartingale. Critical values for the new tests are obtained by a multiplier bootstrap approach and we investigate the performance of the tests also under local alternatives. An extensive simulation study shows the finite-sample properties of the new procedures.

In statistical research there usually exists a choice between structurally simpler or more complex models. We argue that, even if a more complex, locally stationary time series model were true, then a simple, stationary time series model may be advantageous to work with under parameter uncertainty. We present a new model choice methodology, where one of two competing approaches is chosen based on its empirical, finite-sample performance with respect to prediction, in a manner that ensures interpretability. A rigorous, theoretical analysis of the procedure is provided. As an important side result we prove, for possibly diverging model order, that the localised Yule-Walker estimator is strongly, uniformly consistent under local stationarity. An R package, forecastSNSTS, is provided and used to apply the methodology to financial and meteorological data in empirical examples. We further provide an extensive simulation study and discuss when it is preferable to base forecasts on the more volatile time-varying estimates and when it is advantageous to forecast as if the data were from a stationary process, even though they might not be.

Feature allocation models generalize classical species sampling models by allowing every observation to belong to more than one species, now called features. Under the popular Bernoulli product model for feature allocation, we assume $n$ observable samples and we consider the problem of estimating the expected number $M_{n}$ of hitherto unseen features that would be observed if one additional individual was sampled. The interest in estimating $M_{n}$ is motivated by numerous applied problems where the sampling procedure is expensive, in terms of time and/or financial resources allocated, and further samples can be only motivated by the possibility of recording new unobserved features. We consider a nonparametric estimator $\hat{M}_{n}$ of $M_{n}$ which has the same analytic form of the popular Good-Turing estimator of the missing mass in the context of species sampling models. We show that $\hat{M}_{n}$ admits a natural interpretation both as a jackknife estimator and as a nonparametric empirical Bayes estimator. Furthermore, we give provable guarantees for the performance of $\hat{M}_{n}$ in terms of minimax rate optimality, and we provide with an interesting connection between $\hat{M}_{n}$ and the Good-Turing estimator for species sampling. Finally, we derive non-asymptotic confidence intervals for $\hat{M}_{n}$, which are easily computable and do not rely on any asymptotic approximation. Our approach is illustrated with synthetic data and SNP data from the ENCODE sequencing genome project.

A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi}$ on $\mathbb{R}^{d}$ known up to a normalizing constant. Moreover, $-\log\hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally Hölder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+\beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.

Let $A:[0,1]\rightarrow\mathbb{H}_{m}$ (the space of Hermitian matrices) be a matrix valued function which is low rank with entries in Hölder class $\Sigma(\beta,L)$. The goal of this paper is to study statistical estimation of $A$ based on the regression model $\mathbb{E}(Y_{j}|\tau_{j},X_{j})=\langle A(\tau_{j}),X_{j}\rangle,$ where $\tau_{j}$ are i.i.d. uniformly distributed in $[0,1]$, $X_{j}$ are i.i.d. matrix completion sampling matrices, $Y_{j}$ are independent bounded responses. We propose an innovative nuclear norm penalized local polynomial estimator and establish an upper bound on its point-wise risk measured by Frobenius norm. Then we extend this estimator globally and prove an upper bound on its integrated risk measured by $L_{2}$-norm. We also propose another new estimator based on bias-reducing kernels to study the case when $A$ is not necessarily low rank and establish an upper bound on its risk measured by $L_{\infty}$-norm. We show that the obtained rates are all optimal up to some logarithmic factor in minimax sense. Finally, we propose an adaptive estimation procedure based on Lepskii’s method and model selection with data splitting technique, which is computationally efficient and can be easily implemented and parallelized on distributed systems.

We complement the theory developed in Preinerstorfer and Pötscher (2016) with further finite sample results on size and power of heteroskedasticity and autocorrelation robust tests. These allow us, in particular, to show that the sufficient conditions for the existence of size-controlling critical values recently obtained in Pötscher and Preinerstorfer (2018) are often also necessary. We furthermore apply the results obtained to tests for hypotheses on deterministic trends in stationary time series regressions, and find that many tests currently used are strongly size-distorted.

Given additional distributional information in the form of moment restrictions, kernel density and distribution function estimators with implied generalised empirical likelihood probabilities as weights achieve a reduction in variance due to the systematic use of this extra information. The particular interest here is the estimation of the density or distribution functions of (generalised) residuals in semi-parametric models defined by a finite number of moment restrictions. Such estimates are of great practical interest, being potentially of use for diagnostic purposes, including tests of parametric assumptions on an error distribution, goodness-of-fit tests or tests of overidentifying moment restrictions. The paper gives conditions for the consistency and describes the asymptotic mean squared error properties of the kernel density and distribution estimators proposed in the paper. A simulation study evaluates the small sample performance of these estimators.

We consider nonlinear regression models that are solely defined by a parametric model for the regression function. The responses are assumed to be missing at random, with the missingness depending on multiple covariates. We propose estimators for expectations of a known function of response and covariates. Our estimator is a nonparametric estimator corrected for the regression function. We show that it is asymptotically efficient in the Hájek and Le Cam sense. Simulations and an example using real data confirm the optimality of our approach.

Motivated by the growing interest in sound forecast evaluation techniques with an emphasis on distribution tails rather than average behaviour, we investigate a fundamental question arising in this context: Can statistical features of distribution tails be elicitable, i.e. be the unique minimizer of an expected score? We demonstrate that expected scores are not suitable to distinguish genuine tail properties in a very strong sense. Specifically, we introduce the class of max-functionals, which contains key characteristics from extreme value theory, for instance the extreme value index. We show that its members fail to be elicitable and that their elicitation complexity is in fact infinite under mild regularity assumptions. Further we prove that, even if the information of a max-functional is reported via the entire distribution function, a proper scoring rule cannot separate max-functional values. These findings highlight the caution needed in forecast evaluation and statistical inference if relevant information is encoded by such functionals.

Mixture models appear in many research areas. In genetic and epidemiology applications, sometimes the mixture proportions may vary but are known. For such data, the existing methods for the underlying component density estimation may produce undesirable results: negative values in the density estimates. In this paper, we propose a maximum smoothed likelihood method to estimate these component density functions. The proposed estimates maximize a smoothed log likelihood function which can inherit all the important properties of probability density functions. A majorization-minimization algorithm is suggested to compute the proposed estimates numerically. We show that, starting from any initial value, the algorithm converges. Furthermore, we establish the asymptotic convergence rate of the $L_{1}$ errors of our proposed estimators. Our method provides a general framework for dealing with many similar mixture model problems. An adaptive procedure is suggested for choosing the bandwidths in our estimation procedure. Simulation studies show that the proposed method is very promising and can be much more efficient than the existing method in terms of the $L_{1}$ errors. A malaria data application shows the advantages of our method over others.

A useful approach for analysing multiple time series is via characterising their spectral density matrix as the frequency domain analog of the covariance matrix. When the dimension of the time series is large compared to their length, regularisation based methods can overcome the curse of dimensionality, but the existing ones lack theoretical justification. This paper develops the first non-asymptotic result for characterising the difference between the sample and population versions of the spectral density matrix, allowing one to justify a range of high-dimensional models for analysing time series. As a concrete example, we apply this result to establish the convergence of the smoothed periodogram estimators and sparse estimators of the inverse of spectral density matrices, namely precision matrices. These results, novel in the frequency domain time series analysis, are corroborated by simulations and an analysis of the Google Flu Trends data.

We consider the minimax estimation problem of a discrete distribution with support size $k$ under locally differential privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number $\epsilon$ measures the privacy level of a privatization scheme.

In our previous work (IEEE Trans. Inform. Theory, 2018), we proposed a family of new privatization schemes and the corresponding estimator. We also proved that our scheme and estimator are order optimal in the regime $e^{\epsilon}\ll k$ under both $\ell_{2}^{2}$ (mean square) and $\ell_{1}$ loss. In this paper, we sharpen this result by showing asymptotic optimality of the proposed scheme under the $\ell_{p}^{p}$ loss for all $1\le p\le 2.$ More precisely, we show that for any $p\in[1,2]$ and any $k$ and $\epsilon,$ the ratio between the worst-case $\ell_{p}^{p}$ estimation loss of our scheme and the optimal value approaches $1$ as the number of samples tends to infinity. The lower bound on the minimax risk of private estimation that we establish as a part of the proof is valid for any loss function $\ell_{p}^{p},p\ge 1.$

The study of the causal relationships in a stochastic process $(Y_{t},Z_{t})_{t\in\mathbb{Z}}$ is a subject of a particular interest in finance and economy. A widely-used approach is to consider the notion of Granger causality, which in the case of first order Markovian processes is based on the joint distribution function of ${{(Y_{t+1},Z_{t})}}$ given ${{Y_{t}}}$. The measures of Granger causality proposed so far are global in the sense that if the relationship between ${{Y_{t+1}}}$ and ${{Z_{t}}}$ changes with the value taken by ${{Y_{t}}}$, this may not be captured. To circumvent this limitation, this paper proposes local causality measures based on the conditional copula of ${{(Y_{t+1},Z_{t})}}$ given ${{Y_{t}}}=x$. Exploiting results by [5] on the asymptotic behavior of two kernel-based conditional copula estimators under $\alpha$-mixing, the asymptotic normality of nonparametric estimators of these local measures is deduced and asymptotically valid confidence intervals are built; tests of local non-causality are also developed. The suitability of the proposed methods is investigated with simulations and their usefulness is illustrated on the time series of Standard & Poor’s 500 prices and trading volumes.

We consider a dynamic version of the stochastic block model, in which the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution is modeled through a hidden Markov chain on the nodes memberships. We prove the consistency (as the number of nodes and time steps increase) of the maximum likelihood and variational estimators of the model parameters, and obtain upper bounds on the rates of convergence of these estimators. We also explore the particular case where the number of time steps is fixed and connectivity parameters are allowed to vary.

Maximizing the likelihood has been widely used for estimating the unknown covariance parameters of spatial Gaussian processes. However, evaluating and optimizing the likelihood function can be computationally intractable, particularly for large number of (possibly) irregularly spaced observations, due to the need to handle the inverse of ill-conditioned and large covariance matrices. Extending the “inversion-free” method of Anitescu, Chen and Stein [1], we investigate a broad class of covariance parameter estimation based on inversion-free surrogate losses and block diagonal approximation schemes of the covariance structure. This class of estimators yields a spectrum for negotiating the trade-off between statistical accuracy and computational cost. We present fixed-domain asymptotic properties of our proposed method, establishing $\sqrt{n}$-consistency and asymptotic normality results for isotropic Matern Gaussian processes observed on a multi-dimensional and irregular lattice. Simulation studies are also presented for assessing the scalability and statistical efficiency of the proposed algorithm for large data sets.

For sets $\mathcal{Q}$ and $\mathcal{Y}$, the generalized Fréchet mean $m\in \mathcal{Q}$ of a random variable $Y$, which has values in $\mathcal{Y}$, is any minimizer of $q\mapsto \mathbb{E}[\mathfrak{c}(q,Y)]$, where $\mathfrak{c}\colon \mathcal{Q}\times \mathcal{Y}\to \mathbb{R}$ is a cost function. There are little restrictions to $\mathcal{Q}$ and $\mathcal{Y}$. In particular, $\mathcal{Q}$ can be a non-Euclidean metric space. We provide convergence rates for the empirical generalized Fréchet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Fréchet means, we do not require a finite diameter of the $\mathcal{Q}$ or $\mathcal{Y}$. Instead, we assume an inequality, which we call quadruple inequality. It generalizes an otherwise common Lipschitz condition on the cost function. This quadruple inequality is known to hold in Hadamard spaces. We show that it also holds in a suitable way for certain powers of a Hadamard-metric.

Matrix factorization is a powerful data analysis tool. It has been used in multivariate time series analysis, leading to the decomposition of the series in a small set of latent factors. However, little is known on the statistical performances of matrix factorization for time series. In this paper, we extend the results known for matrix estimation in the i.i.d setting to time series. Moreover, we prove that when the series exhibit some additional structure like periodicity or smoothness, it is possible to improve on the classical rates of convergence.

Clusters of different objects are of great interest in many fields, such as agriculture and ecology. One kind of clustering methods is very different from the traditional statistical clustering analysis, which is based on discrete data points. This method of clustering defines clusters as the connected areas where a well-defined spatial random field is above certain threshold. The statistical properties, especially the distributional properties, of the defined clusters are vital for the studies of related fields. However, the available statistical techniques for analyzing clustering models are not applicable to these problems. We study the distribution properties of the clusters by defining a distribution function of the clusters rigorously and providing methods to estimate the spatial distribution function. Our results are illustrated by numerical experiments and an application to a real world problem.

Ratios of medians or other suitable quantiles of two distributions are widely used in medical research to compare treatment and control groups or in economics to compare various economic variables when repeated cross-sectional data are available. Inspired by the so-called growth incidence curves introduced in poverty research, we argue that the ratio of quantile functions is a more appropriate and informative tool to compare two distributions. We present an estimator for the ratio of quantile functions and develop corresponding simultaneous confidence bands, which allow to assess significance of certain features of the quantile functions ratio. Derived simultaneous confidence bands rely on the asymptotic distribution of the quantile functions ratio and do not require re-sampling techniques. The performance of the simultaneous confidence bands is demonstrated in simulations. Analysis of expenditure data from Uganda in years 1999, 2002 and 2005 illustrates the relevance of our approach.

We consider the problem of estimating smooth integrated functionals of a monotone nonincreasing density $f$ on $[0,\infty)$ using the nonparametric maximum likelihood based plug-in estimator. We find the exact asymptotic distribution of this natural (tuning parameter-free) plug-in estimator, properly normalized. In particular, we show that the simple plug-in estimator is always $\sqrt{n}$-consistent, and is additionally asymptotically normal with zero mean and the semiparametric efficient variance for estimating a subclass of integrated functionals. To the best of our knowledge, this is the first time that such a large class of integrated functionals based on the Grenander estimator is treated in a unified fashion. Compared to the previous results on this topic (see e.g., Nickl (2008), Giné and Nickl (2016, Chapter 7), Jankowski (2014), and Söhl (2015)) our results hold for a much larger class of functionals (which include linear and non-linear functionals) under less restrictive assumptions on the underlying $f$ — we do not require $f$ to be (i) smooth, (ii) bounded away from $0$, or (iii) compactly supported. Further, when $f$ is the uniform distribution on a compact interval we explicitly characterize the asymptotic distribution of the plug-in estimator — which now converges at a non-standard rate — thereby extending the results in Groeneboom and Pyke (1983) for the case of the quadratic functional.

The purpose of this paper is to construct confidence intervals for the regression coefficients in the Fine-Gray model for competing risks data with random censoring, where the number of covariates can be larger than the sample size. Despite strong motivation from biomedical applications, a high-dimensional Fine-Gray model has attracted relatively little attention among the methodological or theoretical literature. We fill in this gap by developing confidence intervals based on a one-step bias-correction for a regularized estimation. We develop a theoretical framework for the partial likelihood, which does not have independent and identically distributed entries and therefore presents many technical challenges. We also study the approximation error from the weighting scheme under random censoring for competing risks and establish new concentration results for time-dependent processes. In addition to the theoretical results and algorithms, we present extensive numerical experiments and an application to a study of non-cancer mortality among prostate cancer patients using the linked Medicare-SEER data.

We consider the problem of parameter estimation for the partially observed linear stochastic differential equation. We assume that the unobserved Ornstein-Uhlenbeck process depends on some unknown parameter and estimate the unobserved process and the unknown parameter simultaneously. We construct the One-step MLE-process for the estimator of the parameter and describe its large sample asymptotic properties, including consistency and asymptotic normality. Using the Kalman-Bucy filtering equations we construct recurrent estimators of the state and the parameter.

The present paper obtains a complete description of the limit distributions of sample covariances in $N\times n$ panel data when $N$ and $n$ jointly increase, possibly at different rate. The panel is formed by $N$ independent samples of length $n$ from random-coefficient AR(1) process with the tail distribution function of the random coefficient regularly varying at the unit root with exponent $\beta >0$. We show that for $\beta\in (0,2)$ the sample covariances may display a variety of stable and non-stable limit behaviors with stability parameter depending on $\beta$ and the mutual increase rate of $N$ and $n$.

Within a nonparametric framework, we consider the problem of testing the equality of marginal distributions for a sequence of independent and identically distributed bivariate data, with unobservable order in each pair. In this case, it is not possible to construct the corresponding empirical distributions functions and yet this article shows that a systematic approach to hypothesis testing is possible and provides an empirical process on which inference can be based. Furthermore, we identify the linear statistics that are asymptotically optimal for testing the hypothesis of equal marginal distributions against contiguous alternatives. Finally, we exhibit an interesting property of the proposed stochastic process: local alternatives of dependence can also be detected.

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the supremum distance between the Grenander distribution estimator and the empirical distribution may still be of order $O(n^{-2/3}(\log n)^{2/3})$ almost surely, which reduces to an existing result of Kiefer and Wolfowitz when $f$ has bounded support. We further refine this result by allowing $F$ to be not strictly concave or even non-concave and instead requiring it be “asymptotically” strictly concave. Building on these results, we then develop a test of concavity of $F$ or equivalently monotonicity of $f$, which is shown to have asymptotically pointwise level control under the entire null as well as consistency under any fixed alternative. In fact, we show that our test has local size control and nontrivial local power against any local alternatives that do not approach the null too fast, which may be of interest given the irregularity of the problem. Extensions to settings involving testing concavity/convexity/monotonicity are discussed.

Active learning is a type of sequential design for supervised machine learning, in which the learning algorithm sequentially requests the labels of selected instances from a large pool of unlabeled data points. The objective is to produce a classifier of relatively low risk, as measured under the $0$-$1$ loss, ideally using fewer label requests than the number of random labeled data points sufficient to achieve the same. This work investigates the potential uses of surrogate loss functions in the context of active learning. Specifically, it presents an active learning algorithm based on an arbitrary classification-calibrated surrogate loss function, along with an analysis of the number of label requests sufficient for the classifier returned by the algorithm to achieve a given risk under the $0$-$1$ loss. Interestingly, these results cannot be obtained by simply optimizing the surrogate risk via active learning to an extent sufficient to provide a guarantee on the $0$-$1$ loss, as is common practice in the analysis of surrogate losses for passive learning. Some of the results have additional implications for the use of surrogate losses in passive learning.

Frequentist conditions for asymptotic consistency of Bayesian procedures with i.i.d. data focus on lower bounds for prior mass in Kullback-Leibler neighbourhoods of the data distribution. The goal of this paper is to investigate the flexibility in these criteria. We derive a versatile new posterior consistency theorem, which is used to consider Kullback-Leibler consistency and indicate when it is sufficient to have a prior that charges metric balls instead of KL-neighbourhoods. We generalize our proposal to sieved models with Barron’s negligible prior mass condition and to separable models with variations on Walker’s condition. Results are also applied in semi-parametric consistency: support boundary estimation is considered explicitly and consistency is proved in a model for which Kullback-Leibler priors do not exist. As a further demonstration of applicability, we consider metric consistency at a rate: under a mild integrability condition, the second-order Ghosal-Ghosh-van der Vaart prior mass condition can be relaxed to a lower bound for ordinary KL-neighbourhoods. The posterior rate is derived in a parametric model for heavy-tailed distributions in which the Ghosal-Ghosh-van der Vaart condition cannot be satisfied by any prior.

In this paper we investigate a semiparametric testing approach to answer if the parametric family allocated to the unknown density of a two-component mixture model with one known component is correct or not. Based on a semiparametric estimation of the Euclidean parameters of the model (free from the null assumption), our method compares pairwise the Fourier’s type coefficients of the model estimated directly from the data with the ones obtained by plugging the estimated parameters into the mixture model. These comparisons are incorporated into a sum of square type statistic which order is controlled by a penalization rule. We prove under mild conditions that our test statistic is asymptotically $\chi^{2}_{1}$-distributed and study its behavior, both numerically and theoretically, under different types of alternatives including contiguous nonparametric alternatives. We discuss the counterintuitive, from the practitioner point of view, lack of power of the maximum likelihood version of our test in a neighborhood of challenging non-identifiable situations. Several level and power studies are numerically conducted on models close to those considered in the literature, such as in McLachlan et al. [21], to validate the suitability of our approach. We also implement our testing procedure on the Carina galaxy real dataset which low luminosity mixes with the one of its companion Milky Way. Finally we discuss possible extensions of our work to a wider class of contamination models.

We study the problem of distributional approximations to high-dimensional non-degenerate $U$-statistics with random kernels of diverging orders. Infinite-order $U$-statistics (IOUS) are a useful tool for constructing simultaneous prediction intervals that quantify the uncertainty of ensemble methods such as subbagging and random forests. A major obstacle in using the IOUS is their computational intractability when the sample size and/or order are large. In this article, we derive non-asymptotic Gaussian approximation error bounds for an incomplete version of the IOUS with a random kernel. We also study data-driven inferential methods for the incomplete IOUS via bootstraps and develop their statistical and computational guarantees.

A new cluster analysis method, $K$-quantiles clustering, is introduced. $K$-quantiles clustering can be computed by a simple greedy algorithm in the style of the classical Lloyd’s algorithm for $K$-means. It can be applied to large and high-dimensional datasets. It allows for within-cluster skewness and internal variable scaling based on within-cluster variation. Different versions allow for different levels of parsimony and computational efficiency. Although $K$-quantiles clustering is conceived as nonparametric, it can be connected to a fixed partition model of generalized asymmetric Laplace-distributions. The consistency of $K$-quantiles clustering is proved, and it is shown that $K$-quantiles clusters correspond to well separated mixture components in a nonparametric mixture. In a simulation, $K$-quantiles clustering is compared with a number of popular clustering methods with good results. A high-dimensional microarray dataset is clustered by $K$-quantiles.

Taking a multidimensional time-homogeneous dynamical system and adding a randomly perturbed time-dependent deterministic signal to some of its components gives rise to a high-dimensional system of stochastic differential equations which is driven by possibly very low-dimensional noise. Equations of this type are commonly used in biology for modeling neurons or in statistical mechanics for certain Hamiltonian systems. Assuming that the signal depends on an unknown shape parameter $\vartheta $ and also has an unknown periodicity $T$, we prove Local Asymptotic Normality (LAN) jointly in $\vartheta $ and $T$ for the statistical experiment arising from (partial) observation of this diffusion in continuous time. The local scale turns out to be $n^{-1/2}$ for $\vartheta $ and $n^{-3/2}$ for $T$. Our approach unifies and extends existing results on LAN in variants of the signal in noise model where the parameters $\vartheta $ and $T$ are treated separately. Consequently, we can establish the same efficiency bounds in our more complex model and make use of efficient estimators known from these submodels.

This paper proposes inferential methods for high-dimensional repeated measures in factorial designs. High-dimensional refers to the situation where the dimension is growing with sample size such that either one could be larger than the other. The most important contribution relates to high-accuracy of the methods in the sense that p-values, for example, are accurate up to the second-order. Second-order accuracy in sample size as well as dimension is achieved by obtaining asymptotic expansion of the distribution of the test statistics, and estimation of the parameters of the approximate distribution with second-order consistency. The methods are presented in a unified and succinct manner that it covers general factorial designs as well as any comparisons among the cell means. Expression for asymptotic powers are derived under two reasonable local alternatives. A simulation study provides evidence for a gain in accuracy and power compared to limiting distribution approximations and other competing methods for high-dimensional repeated measures analysis. The application of the methods are illustrated with a real-data from Electroencephalogram (EEG) study of alcoholic and control subjects.

This paper focuses on modeling the dynamic attributes of a dynamic network with a fixed number of vertices. These attributes are considered as time series which dependency structure is influenced by the underlying network. They are modeled by a multivariate doubly stochastic time series framework, that is we assume linear processes for which the coefficient matrices are stochastic processes themselves. We explicitly allow for dependence in the dynamics of the coefficient matrices as well as between the two stochastic processes driving the time series. This framework allows for a separate modeling of the attributes and the underlying network. In this setting, we define network autoregressive models and discuss their stationarity conditions. Furthermore, an estimation approach is discussed in a low- and high-dimensional setting and how this can be applied to forecasting. The finite sample behavior of the forecast approach is investigated. This approach is applied to real data whereby the goal is to forecast the GDP of $33$ economies.

High-dimensional auto-regressive models provide a natural way to model influence between $M$ actors given multi-variate time series data for $T$ time intervals. While there has been considerable work on network estimation, there is limited work in the context of inference and hypothesis testing. In particular, prior work on hypothesis testing in time series has been restricted to linear Gaussian auto-regressive models. From a practical perspective, it is important to determine suitable statistical tests for connections between actors that go beyond the Gaussian assumption. In the context of high-dimensional time series models, confidence intervals present additional estimators since most estimators such as the Lasso and Dantzig selectors are biased which has led to de-biased estimators. In this paper we address these challenges and provide convergence in distribution results and confidence intervals for the multi-variate AR(p) model with sub-Gaussian noise, a generalization of Gaussian noise that broadens applicability and presents numerous technical challenges. The main technical challenge lies in the fact that unlike Gaussian random vectors, for sub-Gaussian vectors zero correlation does not imply independence. The proof relies on using an intricate truncation argument to develop novel concentration bounds for quadratic forms of dependent sub-Gaussian random variables. Our convergence in distribution results hold provided $T=\Omega((s\vee \rho)^{2}\log^{2}M)$, where $s$ and $\rho$ refer to sparsity parameters which matches existed results for hypothesis testing with i.i.d. samples. We validate our theoretical results with simulation results for both block-structured and chain-structured networks.

This paper concerns the dependence structure of a random pair $(Y_{1},Y_{2})$ conditionally upon a covariate $X$ in the case where the variable $Y_{1}$ is subject to random right censoring. The dependence structure is described by a conditional copula for which we propose a nonparametric estimator. We establish the asymptotic properties of the proposed estimator and investigate its finite sample behavior in a numerical study. The methodology is illustrated through a real data example featuring patients with advanced lung cancer.

We define a modified Wasserstein distance for distribution clustering which inherits many of the properties of the Wasserstein distance but which can be estimated easily and computed quickly. The modified distance is the sum of two terms. The first term — which has a closed form — measures the location-scale differences between the distributions. The second term is an approximation that measures the remaining distance after accounting for location-scale differences. We consider several forms of approximation with our main emphasis being a tangent space approximation that can be estimated using nonparametric regression and leads to fast and easy computation of barycenters which otherwise would be very difficult to compute. We evaluate the strengths and weaknesses of this approach on simulated and real examples.

The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive the distributional limits of the empirical Sinkhorn divergence and its centered version (Sinkhorn loss). We also propose a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions. Our work is inspired by the results of Sommerfeld and Munk in [33] on the asymptotic distribution of empirical Wasserstein distance on finite space using unregularized transportation costs. Incidentally we also analyze the asymptotic distribution of entropy-regularized Wasserstein distances when the regularization parameter tends to zero. Simulated and real datasets are used to illustrate our approach.

In this paper, we investigate quasi-maximum likelihood (QML) estimation for the parameters of a cointegrated solution of a continuous-time linear state space model observed at discrete time points. The class of cointegrated solutions of continuous-time linear state space models is equivalent to the class of cointegrated continuous-time ARMA (MCARMA) processes. As a start, some pseudo-innovations are constructed to be able to define a QML-function. Moreover, the parameter vector is divided appropriately in long-run and short-run parameters using a representation for cointegrated solutions of continuous-time linear state space models as a sum of a Lévy process plus a stationary solution of a linear state space model. Then, we establish the consistency of our estimator in three steps. First, we show the consistency for the QML estimator of the long-run parameters. In the next step, we calculate its consistency rate. Finally, we use these results to prove the consistency for the QML estimator of the short-run parameters. After all, we derive the limiting distributions of the estimators. The long-run parameters are asymptotically mixed normally distributed, whereas the short-run parameters are asymptotically normally distributed. The performance of the QML estimator is demonstrated by a simulation study.

This paper presents a class of new algorithms for distributed statistical estimation that exploit divide-and-conquer approach. We show that one of the key benefits of the divide-and-conquer strategy is robustness, an important characteristic for large distributed systems. We establish connections between performance of these distributed algorithms and the rates of convergence in normal approximation, and prove non-asymptotic deviations guarantees, as well as limit theorems, for the resulting estimators. Our techniques are illustrated through several examples: in particular, we obtain new results for the median-of-means estimator, and provide performance guarantees for distributed maximum likelihood estimation.

Two-sample testing is a fundamental problem in statistics. Despite its long history, there has been renewed interest in this problem with the advent of high-dimensional and complex data. Specifically, in the machine learning literature, there have been recent methodological developments such as classification accuracy tests. The goal of this work is to present a regression approach to comparing multivariate distributions of complex data. Depending on the chosen regression model, our framework can efficiently handle different types of variables and various structures in the data, with competitive power under many practical scenarios. Whereas previous work has been largely limited to global tests which conceal much of the local information, our approach naturally leads to a local two-sample testing framework in which we identify local differences between multivariate distributions with statistical confidence. We demonstrate the efficacy of our approach both theoretically and empirically, under some well-known parametric and nonparametric regression methods. Our proposed methods are applied to simulated data as well as a challenging astronomy data set to assess their practical usefulness.

Variable selection methods have been developed in linear regression to provide sparse solutions. Recent studies have focused on further interpretations on the sparse solutions in terms of false positive control. In this paper, we consider false negative control for variable selection with the goal to efficiently select a high proportion of relevant predictors. Different from existing studies in power analysis and sure screening, we propose to directly estimate the false negative proportion (FNP) of a decision rule and select the smallest subset of predictors that has the estimated FNP less than a user-specified control level. The proposed method is adaptive to the user-specified control level on FNP by selecting less candidates if a higher level is implemented. On the other hand, when data has stronger effect size or larger sample size, the proposed method controls FNP more efficiently with less false positives. New analytic techniques are developed to cope with the major challenge of FNP control when relevant predictors cannot be consistently separated from irrelevant ones. Our numerical results are in line with the theoretical findings.

Varying coefficient models are useful in applications where the effect of the covariate might depend on some other covariate such as time or location. Various applications of these models often give rise to case-specific prior distributions for the parameter(s) describing how much the coefficients vary. In this work, we introduce a unified view of varying coefficients models, arguing for a way of specifying these prior distributions that are coherent across various applications, avoid overfitting and have a coherent interpretation. We do this by considering varying coefficients models as a flexible extension of the natural simpler model and capitalising on the recently proposed framework of penalized complexity (PC) priors. We illustrate our approach in two spatial examples where varying coefficient models are relevant.

This paper considers partial linear regression models when neither the response variable nor the covariates can be directly observed, but are instead measured with both multiplicative and additive distortion measurement errors. We propose conditional variance estimation methods to calibrate the unobserved variables. A profile least-squares estimator associated with the asymptotic results and confidence intervals is then proposed. To do hypothesis testing of the parameters, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties of the estimator and the test statistic are also established. Further, we employ the smoothly clipped absolute deviation penalty to select relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Estimation, hypothesis testing, and variable selection are discussed under the scenario of multiplicative distortion alone. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its applicability.

A case-control family study is a study where individuals with a disease of interest (case probands) and individuals without the disease (control probands) are randomly sampled from a well-defined population. Possibly right-censored age at onset and disease status are observed for both probands and their relatives. Correlation among the outcomes within a family is induced by factors such as inherited genetic susceptibility, shared environment, and common behavior patterns. For this setting, we present a nonparametric estimator of the marginal survival function, based on local linear estimation of conditional survival functions. Asymptotic theory for the estimator is provided, making this paper the first to present for this data setting a fully nonparametric estimator with proven consistency. Simulation results are presented showing that the method performs well. The method is illustrated on data from a prostate cancer study.