We study asymptotic minimax problems for estimating a $d$-dimensional regression parameter over spheres of growing dimension ($d\to\infty$). Assuming that the data follows a linear model with Gaussian predictors and errors, we show that ridge regression is asymptotically minimax and derive new closed form expressions for its asymptotic risk under squared-error loss. The asymptotic risk of ridge regression is closely related to the Stieltjes transform of the Marčenko–Pastur distribution and the spectral distribution of the predictors from the linear model. Adaptive ridge estimators are also proposed (which adapt to the unknown radius of the sphere) and connections with equivariant estimation are highlighted. Our results are mostly relevant for asymptotic settings where the number of observations, $n$, is proportional to the number of predictors, that is, $d/n\to\rho\in(0,\infty)$.

Motivated by the problem of testing tetrad constraints in factor analysis, we study the large-sample distribution of Wald statistics at parameter points at which the gradient of the tested constraint vanishes. When based on an asymptotically normal estimator, the Wald statistic converges to a rational function of a normal random vector. The rational function is determined by a homogeneous polynomial and a covariance matrix. For quadratic forms and bivariate monomials of arbitrary degree, we show unexpected relationships to chi-square distributions that explain conservative behavior of certain Wald tests. For general monomials, we offer a conjecture according to which the reciprocal of a certain quadratic form in the reciprocals of dependent normal random variables is chi-square distributed.

We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on $\mathbb{R}^{N}$; (ii) nonreversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on $\mathbb{R}^{N}$ in a number of steps which is independent of $N$. We also present the underlying theory for the limiting nonreversible diffusion on Hilbert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm.

Introduced is the notion of minimality for spectral representations of sum- and max-infinitely divisible processes and it is shown that the minimal spectral representation on a Borel space exists and is unique. This fact is used to show that a stationary, stochastically continuous, sum- or max-i.d. random process on $\mathbb{R}^{d}$ can be generated by a measure-preserving flow on a $\sigma$-finite Borel measure space and that this flow is unique. This development makes it possible to extend the classification program of Rosiński (Ann. Probab. 23 (1995) 1163–1187) with a unified treatment of both sum- and max-infinitely divisible processes. As a particular case, a characterization of stationary, stochastically continuous, union-infinitely divisible random measurable subsets of $\mathbb{R}^{d}$ is obtained. Introduced and classified are several new max-i.d. random field models including fields of Penrose type and fields associated to Poisson line processes.

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax optimal under minimal and natural conditions. This closes an important gap in the literature. Optimal adaptive estimation is obtained by a data-driven bandwidth choice. As a side result, we obtain optimal rates for the plug-in estimation of distribution functions with unknown error distributions. The method is applied to a real data example.

In this paper, we consider nonparametric finite translation hidden Markov models, or more generally finite translation mixtures with dependent latent variables. We prove that all the parameters of the model are identifiable as soon as the matrix that defines the joint distribution of two consecutive latent variables is non-singular and the translation parameters are distinct. Under this assumption, we provide a consistent estimator of the number of populations, of the translation parameters and of the distribution of two consecutive latent variables, which we prove to be asymptotically normally distributed under mild dependency assumptions. We propose a nonparametric estimator of the unknown translated density. In case the latent variables form a Markov chain, we prove that this estimator is minimax adaptive over regularity classes of densities.

Pickands dependence functions characterize bivariate extreme value copulas. In this paper, we study the class of polynomial Pickands functions. We provide a solution for the characterization of such polynomials of degree at most $m+2$, $m\geq0$, and show that these can be parameterized by a vector in $\mathbb{R}^{m+1}$ belonging to the intersection of two ellipsoids. We also study the class of Bernstein approximations of order $m+2$ of Pickands functions which are shown to be (polynomial) Pickands functions and parameterized by a vector in $\mathbb{R}^{m+1}$ belonging to a polytope. We give necessary and sufficient conditions for which a polynomial Pickands function is in fact a Bernstein approximation of some Pickands function. Approximation results of Pickands dependence functions by polynomials are given. Finally, inferential methodology is discussed and comparisons based on simulated data are provided.

This work is concerned with the detection of a mixture distribution from a $\mathbb{R}$-valued sample. Given a sample $X_{1},\dots,X_{n}$ and an even density $\phi$, our aim is to detect whether the sample distribution is $\phi(\cdot-\mu)$ for some unknown mean $\mu$, or is defined as a two-component mixture based on translations of $\phi$. We propose a procedure which is based on several spacings of the order statistics, which provides a level-$\alpha$ test for all $n$. Our test is therefore a multiple testing procedure and we prove from a theoretical and practical point of view that it automatically adapts to the proportion of the mixture and to the difference of the means of the two components of the mixture under the alternative. From a theoretical point of view, we prove the optimality of the power of our procedure in various situations. A simulation study shows the good performances of our test compared with several classical procedures.

In this paper, we study the quenched central limit theorem for the discrete Fourier transform. We show that the Fourier transform of a stationary ergodic process, suitable centered and normalized, satisfies the quenched CLT conditioned by the past sigma algebra. For functions of Markov chains with stationary transitions, this means that the CLT holds with respect to the law of the chain started at a point for almost all starting points. It is necessary to emphasize that no assumption of irreducibility with respect to a measure or other regularity conditions are imposed for this result. We also discuss necessary and sufficient conditions for the validity of quenched CLT without centering. The results are highly relevant for the study of the periodogram of a Markov process with stationary transitions which does not start from equilibrium. The proofs are based of a nice blend of harmonic analysis, theory of stationary processes, martingale approximation and ergodic theory.

This paper presents a study of the asymptotical behavior of the empirical distribution function (e.d.f.) of Gaussian vector components, whose correlation matrix $\Gamma^{(m)}$ is dimension-dependent. By contrast with the existing literature, the vector is not assumed to be stationary. Rather, we make a “vanishing second order” assumption ensuring the covariance matrix $\Gamma^{(m)}$ is not too far from the identity matrix, while the behavior of the e.d.f. is affected by $\Gamma^{(m)}$ only through the sequence $\gamma_{m}=m^{-2}\sum_{i\neq j}\Gamma_{i,j}^{(m)}$, as $m$ grows to infinity. This result recovers some of the previous results for stationary long-range dependencies while it also applies to various, high-dimensional, non-stationary frameworks, for which the most correlated variables are not necessarily close to each other. Finally, we present an application of this work to the multiple testing problem, which was the initial statistical motivation for developing such a methodology.

We study the problem of estimating the one-point specification probabilities in non-necessary finite discrete random fields from partially observed independent samples. Our procedures are based on model selection by minimization of a penalized empirical criterion. The selected estimators satisfy sharp oracle inequalities in $L_{2}$-risk.

We also obtain theoretical results on the slope heuristic for this problem, justifying the slope algorithm to calibrate the leading constant in the penalty. The practical performances of our methods are investigated in two simulation studies. We illustrate the usefulness of our approach by applying the methods to a multi-unit neuronal data from a rat hippocampus.

Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a $d$-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are established for the cases of positive strong dependence, short range dependence, and negative dependence. We provide approximations to asymptotic variances that reveal differential rates of convergence under the three types of dependence. Further, in contrast to the one dimensional (i.e., the time series) case, it is shown that the form of the asymptotic variance in dimensions $d>1$ critically depends on the geometry of the sampling region under positive strong dependence and under negative dependence and that there can be non-trivial edge-effects under negative dependence for $d>1$. Precise conditions for the presence of edge effects are also given.

It was shown in Mishura et al. (Stochastic Process. Appl. 123 (2013) 2353–2369), that any random variable can be represented as improper pathwise integral with respect to fractional Brownian motion. In this paper, we extend this result to cover a wide class of Gaussian processes. In particular, we consider a wide class of processes that are Hölder continuous of order $\alpha>1/2$ and show that only local properties of the covariance function play role for such results.

Density regression provides a flexible strategy for modeling the distribution of a response variable $Y$ given predictors $\mathbf{X}=(X_{1},\ldots,X_{p})$ by letting that the conditional density of $Y$ given $\mathbf{X}$ as a completely unknown function and allowing its shape to change with the value of $\mathbf{X}$. The number of predictors $p$ may be very large, possibly much larger than the number of observations $n$, but the conditional density is assumed to depend only on a much smaller number of predictors, which are unknown. In addition to estimation, the goal is also to select the important predictors which actually affect the true conditional density. We consider a nonparametric Bayesian approach to density regression by constructing a random series prior based on tensor products of spline functions. The proposed prior also incorporates the issue of variable selection. We show that the posterior distribution of the conditional density contracts adaptively at the truth nearly at the optimal oracle rate, determined by the unknown sparsity and smoothness levels, even in the ultra high-dimensional settings where $p$ increases exponentially with $n$. The result is also extended to the anisotropic case where the degree of smoothness can vary in different directions, and both random and deterministic predictors are considered. We also propose a technique to calculate posterior moments of the conditional density function without requiring Markov chain Monte Carlo methods.

This paper deals with the concept of equivalence between direct and indirect effects of a treatment on a response using two sets of intermediate variables and covariates. First, we provide criteria for testing whether two sets of variables can estimate the same direct and indirect effects. Next, based on the proposed criteria, we discuss the variable selection problem from the viewpoint of estimation accuracy of direct and indirect effects, and show that selecting a set of variables that has a direct effect on a response cannot always improve estimation accuracy, which is contrary to the situation found in linear regression models. These results enable us to judge whether different sets of variables can yield the same direct and indirect effects and thus help us select appropriate variables to estimate direct and indirect effects with cost reduction or estimation accuracy.

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.

We introduce a general form of sequential Monte Carlo algorithm defined in terms of a parameterized resampling mechanism. We find that a suitably generalized notion of the Effective Sample Size (ESS), widely used to monitor algorithm degeneracy, appears naturally in a study of its convergence properties. We are then able to phrase sufficient conditions for time-uniform convergence in terms of algorithmic control of the ESS, in turn achievable by adaptively modulating the interaction between particles. This leads us to suggest novel algorithms which are, in senses to be made precise, provably stable and yet designed to avoid the degree of interaction which hinders parallelization of standard algorithms. As a byproduct, we prove time-uniform convergence of the popular adaptive resampling particle filter.

We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the $Z$-component by (Ann. Appl. Probab. 12 (2002) 1390–1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.

The main message in this paper is that there are surprisingly many different Brownian bridges, some of them – familiar, some of them – less familiar. Many of these Brownian bridges are very close to Brownian motions. Somewhat loosely speaking, we show that all the bridges can be conveniently mapped onto each other, and hence, to one “standard” bridge.

The paper shows that, a consequence of this, we obtain a unified theory of distribution free testing in $\mathbb{R}^{d}$, both for discrete and continuous cases, and for simple and parametric hypothesis.

In this paper, we study the differentiability of implicitly defined functions which we encounter in the profile likelihood estimation of parameters in semi-parametric models. Scott and Wild (Biometrika 84 (1997) 57–71; J. Statist. Plann. Inference 96 (2001) 3–27) and Murphy and van der Vaart (J. Amer. Statist. Assoc. 95 (2000) 449–485) developed methodologies that can avoid dealing with such implicitly defined functions by parametrizing parameters in the profile likelihood and using an approximate least favorable submodel in semi-parametric models. Our result shows applicability of an alternative approach presented in Hirose (Ann. Inst. Statist. Math. 63 (2011) 1247–1275) which uses the direct expansion of the profile likelihood.

This paper presents a general asymptotic theory of sequential Bayesian estimation giving results for the strongest, almost sure convergence. We show that under certain smoothness conditions on the probability model, the greedy information gain maximization algorithm for adaptive Bayesian estimation is asymptotically optimal in the sense that the determinant of the posterior covariance in a certain neighborhood of the true parameter value is asymptotically minimal. Using this result, we also obtain an asymptotic expression for the posterior entropy based on a novel definition of almost sure convergence on “most trials” (meaning that the convergence holds on a fraction of trials that converges to one). Then, we extend the results to a recently published framework, which generalizes the usual adaptive estimation setting by allowing different trial placements to be associated with different, random costs of observation. For this setting, the author has proposed the heuristic of maximizing the expected information gain divided by the expected cost of that placement. In this paper, we show that this myopic strategy satisfies an analogous asymptotic optimality result when the convergence of the posterior distribution is considered as a function of the total cost (as opposed to the number of observations).