Asymptotic statistics

Asymptotic statistics is concerned with studying the behaviour of statistical methods for large sample sizes. This has several applications in many different areas within statistics, which is reflected by the diversity of the research on asymptotic statistics within the Statistics group. This research area can be seen as the overhead topic, which is present by the majority of the group.\(\)

Researchers

Prof.dr.ir. G. (Geurt) Jongbloed

Dr. A.F.F. (Alexis) Derumigny

Dr. H.P. (Rik) Lopuhaä

Dr. N. (Nestor) Parolya

Dr. J. (Jakob) Söhl

Prof.dr. A.W. (Aad) van der Vaart

Books

2023 - A. W. van der Vaart, Jon A. Wellner - Springer International Publishing

Weak Convergence and Empirical Processes

2012, Aad van der Vaart, Cambridge University Press

Asymptotic Statistics

This book is an introduction to the field of asymptotic statistics. The treatment is both practical and mathematically rigorous. In addition to most of the standard topics of an asymptotics course, including likelihood inference, M-estimation, the theory of asymptotic efficiency, U-statistics, and rank procedures, the book also presents recent research topics such as semiparametric models, the bootstrap, and empirical processes and their applications. The topics are organized from the central idea of approximation by limit experiments, which gives the book one of its unifying themes. This entails mainly the local approximation of the classical i.i.d. set up with smooth parameters by location experiments involving a single, normally distributed observation. Thus, even the standard subjects of asymptotic statistics are presented in a novel way. Suitable as a graduate or Master's level statistics text, this book will also give researchers an overview of research in asymptotic statistics.

Key Publications

Francesco Gili, Geurt Jongbloed, Aad van der Vaart

Adaptive and Efficient Isotonic Estimation in Wicksell's Problem

We consider nonparametric estimation in Wicksell's problem which has relevant applications in astronomy for estimating the distribution of the positions of the stars in a galaxy given projected stellar positions and in material sciences to determine the 3D microstructure of a material, using its 2D cross sections. In the classical setting, we study the isotonized version of the plug-in estimator (IIE) for the underlying cdf \(F\) of the spheres' squared radii. This estimator is fully automatic, in the sense that it does not rely on tuning parameters, and we show it is adaptive to local smoothness properties of the distribution function \(F\) to be estimated. Moreover, we prove a local asymptotic minimax lower bound in this non-standard setting, with \(\sqrt{(\log n)/n}\)-asymptotics and where the functional \(F\) to be estimated is not regular. Combined, our results prove that the isotonic estimator (IIE) is an adaptive, easy-to-compute, and efficient estimator for estimating the underlying distribution function \(F\).

2024 - Johannes Heiny, Nestor Parolya - Ann. Inst. H. Poincaré Probab. Statist.

Log determinant of large correlation matrices under infinite fourth moment

In this paper, we show the central limit theorem for the logarithmic determinant of the sample correlation matrix \(\mathbf{R}\) constructed from the \((p\times n)\)-dimensional data matrix \(\mathbf{X}\) containing independent and identically distributed random entries with mean zero, variance one and infinite fourth moments. Precisely, we show that for \(p/n\to \gamma\in (0,1)\) as \(n,p\to \infty\) the logarithmic law

\(\begin{equation*} \frac{\log \det \mathbf{R} -(p-n+\frac{1}{2})\log(1-p/n)+p-p/n}{\sqrt{-2\log(1-p/n)- 2 p/n}} \overset{d}{\rightarrow} N(0,1)\, \end{equation*}\)

is still valid if the entries of the data matrix \(\mathbf{X}\) follow a symmetric distribution with a regularly varying tail of index \(\alpha\in (3,4)\). The latter assumptions seem to be crucial, which is justified by the simulations: if the entries of \(\mathbf{X}\) have the infinite absolute third moment and/or their distribution is not symmetric, the logarithmic law is not valid anymore. The derived results highlight that the logarithmic determinant of the sample correlation matrix is a very stable and flexible statistic for heavy-tailed big data and open a novel way of analysis of high-dimensional random matrices with self-normalized entries.

2022 - Hanne Kekkonen - Inverse Problems

Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

We consider the statistical non-linear inverse problem of recovering the absorption term \(f > 0\) in the heat equation

\(\begin{equation*}\begin{cases}{\partial }_{t}u-\frac{1}{2}{\Delta}u+fu=0\quad \hfill & \quad \text{on}\;\mathcal{O}\times (0,\mathbf{T})\hfill \\ u=g\quad \hfill & \quad \text{on}\;\partial \mathcal{O}\times (0,\mathbf{T})\hfill \\ u(\cdot ,0)={u}_{0}\quad \hfill & \quad \text{on}\;\mathcal{O},\hfill \end{cases}\end{equation*}\)

where \(\mathcal{O}\in {\mathbb{R}}^{d}\) is a bounded domain, \(T < \infty\) is a fixed time, and g, u0 are given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution uf on \(\mathcal{O}\times (0,\mathbf{T})\). We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

2022 - Alexis Derumigny, Jean-David Fermanian - Electronic Journal of Statistics

Conditional empirical copula processes and generalized measures of association

We study the weak convergence of conditional empirical copula processes indexed by general families of conditioning events that have non zero probabilities. Moreover, we also study the case where the conditioning events are chosen in a data-driven way. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general multivariate measures of association, possibly given some fixed or random conditioning events. By applying our theoretical results, we prove the asymptotic normality of the estimators of such measures. We illustrate our results with financial data.

2019 - Taras Bodnar, Holger Dette, Nestor Parolya - The Annals of Statistics

Testing for independence of large dimensional vectors

In this paper, new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose, we study the weak convergence of linear spectral statistics of central and (conditionally) noncentral Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) noncentral Fisher matrices is derived which is then used to analyse the power of the tests under the alternative.

The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector. On the other hand, the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.

2018 - Cécile Durot, Hendrik P. Lopuhaä - Statistical Science

Limit Theory in Monotone Function Estimation

We give an overview of the different concepts and methods that are commonly used when studying the asymptotic properties of isotonic estimators. After introducing the inverse process, we illustrate its use in establishing weak convergence of the estimators at a fixed point and also weak convergence of global distances, such as the \(\mathbb{L}_p\)-distance and supremum distance. Furthermore, we discuss the developments on smooth isotonic estimation.

2017 - Hélène Boistard, Hendrik P. Lopuhaä, Anne Ruiz-Gazen - The Annals of Statistics

Functional central limit theorems for single-stage sampling designs

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz–Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.

2014 - Jakob Söhl - Finance and Stochastics

Confidence sets in nonparametric calibration of exponential Lévy models

Confidence intervals and joint confidence sets are constructed for the nonparametric calibration of exponential Lévy models based on prices of European options. To this end, we show joint asymptotic normality in the spectral calibration method for the estimators of the volatility, the drift, the jump intensity and the Lévy density at finitely many points.

2013 - Piet Groeneboom, Geurt Jongbloed - Bernoulli

Testing monotonicity of a hazard: Asymptotic distribution theory

Two test statistics are introduced to test the null hypotheses that the sampling distribution has an increasing hazard rate on a specified interval \([0,a]\). These statistics are empirical \(L_1\)-type distances between the isotonic estimates, which use the monotonicity constraint, and either the empirical distribution function or the empirical cumulative hazard. They measure the excursions of the empirical estimates with respect to the isotonic estimates, owing to local non-monotonicity. Asymptotic normality of the test statistics, if the hazard is strictly increasing on \([0,a]\), is established under mild conditions. This is done by first approximating the global empirical distance by a distance with respect to the underlying distribution function. The resulting integral is treated as sum of increasingly many local integrals to which a central limit theorem can be applied. The behavior of the local integrals is determined by a canonical process, the difference between the stochastic process \(x\mapsto W(x)+x^2\), where \(W\) is standard two-sided Brownian motion, and its greatest convex minorant.

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