Computer-intensive statistical methods

The saddlepoint approximation was introduced into statistics in 1954 by Henry E. Daniels. This basic result on approximating the density function of the sample mean has been generalized to many situations. The accuracy of this approximation is very good, particularly in the tails of the distribution and for small sample sizes, compared with normal or Edgeworth approximation methods. Before applying saddlepoint approximations to the bootstrap, this thesis will focus on saddlepoint approximations for the distribution of quadratic forms in normal variables and for the distribution of the waiting time in the coupon collector's problem. Both developments illustrate the modern art of statistics relying on the computer and embodying both numeric and analytic approximations. Saddlepoint approximations are extremely accurate in both cases. This is underlined in the first development by means of an extensive study and several applications to nonparametric regression, and in the second by several examples, including the exhaustive bootstrap seen from a collector's point of view. The remaining part of this thesis is devoted to the use of saddlepoint approximations in order to replace the computer-intensive bootstrap. The recent massive increases in computer power have led to an upsurge in interest in computer-intensive statistical methods. The bootstrap is the first computer-intensive method to become widely known. It found an immediate place in statistical theory and, more slowly, in practice. The bootstrap seems to be gaining ground as the method of choice in a number of applied fields, where classical approaches are known to be unreliable, and there is sustained interest from theoreticians in its development. But it is known that, for accurate approximations in the tails, the nonparametric bootstrap requires a large number of replicates of the statistic. As this is time-intensive other methods should be considered. Saddlepoint methods can provide extremely accurate approximations to resampling distributions. As a first step I develop fast saddlepoint approximations to bootstrap distributions that work in the presence of an outlier, using a saddlepoint mixture approximation. Then I look at robust M-estimates of location like Huber's M-estimate of location and its initially MAD scaled version. One peculiarity of the current literature is that saddlepoint methods are often used to approximate the density or distribution functions of bootstrap estimators, rather than related pivots, whereas it is the latter which are more relevant for inference. Hence the aim of the final part of this thesis is to apply saddlepoint approximations to the construction of studentized confidence intervals based on robust M-estimates. As examples I consider the studentized versions of Huber's M-estimate of location, of its initially MAD scaled version and of Huber's proposal 2. In order to make robust inference about a location parameter there are three types of robustness one would like to achieve: robustness of performance for the estimator of location, robustness of validity and robustness of efficiency for the resulting confidence interval method. Hence in the context of studentized bootstrap confidence intervals I investigate these in more detail in order to give recommendations for practical use, underlined by an extensive simulation study.

Statistical Applications of Random Matrix Theory: Comparison of Two Populations

Rémy Mariétan

During the last twenty years, Random matrix theory (RMT) has produced numerous results that allow a better understanding of large random matrices. These advances have enabled interesting applications in the domain of communication. Although this theory can contribute to many other domains such as brain imaging or genetic research, its has been rarely applied. The main barrier to the adoption of RMT may be the lack of concrete statistical results from probabilistic Random matrix theory. Indeed, direct generalisation of classical multivariate theory to high dimensional assumptions is often difficult and the proposed procedures often assume strong hypotheses on the data matrix such as normality or overly restrictive independence conditions on the data. This thesis proposes a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors corresponding to the number of observed variables. Although the existing theory builds a very good intuition of the behaviour of these matrices, it does not provide enough results to build a satisfactory test for both the power and the robustness. Hence, inspired by spike models, we define the residual spikes and prove many theorems describing the behaviour of many statistics using eigenvectors and eigenvalues in very general cases. For example in the two central theorems of this thesis, the Invariant Angle Theorem and the Invariant Dot Product Theorem. Using numerous generalisations of the theory, this thesis finally proposes a description of the behaviour of a statistic under a null hypothesis. This statistic allows the user to test the equality of two populations, but also other null hypotheses such as the independence of two sets of variables. Finally, the robustness of the procedure is demonstrated for different classes of models and criteria for evaluating robustness are proposed to the reader. Therefore, the major contribution of this thesis is to propose a methodology both easy to apply and having good properties. Secondly, a large number of theoretical results are demonstrated and could be easily used to build other applications.

EPFL2019

Statistical Aspects of Wasserstein Distances

Victor Panaretos, Yoav Zemel

Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catalyze core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: They can be used to derive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of perturbation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and they have recently become an object of inference in themselves. In this review, we provide a snapshot of the main concepts involved in Wasserstein distances and optimal transportation, and a succinct overview of some of their many statistical aspects.

ANNUAL REVIEWS2019

Statistical methods for insect choice experiments

Ingrid Ricard

Olfactometer experiments are used to determine the effect of odours on the behaviour of organisms such as insects or nematodes, and typically result in data comprising many groups of small counts, overdispersed relative to the multinomial distribution. Overdispersion reflects a lack of independence or heterogeneity among individuals and can lead to statistics having larger variances than expected and possible losses of efficiency. In this thesis, some distributions which consist of generalisations of the multinomial distribution have been developed. These models are based on non-homogeneous Markov chain theory, take the overdispersion into account, and potentially provide a physical interpretation of the overdispersion seen in olfactometer data. Some inference aspects are considered, including comparison of the asymptotic relative efficiencies of three different sampling schemes. The fact that the empirical distributions well approximate the corresponding asymptotic distributions is checked. Observable differences in parameter estimates between data generated under different hypotheses are also studied. Finally, different models intended to shed light on various aspects of the data and/or the experiment procedure, are applied to three real olfactometer datasets.

EPFL2008

Statistical Applications of Random Matrix Theory: Comparison of Two Populations

Rémy Mariétan

EPFL2019