Results 1 
6 of
6
Central Limit Theorems for Classical Likelihood Ratio Tests for HighDimensional Normal Distributions
"... For random samples of size n obtained from pvariate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the highdimensional setting. These test statistics have been extensively studied in multivariate analysis and their limiting d ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
For random samples of size n obtained from pvariate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the highdimensional setting. These test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chisquare distributions as n goes to infinity and p remains fixed. In this paper, we consider the highdimensional case where both p and n go to infinity with p/n → y ∈ (0, 1]. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances. We perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing highdimensional data.
OPTIMAL RANKBASED TESTS FOR HOMOGENEITY OF SCATTER
, 806
"... We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly r ..."
Abstract

Cited by 8 (8 self)
 Add to MetaCart
(Show Context)
We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavytailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normalscore version outperforms traditional Gaussian likelihood ratio tests and their pseudoGaussian robustifications under a very broad range of nonGaussian densities including, for instance, all multivariate Student and powerexponential distributions. 1. Introduction. 1.1. Homogeneity of variances and covariance matrices. The assumption of variance homogeneity is central to the theory and practice of univariate
Optimal tests for homogeneity of covariance, scale, and shape
 J. Multivariate Anal
, 2008
"... The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic cont ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic context of multivariate Gaussian models. And the modified likelihood ratio tests (MLRT) that are used in everyday practice are known to be highly sensitive to violations of Gaussian assumptions. In this paper, we provide a complete and systematic study of the problem, and propose test statistics which, while preserving the optimality features of the MLRT under multinormal assumptions, remain valid under unspecified elliptical densities with finite fourthorder moments. As a first step, the Le Cam LAN approach is used for deriving locally and asymptotically optimal testing procedures φ (n) f for any specified mtuple of radial densities f = (f1,..., fm). Combined with an estimation of the m densities f1,..., fm, these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudoGaussian tests—namely, tests that are locally and asymptotically optimal at Gaussian densities while remaining valid under a much broader class of distributions—in general are preferable. We therefore construct two pseudoGaussian modifications of the Gaussian version φ (n) N of the optimal test φ (n) f. The first one, φ
Likelihood Ratio Tests for HighDimensional Normal Distributions
"... In the paper by Jiang and Yang (2013), six classical Likelihood Ratio Test (LRT) statistics are studied under highdimensional settings. Assuming that a random sample of size n is observed from a pdimensional normal population, they derive the central limit theorems (CLTs) when p/n → y ∈ (0, 1], wh ..."
Abstract
 Add to MetaCart
In the paper by Jiang and Yang (2013), six classical Likelihood Ratio Test (LRT) statistics are studied under highdimensional settings. Assuming that a random sample of size n is observed from a pdimensional normal population, they derive the central limit theorems (CLTs) when p/n → y ∈ (0, 1], which are different from the classical chisquare limits as n goes to infinity while p remains fixed. In this paper, by developing a new tool, we prove that the above six CLTs hold in a more applicable setting: p goes to infinity and p < n − c for some 1 ≤ c ≤ 4. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chisquare approximations and discussions are presented afterwards.
The Annals of Statistics OPTIMAL RANKBASED TESTS FOR HOMOGENEITY OF SCATTER
"... We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs, for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectl ..."
Abstract
 Add to MetaCart
(Show Context)
We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs, for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavytailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normalscore version outperforms traditional Gaussian likelihood ratio tests and their pseudoGaussian robustifications under a very broad range of nonGaussian densities including, for instance, all multivariate Student and powerexponential distributions. ∗The authors are also members of ECORE, the recently created association between CORE and ECARES.
BY
, 2011
"... extremely interested in probability theory and the challenges and opportunities that this topic presents in statistics, and many other areas of research. During the time that I have been constructing my thesis, I have had the chance to reconsider, revise, and refine many of the original concepts and ..."
Abstract
 Add to MetaCart
(Show Context)
extremely interested in probability theory and the challenges and opportunities that this topic presents in statistics, and many other areas of research. During the time that I have been constructing my thesis, I have had the chance to reconsider, revise, and refine many of the original concepts and methodologies foundational to my research. I have also had the opportunity to explore some of these concepts in applied settings, which has helped me to integrate my theoretical exploration in ways that have proven useful and practical. The combined research and writing process has been a rich and enlightening one for me, and over time, I have grown and evolved both in my thinking and in my approach to research. As every doctoral student can attest, completion of a dissertation cannot be accomplished in isolation. It involves the support and contribution of many people, both inside and outside the department. I would first like to thank my advisor, Professor Tiefeng Jiang. I am deeply indebted to him for his support and commitment, his enthusiastic guidance, and his insight throughout the research and writing process. He could not have been more generous with his time and effort in directing me toward the