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Likelihood Ratio Tests for High‐Dimensional Normal Distributions
Author(s) -
Jiang Tiefeng,
Qi Yongcheng
Publication year - 2015
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12147
Subject(s) - mathematics , infinity , statistics , limit (mathematics) , square (algebra) , chi square test , central limit theorem , sample size determination , likelihood ratio test , combinatorics , mathematical analysis , geometry
In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high‐dimensional setting. Assuming that a random sample of size n is observed from a p ‐dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi‐square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n . This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi‐square approximations and discussions are presented afterwards.