DeepAI AI Chat
Log In Sign Up

Limiting distributions of the likelihood ratio test statistics for independence of normal random vectors

by   Mingyue Hu, et al.

Consider the likelihood ratio test (LRT) statistics for the independence of sub-vectors from a p-variate normal random vector. We are devoted to deriving the limiting distributions of the LRT statistics based on a random sample of size n. It is well known that the limit is chi-square distribution when the dimension of the data or the number of the parameters are fixed. In a recent work by Qi, Wang and Zhang (Ann Inst Stat Math (2019) 71: 911–946), it was shown that the LRT statistics are asymptotically normal under condition that the lengths of the normal random sub-vectors are relatively balanced if the dimension p goes to infinity with the sample size n. In this paper, we investigate the limiting distributions of the LRT statistic under general conditions. We find out all types of limiting distributions and obtain the necessary and sufficient conditions for the LRT statistic to converge to a normal distribution when p goes to infinity. We also investigate the limiting distribution of the adjusted LRT test statistic proposed in Qi, Wang and Zhang (2019). Moreover, we present simulation results to compare the performance of classical chi-square approximation, normal and non-normal approximation to the LRT statistics, chi-square approximation to the adjusted test statistic, and some other test statistics.


page 9

page 10

page 11

page 12


Asymptotic Distributions for Likelihood Ratio Tests for the Equality of Covariance Matrices

Consider k independent random samples from p-dimensional multivariate no...

Empirical likelihood method for complete independence test on high dimensional data

Given a random sample of size n from a p dimensional random vector, wher...

Likelihood ratio tests for many groups in high dimensions

In this paper we investigate the asymptotic distribution of likelihood r...

Asymptotics for Markov chain mixture detection

Sufficient conditions are provided under which the log-likelihood ratio ...

On new generalized differentials with respect to a set and their applications

The notions and certain fundamental characteristics of the proximal and ...

Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble

Consider the product of m independent n× n random matrices from the sphe...

Chi-Square Goodness-of-Fit Tests for Conditional Distributions

We propose a cross-classification rule for the dependent and explanatory...