On the sphericity test with large-dimensional observations

Qinwen Wang,Jianfeng Yao

doi:10.1214/13-ejs842

Abstract

In this paper, we propose corrections to the likelihood ratio test and John’s test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with a comparison to several existing methods from the literature. Moreover, we also obtain their asymptotic power functions under the alternative of a spiked population model as a specific alternative.

Highlights

Consider a sample Y1, . . . , Yn from a p-dimensional multivariate distribution with covariance matrix Σp
We present the trends of β1(α) and β2(α) in Figure 3 when only one spike a = 2.5 exists
Using recent central limit theorems for eigenvalues of large sample covariance matrices, we are able to find new asymptotic distributions for two major procedures to test the sphericity of a large-dimensional distribution

Summary

Introduction

Consider a sample Y1, . . . , Yn from a p-dimensional multivariate distribution with covariance matrix Σp. If we let n → ∞ while keeping p fixed, classical asymptotic theory indicates that under the null hypothesis, −2 log Ln =⇒ χ2f , a chi-square distribution with degree of freedom f This asymptotic distribution is further refined by the following Box-Bartlett correction (referred as BBLRT):. Following the idea of the Box-Bartlett correction, [19] established an expansion for the distribution function of the statistics T2 (referred as Nagao’s test),. Following the idea of [15], [9] proposed to use a family of well selected U-statistics to test the sphericity; as showed in our simulation study, the powers of our corrected John’s test are slightly higher than this test in most cases.

Large-dimensional corrections

Monte Carlo study

Asymptotic powers: under the spiked population alternative

Generalization to the case when the population mean is unknown

Additional proofs

Concluding remarks