# Some tests criteria for the covariance matrix with fewer observations than the dimension

### Abstract

We consider testing certain hypotheses concerning the covariance matrix Σ when the number of observations *N*=*n*+1 on the *p*-dimensional random vector *x*, distributed as normal, is less than *p*, *n*<*p*, and *n*/*p* goes to zero. Specifically, we consider testing Σ=σ^{2}*I _{p}*, Σ=

*I*, Σ=Λ, a diagonal matrix, and Σ=σ

_{p}^{2}[(1−ρ)

*I*+ρ1

_{p}*1′*

_{p}*], an intraclass correlation structure, where 1′*

_{p}*=(1,1,…,1), is a*

_{p}*p*-row vector of ones, and

*I*is the

_{p}*p*×

*p*identity matrix. The first two tests are the adapted versions of the likelihood ratio tests when

*n*>

*p*,

*p*-fixed, and

*p*/

*n*goes to zero, to the case when

*n*<

*p*,

*n*-fixed, and

*n*/

*p*goes to zero. The third test is the normalized version of Fisher’s

*z*-transformation which is shown to be asymptotically normally distributed as

*n*and

*p*go to infinity (irrespective of the manner). A test for the fourth hypothesis is constructed using the spherecity test for a (

*p*−1)-dimensional vector but this test can only reject the hypothesis, that is, if the hypothesis is not rejected, it may not imply that the hypothesis is true. The first three tests are compared with some recently proposed tests.