Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension

The problem of hypothesis testing concerning the mean vector for high dimensional data has been investigated by many authors. They have proposed several test criteria and obtained their asymptotic distributions, under somewhat restrictive conditions, when both the sample size and the dimension tend to infinity. Indeed, the conditions used by these authors exclude a typical situation where the population covariance matrix has spiked eigenvalues, as for instance, the population covariance matrix with the compound symmetry structure (the variances are the same; the covariances are the same). In this paper, we relax their conditions to include such important cases, obtaining rather non-standard asymptotic distributions which are the convolution of normal and chi-squared distributions for the population covariance matrix with moderate spiked eigenvalues, and obtaining the asymptotic distributions in the form of convolutions of chi-square distributions for the population covariance matrix with quite spiked eigenvalues.