Asymptotic Distribution of Some Test Criteria for the Mean Vector with Fewer Observations than the Dimension
Shota Katayama, Yutaka Kano and Muni S. Srivastava
Abstract
The problem of hypotheses 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.
Paper
http://dx.doi.org/10.1016/j.jmva.2013.01.008
