Open AccessSome tests criteria for the covariance matrix with fewer observations than the dimension
Author(s) -
M. S. Srivastava
Publication year - 2006
Publication title -
acta et commentationes universitatis tartuensis de mathematica./acta et commentationes universitatis tartuensis de mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.276
H-Index - 6
eISSN - 2228-4699
pISSN - 1406-2283
DOI - 10.12697/acutm.2006.10.07
Subject(s) - mathematics , combinatorics , covariance matrix , zero (linguistics) , identity matrix , matrix (chemical analysis) , dimension (graph theory) , exact test , multivariate random variable , statistics , covariance , discrete mathematics , random variable , eigenvalues and eigenvectors , philosophy , linguistics , physics , materials science , quantum mechanics , composite material
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, 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.