Estimation of dispersion parameters

This paper d eals with a topi c in multivariate a nalys is. Consider that a sample of size n+ 1 has been collected from a p-variate normal distribution having dispersion matrix (CT ii'). Let a;r/n denote the usual unbiased estimate of CTW. Further, let O< l<u be constants such that all cha racteristic roots of a matrix having t he Wishart distribution lie in t he interval [I, u] with probability I -a . A t heorem of Roy, Bose, and Gnan adesikan [A nn . Math. Stat. 24. 5 13-536 (1953); Biometrika 44, 399-410 (1957)] may be stated as follows: The probability is 1 -a t hat every principal minor determinant of I-I (aii') -(CT jj') and of (CTii,)-u-l(a jj') is nonnegative. The previous result may be used to prove t he main theore m of the p resent paper. Theorem: T he probability is at least 1 -a t hat t he following system of r e lat ions hold simultaneo us ly : u-lai i ~CT 'i ~I-la ii; j = l , ... , p a nd ICTii,-}f (11-1 + 1-1 ) 0 ii1 ~7W -I -1,-1) ( a iia i';') ! ~' j r= j'.