Distribución del cuadrado de la máxima correlación canónica para tamaños muestrales pequeños

Resumo

. Assume that the normally distributed random vector X of d components is partitioned into two subvectors X~'> and X(» of and q components ¡3 respectively. Suppose also that the two subvectors are not correlated. In this work wc study thc distribution of the largcst squared canonical correlation r12 whcn p, q and the number of observations in the sample N, are rathcr small. We give the cxplicit expressions of the cumulative distribution functions and the computed values of thc sample mean and variance of r1>. We prove that ihere cxists a stochastic order betwcen the largest squared canonical correlations obtained from two diffcrent partitions of the vector X. Morc precisely, r1> mercase stochastically when ihe diflerence between p and q decrease. Since X(1> and X~>~ are uncorrelated the largest squared canonical correlation in the population X12 is ¡ero. Therefore dic mean of r¡2 is the Ñas of r1> when r1> is used to estimate Xi>. Tbc values of the mean and the variance show that the square of thc Ñas is bigger than thc variance in ah the cases. 1.