Full paper in PDF:
$% M. R. Santillán and A. J. Viollaz, Distribución del cuadrado de la máxima correlación canónica para tamaños muestrales pequeños, Rev. Mat. Univ. Complut. Madrid 2 (1989), no. 2, 3, 203–226.%$

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

María R. SANTILLÁN and Aldo J. VIOLLAZ
 
ABSTRACT

Assume that the normally distributed random vector X of d  components is partitioned into two subvectors X(1)  and X(2)  of p  and q  components respectively. Suppose also that the two subvectors are not correlated. In this work we study the distribution of the largest squared canonical correlation  2
r1  when p  , q  and the number of observations in the sample N  , are rather small. We give the explicit expressions of the cumulative distribution functions and the computed values of the sample mean and variance of 2
r1  . We prove that there exists a stochastic order between the largest squared canonical correlations obtained from two different partitions of the vector X  . More precisely,  2
r1  increase stochastically when the difference between p  and q  decrease. Since X(1)  and X(2)  are uncorrelated the largest squared canonical correlation in the population c21  is zero. Therefore the mean of r21  is the bias of r21  when r21  is used to estimate c21  . The values of the mean and the variance show that the square of the bias is bigger than the variance in all the cases.

1980 Mathematics Subject Classification (1985 revision): 46E35