New estimator for functions of the canonical correlation coefficients

Let ρ1,…,ρp be the population canonical correlation coefficients from a normal distribution. This paper considers the estimation of δ1,…,δp, where δi=ρi2/(1−ρi2),i=1,…,p, in a decision theoretic way. Since the distribution of δi's is complicated, two-staged estimation has been a usual method so far; i.e., first find a good estimator of a matrix whose eigenvalues are the δi's, then use its eigenvalues as the estimators of δi's. In this paper we directly estimate δi's and evaluate the estimators with respect to a quadratic loss function. We propose a new class of estimators and prove its dominance over the usual estimator.