Selection of Largest Multiple Correlation Coefficients: Exact Sample Size Case

Abstract

Consider $k(\geqq 2)$ independent $p$-variate $(p \geqq 2)$ normal distributions $N(\mathbf{\mu}_i, \Sigma_i), i = 1,2, \cdots, k$, where the mean vectors $\mathbf{\mu}_i$ and the covariance matrices $\Sigma_i$ are all unknown. Let $\theta_i$ denote for the $i$th distribution the squared population multiple correlation coefficient between the first variate and the set of $(p - 1)$ variates remaining. A procedure based on the natural ordering of the $k$ sample squared multiple correlation coefficients, each computed from a random sample of size $n(\geqq p + 2)$, is considered for the problem of selection of the $t(< k)$ largest $\theta_i$'s. Given $(1 - \theta_{\lbrack k - t\rbrack}) \geqq \delta(1 - \theta_{\lbrack k - t + 1\rbrack})$ and $\theta_{\lbrack k - t + 1\rbrack} \geqq \gamma\theta_{\lbrack k - t\rbrack}$, where $\theta_{\lbrack i\rbrack}$ denotes the $i$th smallest $\theta$ and $\delta > 1$ and $\gamma > 1$ are preassigned constants, it is shown that the probability of a correct selection is minimized for $\theta_{\lbrack i\rbrack} = (\delta - 1)/(\delta\gamma - 1), i = 1, \cdots, k - t$ and $\theta_{\lbrack i\rbrack} = \gamma(\delta - 1)/(\delta\gamma - 1), i = k - t + 1, \cdots, k$. For a given $P^\ast (< 1)$, the exact common sample size $n$ is then determined so that the infimum of the probability of a correct selection is not smaller than $P^\ast$. For $p = 2$, the problem reduces to selecting $t$ largest correlation coefficients from the $k$ bivariate normal distributions.