Some Optimum Confidence Bounds for Roots of Determinantal Equations

Abstract

A problem considered in this paper is to obtain confidence intervals for all characteristic roots of a population covariance matrix $\mathbf{\Sigma}$ in the form $\lbrack ch_m(\mathbf{S})/u, ch_M(\mathbf{S})/l\rbrack$, where $ch_m(\mathbf{S})$ and $ch_M(\mathbf{S})$ are the minimum and maximum characteristic roots, respectively, of a sample covariance matrix $\mathbf{S}$ from a multivariate normal population and $u$ and $l$ are constants. Intervals of this form having probability at least $1 - \epsilon$ can be obtained by basing $u$ and $l$ on certain $\chi^2$-distributions. Among all intervals in a certain class such intervals are shortest. Another problem treated is to obtain confidence intervals for all characteristic roots $ch(\mathbf{\Sigma}_1\mathbf{\Sigma}_2^{-1})$ in the form $\lbrack ch_m(\mathbf{S}_1\mathbf{S}_2^{-1})/U, ch_M(\mathbf{S}_1\mathbf{S}_2^{-1})/L\rbrack$, where $\mathbf{\Sigma}_1$ and $\mathbf{\Sigma}_2$ and $\mathbf{S}_1$ and $\mathbf{S}_2$ are population and sample covariance matrices of two multivariate normal populations, respectively, and $U$ and $L$ are constants, determined from $F$-distributions to give confidence at least $1 - \epsilon$. Such choices of the constants yield shortest intervals within a certain class. Comparison is made with other methods of finding such intervals. Various uses of the intervals are suggested, such as simultaneous intervals for variances and correlation coefficients. Some other confidence intervals for related problems are considered.