Some Optimum Confidence Bounds for Roots of Determinantal Equations

A problem considered in this paper is to obtain confidence intervals for all characteristic roots of a population covariance matrix Σ in the form chₘ(S)/u, chM(S)/l, where chₘ(S) and chM(S) are the minimum and maximum characteristic roots, respectively, of a sample covariance matrix S from a multivariate normal population and u and l are constants. Intervals of this form having probability at least 1 - ε can be obtained by basing u and l on certain χ²-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(Σ₁Σ₂⁻¹) in the form chₘ(S₁S₂⁻¹)/U, chM(S₁S₂⁻¹)/L, where Σ₁ and Σ₂ and S₁ and S₂ 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 - ε. 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.