The Annals of Mathematical Statistics

Some Optimum Confidence Bounds for Roots of Determinantal Equations

T. W. Anderson

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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.

Article information

Source
Ann. Math. Statist., Volume 36, Number 2 (1965), 468-488.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177700158

Digital Object Identifier
doi:10.1214/aoms/1177700158

Mathematical Reviews number (MathSciNet)
MR172397

Zentralblatt MATH identifier
0223.62045

JSTOR
links.jstor.org

Citation

Anderson, T. W. Some Optimum Confidence Bounds for Roots of Determinantal Equations. Ann. Math. Statist. 36 (1965), no. 2, 468--488. doi:10.1214/aoms/1177700158. https://projecteuclid.org/euclid.aoms/1177700158


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