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December, 1953 Simultaneous Confidence Interval Estimation
S. N. Roy, R. C. Bose
Ann. Math. Statist. 24(4): 513-536 (December, 1953). DOI: 10.1214/aoms/1177728912


The work of Neyman on confidence limits and of Fisher on fiducial limits is well known. However, in most applications the interval or limits for only a single parameter or a single function of the parameters has been considered. Recently Scheffe [2] and Tukey [3] have considered special cases of what may be called problems of simultaneous estimation, in which one is interested in giving confidence intervals for a finite or infinite set of parametric functions such that the probability of the parametric functions of the set being simultaneously covered by the corresponding intervals is a preassigned number $1 - \alpha(0 < \alpha < 1).$ In this paper we discuss in Section 1, a set of sufficient conditions under which such simultaneous estimation is possible, and bring out the connection of this with a method of test construction considered by one of the authors in a previous paper [1]. In Section 2 some univariate examples (including the ones due to Scheffe and Tukey) are considered from this point of view. Sections 3 to 6 are concerned with multivariate applications, giving results which are believed to be new. The associated tests all turn out to be the same as in [1] except for the example in Section 4.3 which, in a sense, is a multivariate generalization of Tukey's example (Section 2.2). Section 3 gives the notation and preliminaries for multivariate applications. Section 4 gives confidence bounds on linear functions of means for multivariate normal populations. Sections 5 and 6 give respectively confidence bounds on certain functions of the elements of population covariance matrices and population canonical regressions, from which a chain of simpler consequences would follow by the application of a set of matrix theorems. This has been partly indicated in the present paper and will be more fully discussed in a later paper.


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S. N. Roy. R. C. Bose. "Simultaneous Confidence Interval Estimation." Ann. Math. Statist. 24 (4) 513 - 536, December, 1953.


Published: December, 1953
First available in Project Euclid: 28 April 2007

zbMATH: 0052.15403
MathSciNet: MR60781
Digital Object Identifier: 10.1214/aoms/1177728912

Rights: Copyright © 1953 Institute of Mathematical Statistics

Vol.24 • No. 4 • December, 1953
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