Simultaneous Confidence Interval Estimation

Abstract

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.