Simultaneous Confidence Interval Estimation

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 - (0 < < 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.