Publication Date

9-13-1976

Abstract

A prediction interval procedure uses information contained in a sample together with other relevant information to produce an interval which will contain the next observation to be taken from the population with confidence no less than a pre-determined /J. One prediction inter­val procedure renders another "inadmissible" at the fJ level if (1) both procedures maintain the B confidence level and (2) the former produces intervals which are at least as "short" as, and sometimes "shorter" than, those produced by the latter; "short" is defined in a natural fashion in terms of expected length or expected upper (lower) limit. as is appropriate for two-or one-sided prediction intervals.

In the Empirical Bayes setting we assume that the present normal population is embedded in a sequence of such populations with common variance o2 and with means µ. being randomly drawn from the “prior” distribution N(lambda,tau2). Prediction interval procedures are derived which render the usual procedure (which ignores prior information) inadmissible in these cases: (1)lambda,tau2 and sigma2 are known (this being the “Bayes” procedure); (2)tau2 and sigma2 are known and at least one prior sample is available; (3)c2+sigma2/tau2 is known and at least one prior sample is available; (4) c2 is known to be bounded below by c02 and at least one prior sample is available. The procedures given for cases (2) and (3) are shown to be Empirical Bayes (in a natural sense); that in case (4) is shown to be asymptotic G-mini-max.

Numerical studies of the magnitude of the improvement offered by the new procedures over the old are presented. We conclude that c2 (or its lower bound) must be somewhat larger than 1 for the new procedures to be practically significant. (When sigma2 is unknown, of course, the sj2 can aid considerable in estimating sigma2 even when c2 is completely unknown, or prior distribution has unknown form.)

Finally. The new procedures are adapted to three other common problems: (1) predicting the mean for m future observations; (2) producing a confidence interval for the mean of the present population; and (3) producing an interval which will contain all of the next m observations with confidence B.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

William John Zimmer

Second Committee Member

Herbert Thaddeus Davis III

Third Committee Member

Steven Arthur Pruess

Language

English

Document Type

Dissertation

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