Publication Date

5-5-1967

Abstract

A sequential test of a statistical hypothesis H0 versus H1 is said to be a running test if there is a positive probability that the test will not stop if H0 is true. Tests of this nature were introduced for testing the Bernoulli case by D. A. Darling and Herbert Robbins [1]; an earlier paper of Roger Farrell [2] deals implicitly with the asymptotic expected sample size of such tests for testing the hypothesis Ѳ = 0 in the parameterized family of generalized density functions h(Ѳ)eϴxdµ. Herbert Robbins, in a lecture given at the Sandia Corporation, Albuquerque, New Mexico, observed that running tests have application to process control. Let < Xi >i=1∞ be independent replicas of a random variable X occurring as the by-product of a process and suppose as long as the distribution function of X satisfies some restriction it is desired to have the process continue. H0 would be the assertion that the distribution function of X was in the family of distribution functions satisfying the restrictions and H1 would contain all the undesirable possibilities for the distribution function of X. A running test T which stops with small probability under H0 but stops the process with high probability under H1 would be of value in regulating the process. Running tests can be constructed to have power one against the most varied hypotheses. In this paper a running test for symmetry is constructed which has power one against any non-symmetric alternative at arbitrary level of significance. Power is thus a poor criterion to use in designing running tests; a more natural criterion is the expected number of observations required for the test to terminate under H1. If a test could be found which had the smallest expected number of observations against all of the simple hypotheses which make up H1, it would be the best test. However this is not possible in general as H1 contains too many possible distributions and the variety of possible sequential tests is too large. It is always possible in the case H0 is all continuous symmetric distributions and H1 is all continuous non-symmetric distributions to take a given test at level of significance α and to construct a new level α test with smaller expected sample size against some of the alternatives in H1. However it may prove impossible to improve the order of magnitude of the expected sample size for some measure of small deviations of the hypotheses in H1 from H0. If one can define a measure of deviation from symmetry for which small values of the measure represent small deviations from symmetry and if it can be shown that the order of magnitude of the expected sample size is a minimum for a given running test, it is reasonable to call the test a best running test for symmetry.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Lambert Herman Koopmans

Second Committee Member

Illegible

Third Committee Member

Bernard Epstein

Fourth Committee Member

Richard Clyde Metzler

Project Sponsors

The National Science Foundation Grant GP 2558, N.A.S.A. Grant 290-397-2

Language

English

Document Type

Dissertation

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