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Likelihood Ratio Tests for Sequential k-Decision Problems

Gary Lorden
The Annals of Mathematical Statistics
Vol. 43, No. 5 (Oct., 1972), pp. 1412-1427
Stable URL: http://www.jstor.org/stable/2240064
Page Count: 16
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Likelihood Ratio Tests for Sequential k-Decision Problems
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Abstract

Sequential tests of separated hypotheses concerning the parameter θ of a Koopman-Darmois family are studied from the point of view of minimizing expected sample sizes pointwise in θ subject to error probability bounds. Sequential versions of the (generalized) likelihood ratio test are shown to exceed the minimum expected sample sizes by at most $M \log \log \underline{\alpha}^{-1}$ uniformly in θ, where $\underline{\alpha}$ is the smallest error probability bound. The proof considers the likelihood ratio tests as ensembles of sequential probability ratio tests and compares them with alternative procedures by constructing alternative ensembles, applying a simple inequality of Wald and a new inequality of similar type. A heuristic approximation is given for the error probabilities of likelihood ratio tests, which provides an upper bound in the case of a normal mean.

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