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March, 1964 Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests
R. H. Farrell
Ann. Math. Statist. 35(1): 36-72 (March, 1964). DOI: 10.1214/aoms/1177703731

Abstract

Let R be the set of real numbers, B1 the set of Borel sets of R, and μ a σ-finite nonnegative measure on B1. Let Ω be an open real number interval (which may be infinite). Throughout we consider a Koopman-Darmois family (1){h(θ)exp(θx),θεΩ} of generalized probability density functions on the measure space (R,B1,μ). We consider one sided tests T of the hypothesis θ<0 against the alternative θ>0. In general, in this paper, T will be a sequential procedure. Associated with T is a stopping variable N (mention of the dependence of N on T is usually omitted). N0.N=n means that sampling stopped after n observations and a decision was made. In this context we consider to be an integer, and N= means that sampling does not stop. In the discussion of Section 1 we will assume that if θεΩ and θ0 then Pθ(N<)=1, that is, sampling stops with probability one. The reason for the exclusion of θ=0 will become apparent in Section 1. We will be concerned with two events, decide θ<0, and, decide θ>0. The main result of this paper may be stated as follows. Theorem 1. Suppose (R,B1,μ),Ω, and {h(θ)exp(θx),θεΩ} are as described above. Define μθ=h(θ)xexp(θx)μ(dx),(2)σ2=h(0)x2μ(dx), and assume μ0=0. Suppose 0<α<1 and 0<β<1 and (3)supθ>0Pθ( decide θ<0)β;supθ<0Pθ( decide θ>0)α. Then limsupθ0+μθ2|log|logμθ|1EθN2σ2P0(N=);(4)limsupθ0μθ2|log|logμθ|1EθN2σ2P0(N=). If α+β<1 there is a generalized sequential probability ratio test T with stopping variable N such that for the test T, (5)P0(N=)=1(α+β);(3) holds ; for the test T, (6)limθ0μθ2|log|logμθ|1EθN=2σ2P0(N=). For all tests T, if P0(N=)>0 then limθ0θ2EθN=. In Section 1, (7) and (8), it is shown that P0(N=)1αβ. Consequently the relations (4) and (5) of Theorem 1 are not vacuous. We were led to formulate Theorem 1 by a problem of constructing bounded length confidence intervals. The relationship is explained in Section 2. The proof of Theorem 1 is given in Section 3.

Citation

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R. H. Farrell. "Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests." Ann. Math. Statist. 35 (1) 36 - 72, March, 1964. https://doi.org/10.1214/aoms/1177703731

Information

Published: March, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0156.39306
MathSciNet: MR157459
Digital Object Identifier: 10.1214/aoms/1177703731

Rights: Copyright © 1964 Institute of Mathematical Statistics

Vol.35 • No. 1 • March, 1964
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