Statistics Papers

Document Type

Journal Article

Date of this Version

1983

Publication Source

The Annals of Statistics

Volume

11

Issue

2

Start Page

640

Last Page

653

DOI

10.1214/aos/1176346169

Abstract

Consider the problem of sequentially testing the hypothesis that the mean of a normal distribution of known variance is less than or equal to a given value versus the alternative that it is greater than the given value. Impose the linear combination loss function under which the risk becomes a constant c, times the expected sample size, plus the probability of error. It is known that all admissible tests must be monotone--that is, they stop and accept if Sn, the sample sum at stage n, satisfies Snan; stop and reject if Snbn. In this paper we show that any admissible test must in addition satisfy bnan≤2b¯(c). The bound 2b¯(c) is sharp in the sense that the test with stopping bounds an≡−b¯(c),bnb¯(c) is admissible.

As a consequence of the above necessary condition for admissibility of a sequential test, it is possible to characterize all sequential probability ratio tests (SPRT's) regarding admissibility. In other words necessary and sufficient conditions for the admissibility of an SPRT are given. Furthermore, an explicit numerical upper bound for b¯(c) is provided.

Keywords

sequential tests, SPRT, admissibility, Wiener process

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Date Posted: 27 November 2017

This document has been peer reviewed.