The Annals of Statistics

Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure

Arthur Cohen and Harold B. Sackrowitz

Full-text: Open access

Abstract

The problem of multiple endpoint testing for k endpoints is treated as a 2k finite action problem. The loss function chosen is a vector loss function consisting of two components. The two components lead to a vector risk. One component of the vector risk is the false rejection rate (FRR), that is, the expected number of false rejections. The other component is the false acceptance rate (FAR), that is, the expected number of acceptances for which the corresponding null hypothesis is false. This loss function is more stringent than the positive linear combination loss function of Lehmann [Ann. Math. Statist. 28 (1957) 1–25] and Cohen and Sackrowitz [Ann. Statist. (2005) 33 126–144] in the sense that the class of admissible rules is larger for this vector risk formulation than for the linear combination risk function. In other words, fewer procedures are inadmissible for the vector risk formulation. The statistical model assumed is that the vector of variables Z is multivariate normal with mean vector μ and known intraclass covariance matrix Σ. The endpoint hypotheses are Hii=0 vs Kii>0, i=1,…,k. A characterization of all symmetric Bayes procedures and their limits is obtained. The characterization leads to a complete class theorem. The complete class theorem is used to provide a useful necessary condition for admissibility of a procedure. The main result is that the step-up multiple endpoint procedure is shown to be inadmissible.

Article information

Source
Ann. Statist., Volume 33, Number 1 (2005), 145-158.

Dates
First available in Project Euclid: 8 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1112967702

Digital Object Identifier
doi:10.1214/009053604000000986

Mathematical Reviews number (MathSciNet)
MR2157799

Zentralblatt MATH identifier
1066.62010

Subjects
Primary: 62C10: Bayesian problems; characterization of Bayes procedures 62C15: Admissibility

Keywords
Step-up procedure intraclass correlation Bayes procedures inadmissibility finite action problem Schur convexity complete class

Citation

Cohen, Arthur; Sackrowitz, Harold B. Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure. Ann. Statist. 33 (2005), no. 1, 145--158. doi:10.1214/009053604000000986. https://projecteuclid.org/euclid.aos/1112967702


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