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**Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure.**
*(English)*
Zbl 1066.62010

Summary: The problem of multiple endpoint testing for \(k\) endpoints is treated as a \(2^k\) 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 E. L. Lehmann [Ann. Moth. Stat. 28, 1–25 (1957; Zbl 0078.33402)] and the authors [see the preceding entry, ibid., 126–144 (2005; Zbl 1066.62008)] in the sense that the class of admissible roles 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 \({\mathbf Z}\) is multivariate normal with mean vector \(\mu\) and known intraclass covariance matrix \(\Sigma\). The endpoint hypotheses are \(H_i:\mu_i=0\) vs \(K_i:\mu_i> 0\), \(i=1,\dots,k\). 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.

The statistical model assumed is that the vector of variables \({\mathbf Z}\) is multivariate normal with mean vector \(\mu\) and known intraclass covariance matrix \(\Sigma\). The endpoint hypotheses are \(H_i:\mu_i=0\) vs \(K_i:\mu_i> 0\), \(i=1,\dots,k\). 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.

### MSC:

62C10 | Bayesian problems; characterization of Bayes procedures |

62C15 | Admissibility in statistical decision theory |

62C07 | Complete class results in statistical decision theory |

### Keywords:

step-up procedure; intraclass correlation; inadmissibility; finite action problem; Schur convexity
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\textit{A. Cohen} and \textit{H. B. Sackrowitz}, Ann. Stat. 33, No. 1, 145--158 (2005; Zbl 1066.62010)

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