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June, 1986 On Optimal Decision Rules for Signs of Parameters
Yosef Hochberg, Marc E. Posner
Ann. Statist. 14(2): 733-742 (June, 1986). DOI: 10.1214/aos/1176349950


The problem of deciding the signs of $k$ parameters $(\theta_1, \cdots, \theta_k) \equiv \mathbf{theta}$ based on $(\hat{\theta}_1, \cdots, \hat{\theta}_k) \sim N(\mathbf{\theta,\Sigma})$ such that $p_\mathbf{\theta} \{$\text{any error$\} \leq \alpha \forall \mathbf{\theta}$ is discussed by Bohrer and Schervish (1980). They characterize a desirable class of procedures called locally optimal. For the case $k = 2, \mathbf{\Sigma = I}$, and $\alpha \leq \frac{1}{3}$, they present a particular rule from this class called the double cross. In this paper, we address the problem of selecting a best rule from among all locally optimal rules when $k = 2$ and $\mathbf{\Sigma = I}$. When $\alpha \leq \frac{1}{3}$, the double cross is shown to be an attractive choice. Other rules are obtained for higher values of $\alpha$. We also examine a more general optimization criterion than the one used by Bohrer and Schervish and obtain different optimal rules for several classes of problems. The optimal rule corresponding to one of these classes has no two-decision region. A modification of the formulation is offered under which a well-known rule (with two decision regions) emerges as the unique optimal procedure.


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Yosef Hochberg. Marc E. Posner. "On Optimal Decision Rules for Signs of Parameters." Ann. Statist. 14 (2) 733 - 742, June, 1986.


Published: June, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0619.62065
MathSciNet: MR840526
Digital Object Identifier: 10.1214/aos/1176349950

Primary: 62J15

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 2 • June, 1986
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