## The Annals of Statistics

- Ann. Statist.
- Volume 14, Number 2 (1986), 733-742.

### On Optimal Decision Rules for Signs of Parameters

Yosef Hochberg and Marc E. Posner

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 14, Number 2 (1986), 733-742.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349950

**Digital Object Identifier**

doi:10.1214/aos/1176349950

**Mathematical Reviews number (MathSciNet)**

MR840526

**Zentralblatt MATH identifier**

0619.62065

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J15: Paired and multiple comparisons

**Keywords**

Three decision rule locally optimal generalized optimization functions

#### Citation

Hochberg, Yosef; Posner, Marc E. On Optimal Decision Rules for Signs of Parameters. Ann. Statist. 14 (1986), no. 2, 733--742. doi:10.1214/aos/1176349950. https://projecteuclid.org/euclid.aos/1176349950