## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 36, Number 6 (1965), 1743-1752.

### Rate of Convergence in the Compound Decision Problem for Two Completely Specified Distributions

J. F. Hannan and J. R. Van Ryzin

#### Abstract

Simultaneous consideration of $n$ statistical decision problems having identical generic structure constitutes a compound decision problem. The risk of a compound decision problem is defined as the average risk of the component problems. When the component decisions are between two fully specified distributions $P_0$ and $P_1, P_0 \neq P_1$, Hannan and Robbins [2] give a decision function whose risk is uniformly close (for $n$ large) to the risk of the best "simple" procedure based on knowing the proportion of component problems in which $P_1$ is the governing distribution. This result was motivated by heuristic arguments and an example (component decisions between $N(-1, 1)$ and $N(1, 1))$ given by Robbins [4]. In both papers, the decision functions for the component problems depended on data from all $n$ problems. The present paper considers, as in Hannan and Robbins [2], compound decision problems in which the component decisions are between two distinct completely specified distributions. The decision functions considered are those of [2]. The improvement is in the sense that a convergence order of the bound is obtained in Theorem 1. Higher order bounds are attained in Theorems 2 and 3 under certain continuity assumptions on the induced distribution of a suitably chosen function of the likelihood ratio of the two distributions.

#### Article information

**Source**

Ann. Math. Statist., Volume 36, Number 6 (1965), 1743-1752.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177699802

**Digital Object Identifier**

doi:10.1214/aoms/1177699802

**Mathematical Reviews number (MathSciNet)**

MR184340

**Zentralblatt MATH identifier**

0161.37901

**JSTOR**

links.jstor.org

#### Citation

Hannan, J. F.; Ryzin, J. R. Van. Rate of Convergence in the Compound Decision Problem for Two Completely Specified Distributions. Ann. Math. Statist. 36 (1965), no. 6, 1743--1752. doi:10.1214/aoms/1177699802. https://projecteuclid.org/euclid.aoms/1177699802