April, 1966 The Compound Decision Problem with $m \times n$ Finite Loss Matrix
J. R. Van Ryzin
Ann. Math. Statist. 37(2): 412-424 (April, 1966). DOI: 10.1214/aoms/1177699523

## 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 $P_1$ and $P_2, P_1 \neq P_2$, Hannan and Robbins  give a decision function whose risk is uniformly close (for $N$ large) to the risk of a best "simple" procedure based on knowing the proportion of component problems in which $P_2$ is the governing distribution. This result was motivated by heuristic arguments and an example (component decisions between $\mathfrak{N}(-1, 1)$ and $\mathfrak{N}(1, 1)$ given by Robbins . In both papers, the decision functions for the component problems depended on data from all $N$ problems. This paper generalizes and strengthens a result of Hannan and Robbins (Theorem 4, ) to the case where each component problem consists of making one of $n$ decisions based on an observation from one of $m$ distributions. Specifically, we find upper bounds for the difference in the risks (the regret function) of a certain compound procedure and a best "simple" procedure which is Bayes against the empirical distribution on the component parameter space. Theorem 2 gives sufficient conditions for a uniform (in parameter sequences) bound on the regret function of order $N^{-\frac{1}{2}}$, while Theorem 3 states sufficient conditions for a uniform bound of order $N^{-1}$. For $m = n = 2$, Theorem 2 furnishes a strengthening of Theorem 4 of . More extensive results for the case $m = n = 2$ are given in a paper by Hannan and Van Ryzin . Please note that the case considered here makes the $N$-decisions after the data from all $N$ problems are available. The sequential case ($k$th decision after observations $1, 2, \cdots, k, k = 1, \cdots, N$) is treated by Hannan in  and by Samuel in .

## Citation

J. R. Van Ryzin. "The Compound Decision Problem with $m \times n$ Finite Loss Matrix." Ann. Math. Statist. 37 (2) 412 - 424, April, 1966. https://doi.org/10.1214/aoms/1177699523

## Information

Published: April, 1966
First available in Project Euclid: 27 April 2007

zbMATH: 0173.46303
MathSciNet: MR193723
Digital Object Identifier: 10.1214/aoms/1177699523 