The Compound Decision Problem with m × n Finite Loss Matrix

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 P₁ and P₂, P₁ \≠ P₂, Hannan and Robbins [5] 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 P₂ is the governing distribution. This result was motivated by heuristic arguments and an example (component decisions between (-1, 1) and (1, 1) given by Robbins [8]. 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, [5]) 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-\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 [5]. More extensive results for the case $m = n = 2$ are given in a paper by Hannan and Van Ryzin [6]. Please note that the case considered here makes the N-decisions after the data from all N problems are available. The sequential case (kth decision after observations 1, 2, \⋯, k, k = 1, \⋯, N) is treated by Hannan in [3] and by Samuel in [10].