Bayesian Look Ahead Sampling Methods to Allocate up to M Observations among k Populations

In this paper, we study the Bayesian look ahead sampling methods for allocating up to M observations among k populations to select the best population(s). First, we investigated the properties of fixed sample-size sampling algorithm proposed by Professor Klaus J. Miescke, which always draws fixed number of observations at the next step. Then we proposed and studied a m-truncated sampling algorithm, which draws up to m observations sequentially. Based on these two algorithms, respectively, two Bayesian look-ahead sampling methods for allocating up to M observations among k populations are developed. To investigate the properties of and compare these two methods, we implement them to allocate up to M observations among k normal distributions with the same variance or k binomial populations to select the best population. For given values of M, the Bayes risks of these two methods are calculated or estimated. The smaller the Bayes risk, the better the method. It turns out that when the sampling cost is large compared with the decision loss, the second method is better than the first. When the sampling cost is not very large, then in the normal case the two methods are comparable, with one method occasionally better than the other. On the other hand, in the binomial case, the second method dominates most of the time. These two methods are then applied in various other situations.