Selecting among the multinomial losers
Pinyuen Chen, Shanker Shyam S. Panchapakesan, Milton Sobel · Sequential Analysis · 1994
In a multinomial setting with a fixed number k of cells. the problem of screening out cells to find the "best" cell, i.e., the one with the smallest cell probability, or looking for a (small) subset of cells containing the best cell is revisited. An inverse sampling procedure is used, unlike past work on this problem ([l], [2], [3], and [4]). Finding the cell with the smallest cell probability is clearly more difficult than finding the one with the argest cell probability. The proposed procedure takes one observation at a time (as usual) and igns a zero to all those (and only those) k - 1 cells into which the observation does not fall Sampling continues sequentially and stops as soon as any one cell has accumulated r zeros. For any given integer c (with 0 ≤ c < r), we put into the selected subset (SS) all those cells with at least r - c zeros and assert that this selected subset contains the best cell. It is important to note that for the slippage configuration (SC) we can attain any specified lower bound P∗ for the probability P(SCB) that the SS contains the best cell by increasing r and need not increase the value of c. Of principal interest is the case c = 0; the reason is that (i) for c = 0 the procedure is somewhat inore efficient as will be apparent later. especially after viewing the tabled results and (ii) for c = 0 our procedure never selects a subset containing all the cells. Using the SC we determine the smallest alue of r that satisfies a preassigned lower bound P∗ for P(SCB). Two different is of a correct selection are considered, both related to (but stinct from) the probability P(SCB) that the SS contains the best cell. The results of this new procedure are numerically mpared with those in the references cited above using randomization to make the comparisons fair nd reasonable. If the other procedure is a fixed sample size procedure using N observations, en we wish to randomize between some r – 1 and the next integer r so that the resulting (N) for a proposed procedure will be (exactly) equal to the N-value for the other procedure. The proposed SAML (selecting among the omial losers) procedure turns out to be more efficient d, for at least one of the criteria, uniformly more efficient for all values of the specified parameters. Later we will make the conjecture based on numerical evidence that under the ive model (which was used in [1] for selecting the cell with the smallest cell probability) the SC is least favorable (LFC).