A Sequential Procedure for Selecting the Most Probable Multinomial Event

Abstract

This paper deals with a sequential sampling procedure for selecting from a given multinomial distribution with K .cells,-the cell with the largest probability of occurrence. Observations being taken sequentially from the given distribution, the sampling is terminated when the largest count in any cell is equal to N or when the difference between the largest and the 4 next largest cell counts is equal to r, where r and N are given positive integers. The Cell with the largest count on the termination of sampling is selected as the most probable event. I The probability of a correct selection and the expected total number of observations are given.

Truncated selection procedures for the most probable event and the least probable event

“…The stopping rule for both procedures uses truncation of the procedure studied by Ramey and Alam (1979, Biometrika, 66, 171-173). A property of the least favorable configuration of the proposed procedures is proved, which partially solves a conjecture given in Ramey and Alam (1979). The proposed procedures are compared with other procedures which have been considered in the literature and are found to be better in certain respects.…”

“…Their procedure was the combination of stopping rules considered by Cacoullos and Sobel (1966) and Alam (1971). Cacoullos and Sobel's procedure stops sampling when the frequency of any cell reaches a given positive integer M. Alam's procedure stops sampling when the difference between the largest frequency and the second largest frequency is equal to a given positive integer r. The combination of these two stopping rules, which was studied by Ramey and Alam (1979), stops sampling when either of these two stopping criteria is satisfied with the sample size bounded by kM -k + 1. When sampling is terminated, the procedure selects the cell with the unique largest cell frequency as corresponding to the most probable event.…”

“…(The result can also be applied to curtailing the fixed sample size procedure for selecting the least probable cell.) Ramey and Alam (1979) conjectured that (3.1) is the LFC for their procedure. Since our procedure R1 has stopping rule which is a combination of curtailed procedure and Ramey and Alam's procedure, we infer that the LFC for R1 is also (3.1).…”

The Slippage Configuration Is Always the Least Favorable Configuration for Two Alternatives

“…The SC has been proved to be the LFC for the following procedures: The BEM procedure of Bechhofer, Elmaghraby, andMorse (1959, proved by Kesten and, the BK curtailed sequential procedure of Bechhofer and Kulkarni (1984), the CS inverse sampling procedure of Cacoullos and Sobel (1966), Alam's (1971) fixed difference procedure (proved by Alam for k = 2 and by Bhandari and Ali [1994] for k ≥ 3), Bechhofer, Kiefer, and Sobel's (1968) unbounded procedure BKS (proved also by Levin, 1984), and Chen's (1992) curtailed inverse-sampling procedure. The SC has been conjectured to be the LFC for Bechhofer and Goldsman's truncated BKS procedure in Bechhofer andGoldsman (1985b, 1986), the RA procedure of Ramey and Alam (1979) and Bechhofer and Goldsman (1985a), Chen's (1992) C (also called RA ) truncation curtailment modification of RA, and the recently proposed linear programming-based optimal randomized and integer-programming-based optimal deterministic procedures in Tollefson (2012) and Tollefson et al (2014a,b). For k = 2 alternatives the conjecture for the RA and C procedures has been proved in Ramey and Alam (1979) and Chen (1992), respectively.…”

“…The SC has been conjectured to be the LFC for Bechhofer and Goldsman's truncated BKS procedure in Bechhofer andGoldsman (1985b, 1986), the RA procedure of Ramey and Alam (1979) and Bechhofer and Goldsman (1985a), Chen's (1992) C (also called RA ) truncation curtailment modification of RA, and the recently proposed linear programming-based optimal randomized and integer-programming-based optimal deterministic procedures in Tollefson (2012) and Tollefson et al (2014a,b). For k = 2 alternatives the conjecture for the RA and C procedures has been proved in Ramey and Alam (1979) and Chen (1992), respectively.…”