 ## Optimal and suboptimal Bayes Procedures for selecting the best category in multinomial distribution

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Optimal and suboptimal Bayes Procedures for selecting the best category in multinomial distribution

University of Babylon, Iraq
Department of Mathematics

Abstract

This paper deals with multinomial selection problem. Optimal Bayesian sequential and fixed sample size procedures are proposed for selecting the best (i.e largest probability) multinomial cell. These are constructed using Bayesian decision-theoretic approach in conjunction with the dynamic programming technique.

Bayes risks are used for comparison between these procedures. Suboptimal Bayesian sequential methods are also considered and their performance is studies using Monte Carlo simulation. Performance characteristics such as the probability of correct selection expected sample size are evaluated assuming a maximum sample size.

Single observation sequential rule as well as rule when group of observations are taken and fixed sample size rule are discussed. Some concluding remarks and suggestions for future work are also included.

1. Introduction

During the early fifties, it was pointed out by several researcher that testing the homogeneity of population means or variances is not satisfactory solution to a comparison of performance of several populations. One may wish to either rank them according to their performance or select one or more from among them for future use or future evaluation. These problems are known as ranking and selection problem.

Consider a multinomial distribution which is characterized by k events (cells) with probability vector , where  is the probability of the event  with . Let  be respective frequencies in k cells of the distribution with . Further, let denote the ordered values of the . It is assumed that the values of  and of the  is completely unknown. The goal of the experimenter is to select the most probable event, that, is the event associated with , also called the best cell. A correct selection, denoted by CS, is defined as the selection of the best cell. According to this formulation we have a multinomial-decision selection problem.

The statistical formulation as stated above is typical of many practical problems arising in various fields of applications, that is, there are many situations where the multinomial distribution applies and the goal of practical as well as theoretical interest is to select the category that has the best probability (the best cell).Some applications are as follows.

In social survey subjects might be or asked whether they agree, disagree or have no opinion about particular political the response most likely to be chosen by a randomly sampled individual.

In marketing research, these procedures can be used to determine the most popular brand of a given product,

A manufacturer would like to know which of three potential plant l maximize expected revenue,

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• Optimal , suboptimal ,Bayes selecting , multinomial distribution
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