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bayesian fixed sample size procedure for selecting the least prbable event in multinomial distribution

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5/12/2011 5:23:29 PM
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Bayesian Fixed Sample Size Procedure for Selecting the Least   Probable  Event in Multinomal Distribution
Saad  A.Madhi  and  Kawther  F .Hamza
Dept .of Mathematics    College of Education.
In this paper, a fixed sample size prosedure for selecting the  least probable event (i.e ,the cell with smallest probability ) in multinomial distribution is given .Bayesian decision –theoretic approach is used to construct this procedure.Bayes  risks of taking decisions under linear losses and Dirichelet priors are derived .
1. Introduction
Consider the multinomial distribution with k cells and unknown probabilities of an observation in the ith cell   ,(i=1,2,….,k),where     . It is required to find the cell with the smallest (least) probability (best cell in this sense).There are many practical situations where a solution to this problem  is required . For example ,we have a sample of blood from each of a large number of populations in a certain city ,and each person is classifed as type A, type B, type AB, or type O,since we wish to identify the blood type that  is  more rare ,the  goal   is  to  determine  which  blood  type  occurs  least  frequently among these persons . 
The problem of selecting the smallest  cell probability has been considered   by Alam and Thompson (1972) using indefference zone  approach  . According to this approach ,the cell with smallest count is selected as the least probable event ,with ties broken by randomization .
Let (1/k < <1) and  c ,(0<c<1/k-1) by given .the smallest count should be determined for probability of correct select selection  whenever   ,   (i=1, 2, …, k).                                        
This paper deals with Bayesian fixed sample size procedure for selecting  the least cell probability in multinomial distribution whose parameters are distributed a prior  according to a Dirichlet distribution. Section 2 contains the formulation of the problem .Section 3 ,presents the prior and posterior probabilities .In section 4 ,we develop a procedure for selecting the smallest cell probability in multinomial distribution using a Bayesian  Decision – theoretic approach . Section 5 contains  some concluding remarks and future works .

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  • bayesian fixed sample , selecting the least prbable , multinomial distribution