Bayesian Fixed Sample Size Procedure for Selecting the Better of Two Poisson Populations With General Loss Function

In this paper an optimal (Bayesian) fixed sample size procedure for selecting the better of two Poisson populations is proposed and studied . Bayesian decision-theoretic approach with general loss function and Gamma priors are used to construct this procedure . The numerical result of this procedure are given with different loss functions constant , linear and quadratic , in one equation we can obtain the Bayes risk for the three types of the loss functions : constant , linear and quadratic . in this paper the numerical results are given by using Math Works Matlab ver. 7.10.0 .


1-Introduction
The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in the specified unit of space or time , for example the number of phone calls received by a telephone operator in 10-minutes or the term frequencies in a given document .[see ] A common problem that arises in practice is the selection of the better of two Poisson populations with unknown parameters .
Formally , we can state the problem as follows .Consider two independent Poisson populations Many researchers have considered this problem under different types of formulations .Goel (1972) studied the problem of selecting a subset of k Poisson populations which includes the best , i.e. the one having the largest value of the parameter .Gupta and Nagel (1971) proposed a randomize selection rule for Poisson distribution .Alam and Thompson (1973) proposed a procedure to select simultaneously the population associated with largest parameter and estimate this parameter .Gupta and Huang (1975) considered the selection from k Poisson populations a variable size subset including that population with the largest parameter when (equal) sample sizes are taken .Gupta and Wong (1977) discussed the problem of selecting a subset of k different Poisson processes including the best which is associated with the largest value of the mean rate .
Gupta , Leong and Wong (1979) considered the problem of selecting a subset of k Poisson populations including the best which is associated with the smallest value of the parameter .
Liang and Panchapakesan (1987) derived a Bayes rule having the isotonic property for selecting the Poisson populations superior to a control population under general loss function .Gupta and Liang (1991) proposed an empirical Bayes method for Poisson selection problem , where the goal is to select all good populations and exclude all load populations .
Gupta and Liang (1999) studied the problem of selecting the most reliable Poisson population from among k competitors provided it is better than a control using nonparametric Bayes approach .Madhi and Hathoot (2005) proposed a Bayesian fixed sample size procedures for this problem .λ by using Bayesian decisiontheoretic framework with Gamma prior and with general loss function .

3-Basic Definitions and Concepts 3-1-Statistical Decision Theory (i) Basic Ideas [see ]
Statistics December be consider as the science of decision making in the presence of uncertainty .The problems of statistical inferences can fit into the decision theory framework , for example , testing of a hypothesis H o against a hypothesis H 1 December be regarded as a decision between two actions (i) accepting H o or (ii) accepting H 1 .
In decision problems , the state of nature is unknown , but a decision maker must be made -a decision whose consequences depend on the unknown state of nature .Such a problem is a statistical decision problem when there are data that give partial information a bout the unknown state .
The basic elements of a statistical decision problem can be formalized mathematically as follows: A set A , the action space , consisting of all possible actions , A ∈ a , available to the decision maker ; a set Ω , the parameter space , consisting of all possible 'state of the nature' , Ω ∈ θ , one and only one of which obtains or will obtain (this 'true' state being unknown to the decisionmaker) ; a function L , the loss function , having domain a set R x , the range of X , consisting of all the possible realizations ,

4-Solution of the Problem
We term our problem as a two-decision problem and represent it symbolically as population : and population : For parameter λ and action a , the loss function is defined as : For r=0 , we have a constant loss function , for r=1 , we have a linear loss function and for r=2 , we have a quadratic loss function , k 1 ,k 2 give decision losses in units of costs .
Let us suppose that ) ,..., , ( x n If we take r=0 we find from the above equations the posterior expected looses for constant loss function for the two decisions d 1 and d 2 , if we take r=1 we find from the above equations the posterior expected looses for linear loss function for the two decisions , if we take r=2 we find from the above equations the posterior expected looses for quadratic loss function for two decision d 1 and d 2 .
For the two -decision problem considered a above , the Bayesian selection procedure is given as follows : Make

4-Numerical Results and Discussions
This section contains some numerical result about this procedure , we take various sample size n and various priors .We write a program for this procedure from which we give three types of Risk for three types of loss functions (constant , linear and quadratic) .from this numerical result we note that : 1-the procedure is well defined , as we seen in

Conclusions
In this paper we derives a procedure for selecting the best of two Poisson populations employing a decision -theoretic Bayesian frame work with general loss function with Beta prior .
From this paper we note that : 1-In this paper we derive approach for selecting the best of two Poisson populations by using Bayesian decision theory with general loss function .
2-In this procedure we can have Bayes risk for three loss function (constant, linear and quadratic) by using one equation .
3-the Bayes risk for quadratic loss function is less than the Bayes risk for linear and constant loss function .1) and ( 2) we saw that the procedure is well defined .5-if we increase sample size , the Bayes risk will decreases for all loss functions as we saw in figure(1) , figure(2) and figure(3) .
, the decision space, consisting of all possible decisions , D d ∈ , each such decision function d having domain R x and codomain A .
function depends on the outcome x and thus a random variable .Its expected value , i.e. its average over all possible outcomes is called the risk function and is denoted by of degree of belief over Θ .
Our first task in the Bayesian approach is the specification of a prior p.d.f g( λ ) .we take the prior distribution to be a member of the conjugate We derive the stopping (Baye's) risks of decision d 1 and d 2 for general loss function given above and the stopping risk (the posterior expected looses) of making decision d i denoted by ) ; , ( 2-as sample size n increase , the Bayes risk decreases for all loss function .3-The Bayes risk for quadratic loss function is less than the Bayes risk for linear and constant loss functions .4-we generate a random sample from each population by using the function (poissrnd) in Math Works Matlab ver.7.10.0 .

Figure( 1
Figure(1) : The influence of the sample size on the posterior expected loss for constant loss function decision d 1 that is selecting 1

Table ( 2
prior ) : The effect of sample size n on Bayes Risk (R 1 and R 2 ) , for fixed values of Gupta , S.S. and Liang , T.C. (1999).selecting the most reliable Poisson population provided its better than a control : A nonparametric empirical Bayes approach .Technical Report 97-9C .Dept. of statistics , west Lafayette , In USA .8-Liang , T.C. and Panchapakesan , S. (1987).Isotonic selection with respect to a control : A Bayesian approach .Technical report No. 87-24 .Dept. of statistics , Purdue University .9-Madhi , S.A. and Hathoot , S.F. (2005) .Bayesian fixed sample size procedures for selecting the better of two Poisson populations .J. of Babylon University / Pure and applied sciences , vol. 10, No. 3,838-849 .