The Sampling Distribution of the Maximum Likelihood Estimators for the Parameters of Beta-Binomial Distribution

In this paper sampling distributions for the maximum likelihood estimators of the BetaBinomial model are obtained numerically using approximate random numbers from the Beta-Binomial distribution. STATGRAPHICS package is used to obtain the best fitted distributions using Chi-square and Kolmogorov-Smirnov tests.


Introduction
The Beta-Binomial model is one of the oldest discrete probability mixture models and is widely applied in the social, physical, and health sciences, it was formally proposed by Skellam (1948), although the idea was suggested earlier by Pearson (1925) in an experimental investigation of Bayes theorem.A familiar model in many applications is to assume that an observed set of counts is from a Binomial distribution with unit specific probabilities governed by a Beta distribution (the so-called Beta-Binomial model; BBM).
Consider a population in which for each member some event occurs as the outcome of a Bernoulli trial with fixed probability P , given 0 1 P < < , the number of occurrences for x in n trials has the Binomial distribution with density mass function given by ( ; , ) Suppose that P is a random variable which follows a Beta distribution, Using equations (1.1) and (1.2), the Beta-Binomial model will be ( , ) ( ; , , ) , 0,1, 2,..., , , 0 ( , ) 3) The cumulative distribution function c.d.f of X , can be written as follows: B α β is the complete Beta function, n is non-negative integer, and α and β are shape parameters.Currently, there are little studies for estimation of the unknown parameters of the Beta-Binomial distribution.Griffiths (1973) estimated the unknown parameters of Beta-Binomial distribution using maximum likelihood estimation and applied his results to the household distribution of the total numbers of a disease, Lee and Sabavala (1987) developed a Bayesian procedures for the Beta-Binomial model and, used a suitable reparameterization, establishes a conjugatetype property for a Beta family of priors.The transformed parameters have interesting interpretations, especially in marketing applications, and are likely to be more stable proposed a Bayesian estimation and prediction for the Beta-Binomial Model, Tripathi et al. (1994) introduced some alternative methods for estimating the parameters in the Beta Binomial and truncated Beta-Binomial models, Lee and Lio (1999) introduced a note on Bayesian estimation and prediction for the Beta-Binomial model.The main purpose of the paper is to extend the study of Lee and Sabavala (1987) by numerical integration.This method can be used for the general case of trials.When the number of trials is two, the results are similar to those from Lee and Sabavala (1987).Quintana and Wing-Kuen (1996) introduced a Bayesian estimation of Beta-Binomial model by simulating posterior densities and Everson and Braldlow (2002) presented a Bayesian inference for the Beta-Binomial distribution via polynomial expansions.
In this paper we introduced sampling distributions for the maximum likelihood estimators of the Beta-Binomial model numerically using approximate random numbers from the Beta-Binomial.STATGRAPHICS package is used to obtain the best fitted distributions using Chi-square and Kolmogorov-Smirnov tests.

Maximum Likelihood Estimators (MLE)
Lee and Sabavala (1987), used the maximum likelihood method to estimate the unknown parameters for the Beta-Binomial distribution when 2 n = and Lee and Lio (1999) discussed some estimation problem to estimate the unknown reparametrized parameters (π ,θ ) when 2 n ≥ .Lee and Sabavala (1987) estimated the unknown parameters for Beta-Binomial distribution ( , )   α β , using the following likelihood function 0 ( , ) [ ( ; , , )] ...( ) where x f is the number of the units with x occurrences has the Binomial distribution and The first derivatives of (2.2) with respect to α and β are given by 3) to zero, the maximum likelihood estimates α and β for α and β respectively can be obtained as the solution of the following equations These may be solved numerically for obtaining the estimates α and β .
.( 1) Taking the first derivatives of the natural logarithm of the likelihood function (2.4) with respect to π and θ , equating the results to zero and solving these results numerically, the estimates π and θ can be obtained.

Numerical Illustration and Sampling Distribution for the Maximum Likelihood estimators.
A numerical illustration for the maximum likelihood estimators for the Beta-Binomial distribution using Binomial and Poisson approximation will be obtained.MATHCAD program is used to evaluate the ML estimators and to obtain Pearson Coefficient and Pearson family of distribution.A trail will be made to obtain the best fitted distribution to maximum likelihood estimators using STATGRAPHICS package.

(a) Binomial and Poisson Approximation
The Beta-Binomial distribution is a combination of Binomial distribution with probability of Success P .Suppose P is a random variable that follows Beta distribution with shape parameters α and β , respectively, the Beta-Binomial distribution may be obtained from a Binomial distribution.Teerapabolarn (2008) obtained an upper bound on Binomial approximation to the Beta-Binomial distribution when , P α β α α β = + as follows: • If

(b) Maximum Likelihood Illustration and suggested Pearson Family of distributions
A numerical illustration for the maximum likelihood estimators for the Beta-Binomial distribution using Binomial and Poisson approximation will be obtained.MATHCAD program 3 and 4 are used to evaluate the ML estimators using equations (1.3) and (2.1) respectively using the following steps Step 2: A Chi-square goodness-of-fit test for every generated sample (Binomial, Poisson) approximation, is carried out using the following hypotheses H 0: Data follows the Beta-Binomial Distribution.H 1: Data does not follow the Beta-Binomial Distribution. Step

(c) Best Fitted Distribution using Statgraphics package.
Using STATGRAPHICS package and generated random numbers in section 3, the fitted distributions for the maximum likelihood estimators α and β are obtained, the fitted distributions will be tested using Kolmogrov Smirnov and Chi-square goodness-of-fit tests to determine the best fit.Goodness-of-fit test for the maximum likelihood estimators for the Beta-Binomial distribution are obtained using Binomial and Poisson approximation for different , n α and β .We conclude that the maximum likelihood estimators of Beta Binomial distribution using Poisson and Binomial approximation have normally densities when 5 n = , 10 and 15.

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STATGRAPHICS package is used to obtain sampling distribution for the maximum likelihood estimators, all results are listed in table 3 and 4. From these tables we conclude that the sampling distributions for the maximum likelihood estimators of the Beta-Binomial distribution have normal densities.
unknown shape parameters and ( , ) B α β is the complete Beta function.Consider the following simple mixture model

(b)Maximum Likelihood estimators for the unknown reparametrized parameters of the Beta-Binomial distribution when 2 n
≥ .Lee and Lio (1999) estimated the unknown reparametrized parameters( , ) the results gave a good Binomial approximation to the Beta-Binomial distribution if , obtained a different upper bound on Poisson approximation to the Beta-Binomial distribution as follows.upper bound on Poisson approximation to the Beta-Binomial distribution will be ( , , ) ,