ESTIMATION OF PARAMETERS IN THE BETA BINOMIAL MODEL

RAM C. TRIPATHI^{1}, RAMESH C. GUPTA^{2} AND JOHN GURLAND^{3}

1Division of Mathematics, Computer Science and Statistics, The University of Texas, San Antonio, TX 78249, U.S.A. 2Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, U.S.A.

3Department of Statistics, University of Wisconsin, Madison, WI 53706, U.S.A.

(Received July 8, 1991; revised September 28, 1993)

Abstract. This paper contains some alternative methods for estimating the parameters in the beta binomial and truncated beta binomial models. These methods are compared with maximum likelihood on the basis of Asymptotic Relative Efficiency (ARE). For the beta binomial distribution a simple estimator based on moments or ratios of factorial moments has high ARE for most of the parameter space and it is an attractive and viable alternative to computing the maximum likelihood estimator. It is also simpler to compute than an estimator based on the mean and zeros, proposed by Chatfield and Goodhart (1970, Appl. Statist., 19, 240–250), and has much higher ARE for most part of the parameter space. For the truncated beta binomial, the simple estimator based on two moment relations does not behave quite as well as for the BB distribution, but a simple estimator based on two linear relations involving the first three moments and the frequency of “ones” has extremely high ARE. Some examples are provided to illustrate the procedure for the two models.

Key words and phrases: Maximum likelihood, minimum chi-square, asymptotic relative efficiency, truncated beta binomial.