Estimating the Parameters of Mixtures of Binomial Distributions
- 1 June 1964
- journal article
- research article
- Published by JSTOR in Journal of the American Statistical Association
- Vol. 59 (306) , 510
- https://doi.org/10.2307/2283005
Abstract
Let Y 1, ···, Y m be independent, each having distribution where 0 < p 1 < ··· < p r < 1, 0 < α j , < 1, Σα j = 1, and n ≧ 2r − 1. This distribution is a mixture of r binomials. It is shown to be an identifiable mixture under the given restrictions on the parameters. Moment estimators for the 2r − 1 parameters p 1, ···, P r , α1, ···, α r − 1 are constructed and the covariance matrix of the joint asymptotic normal distribution of the estimators is obtained. It is found by comparison with the Cramér-Rao lower bound that the asymptotic efficiency of the moment estimators tends to unity as n → ∞. Efficient estimators for any n ≧2r − 1 are obtained as functions of the moment estimators by two procedures, one involving expansion of a χ2 about the moment estimators to obtain BAN estimators, the other using Fisher's “information” to compute a correction factor for the moment estimators. BAN estimators may also be computed using Neyman's linearization technique. A small empirical study of the behavior of several of these estimators for moderate sample sizes is included.Keywords
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