The Fitting of the Positive Binomial Distribution When Both Parameters Are Estimated From the Sample

Abstract

1. The positive binomial distribution P ( z ) = pXqn-" is usually fitted to samples (consisting of, say, N sampling units) with n known. When the estimation of both n and p must be done from the sample (e.g. Hoel, 1947;Skellam, 1948) this can be efficiently done by putting Z = , 2 ( p = np) and obtaining n as the root of the maximum-likelihood equation: 0 N , c x=j+l = 0 (Haldane, 1941; Fisher, 1941)R-1 j=o 6 -j by iteration (then $ can be obtained as Z/6) (R = max. {xi}, N , = sample frequency of z). This estimate is best for any n , p and N . If 6 < 100, it is probably faster, although less elegant, to find that integer which, when substituted for n into the expression for the likelihood (while substituting 5 for np), maximizes it; this can now be done with the help of recently published tables ( 1947,1949). If 6 > 100, it is very seldom reasonable to assume that the moment estimates of n and p are sacrificing any worth-while information. This can be shown thus:Rewriting Fisher's (1941) expression for E , the efficiency of the method of moments, to apply to the binomial, we get Hence moment estimates sacrifice 'almost all' information if n = 2 , but throw away only negligible information if (i) n > 21, (ii) n > 2, p <A, (iii) n > 2 , ( l / p -1 ) ( n -2 ) > 20.2. ,The above considerations are purely formal; now it must also be considered that n is necessarily a positive integer, 2 R.* This can be conveniently done thus: It is known that 6 is asymptotically normally distributed with p = n, Although u2 is a function of n and p , it can be taken as known, i.e. a function of 6 and 3 only, because we are dealing with large-sample distributions. The problem of estimating n , which is necessarily a positive integer, thus reduces to that discussed by Hammersley (1950). Following * Fisher (1941) pointed out that there is no positive binomial distribution with non-integer exponent, because the expansion of ( p + q ) " (v not integer) contains terms which are < 0. The distributions occurring in certain earlier papers (e.g. Whitaker, 1914) could be adjusted accordingly.8 VOl. 18

An Improved Method of Estimating an Integer-Parameter by Maximum Likelihood

“…Binet 1953;Fisher 1941;and Haldane 1941 It should be emphasized that (3.2) is the result of the old method existing in the literature while (3.1) comes from the method proposed here. We now compare the two methods by considering a numerical example (Binet 1953) If we assume a binomial (N,6) distribution (Binet 1953), the moment estimate of N, that is the nearest integer to X2/(X -S2), iS 12, where S2 = (X, -)2 (n -1) is the sample variance. In order to obtain the MLE of N, Binet (1953) evaluated the likelihood function for N = 11, 12, and 13, obtained the largest value at N = 12, and concluded that the MLE for this data is 12.…”

“…It is clear that L*(N) is a decreasing function of N for N -7, so that Binet's (1953) calculations were in error. The method proposed here does not require evaluation of the likelihood function, and a hand calculator is enough to find the rough range for the root of (3.1) which provides the MLE of N.…”

“…We now compare the two methods by considering a numerical example (Binet 1953) If we assume a binomial (N,6) distribution (Binet 1953), the moment estimate of N, that is the nearest integer to X2/(X -S2), iS 12, where S2 = (X, -)2 (n -1) is the sample variance. In order to obtain the MLE of N, Binet (1953) evaluated the likelihood function for N = 11, 12, and 13, obtained the largest value at N = 12, and concluded that the MLE for this data is 12. In these calculations, however, 6 = i/N is rounded off to three decimal places, which has caused an error in evaluating the likelihood function.…”

“…Several researchers (Binet 1953, Fisher 1941, and Haldane 1941 have investigated the maximum likelihood estimator (MLE) of N in the binomial (N,6) case when both N and 0 are unknown. Feldman and Fox (1968) investigated the same problem assuming 0 to be known.…”

On the Erratic Behavior of Estimators ofNin the BinomialN, pDistribution

“…Related work on the estimation of N in the Binomial problem has been undertaken by Binet (1954), who discussed the relative appropriateness of Binomial, Negative Binomial, and Poisson models; Feldman and Fox (1968), Draper and Guttman (1971), Kahn (1987), Raferty (1988), and Aitkin ( 1992), who suggested Bayesian approaches to inference; DeRiggi (1978), who described the shape of the likelihood function and its ramifications; Casella (1986) and Aitkin and Stasinopoulos (1 989), who discussed several issues of stability; and Blumenthal and Dahiya (198 1) and Aitkin (199 l), who reviewed previous work on the problem. Aitkin's review, part of a much more general account of Bayesian inference, is particularly illuminating in terms of the light that it sheds on the effect of different prior p densities on the integrated likelihood and its interpretations.…”