Estimation of the Negative Binomial Parameter K by Maximum Quasi-Likelihood

We investigate estimation of the parameter, K, of the negative binomial distribution for small samples, using a method-of-moments estimate (MME) and a maximum quasi-likelihood estimate (MQLE). Previous work is reviewed; the importance of indirect estimation of K through its reciprocal, a, and of allowance for negative estimates of K (or a) are discussed. Samples of size 50 are simulated 10,000 times for each of several parameter combinations to examine the properties of the estimates. Samples of sizes 10, 20, 30, and 50 are simulated 1,000 times to investigate the effect of sample size. Both estimators perform reasonably well except when the mean is small and the sample size does not exceed 20. Three examples are given, one of a designed experiment, for which the MQLE is especially suited; confidence limits are derived for the MQLE. Further work along these lines is required for adequate assessment of the usual maximum likelihood estimate.