Interval estimation of the negative binomial dispersion parameter
Inference concerning the negative binomial dispersion parameter, denoted by c, is important in many biological and biomedical investigations. Properties of the maximum-likelihood estimator of c and its bias-corrected version have been studied extensively, mainly, in terms of bias and efficiency [W. W. Piegorsch, Maximum likelihood estimation for the negative binomial dispersion parameter, Biometrics 46 (1990), pp. 863-867; S.J. Clark and J.N. Perry, Estimation of the negative binomial parameter kappa by maximum quasi-likelihood, Biometrics 45 (1989), pp. 309-316; K. K. Saha and S. R. Paul, Bias corrected maximum likelihood estimator of the negative binomial dispersion parameter, Biometrics 61 (2005), pp. 179-185]. However, not much work has been done on the construction of confidence intervals (C.I.s) for c. The purpose of this paper is to study the behaviour of some C. I. procedures for c. We study, by simulations, three Wald type C. I. procedures based on the asymptotic distribution of the method of moments estimate (mme), the maximum-likelihood estimate (mle) and the bias-corrected mle (bcmle) [K. K. Saha and S. R. Paul, Bias corrected maximum likelihood estimator of the negative binomial dispersion parameter, Biometrics 61 (2005), pp. 179-185] of c. All three methods show serious under-coverage. We further study parametric bootstrap procedures based on these estimates of c, which significantly improve the coverage probabilities. The bootstrap C.I.s based on the mle (Boot-MLE method) and the bcmle (Boot-BCM method) have coverages that are significantly better (empirical coverage close to the nominal coverage) than the corresponding bootstrap C. I. based on the mme, especially for small sample size and highly over-dispersed data. However, simulation results on lengths of the C.I.s show evidence that all three bootstrap procedures have larger average coverage lengths. Therefore, for practical data analysis, the bootstrap C. I. Boot-MLE or Boot-BCM should be used, although Boot-MLE method seems to be preferable over the Boot-BCM method in terms of both coverage and length. Furthermore, Boot-MLE needs less computation than Boot-BCM.
dispersion parameter maximum-likelihood estimator simulation method of moment estimator bootstrap confidence interval bias-corrected maximum-likelihood estimator negative binomial model wald type confidence interval