Complete sufficiency and maximum likelihood estimation for the two- parameter negative binomial distribution (Q1082004)

The estimation problem of both parameters in the two-parameter negative binomial distribution is discussed in the finite sample case. It is shown that the minimal sufficient statistic is not complete, and that the maximum likelihood estimator (mle) for the shape parameter k exists if the sample variance is strictly larger than the sample mean. Contours and 3-dimensional plots of the log-likelihood function indicate that the mle for k is highly non-robust.