Binomial and Censored Sampling in Estimation and Decision Making for the Negative Binomial Distribution

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Abstract
The estiamtion of the mean of a negative binomial distribution from binomial data (the number of samples units out of a given total with less than or equal to a predetermined number, t, of individuals) is studied. A theorem is proved to show that, asymptotically, the estimate is an increasing function of t if the assumed value of the exponent, k, of the distribution is greater than the true value, and a decreasing function if the assumed value is less than the true value. If sampling is undertaken for decision making, the theorem can indicate a suitable value of t such that the procedure is reasonably robust for k. A censored sampling scheme is suggested for estimating the mean and k based on frequencies of samples with 0, 1, 2, .., t, and more than t individuals. It is shown that t need not be large for relatively accurate and efficient estimation.