Approximate Confidence Intervals for a Parameter of the Negative Hypergeometric Distribution

The negative hypergeometric distribution is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. Thus, sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. Assuming the total number of units in the population is known but the number of each type is not, we consider the problem of estimating this unknown parameter. We investigate the maximum likelihood estimator and an unbiased estimator for the parameter. We use the method of Taylor’s series to develop five approximations for the variance of the parameter estimators. We then propose five large sample confidence intervals for the parameter. Based on these results, we simulated a large number of samples from various negative hypergeometric distributions to investigate performance of three of these formulas. We evaluate their performance in terms of empirical probability of parameter coverage and confidence interval length. The unbiased estimator is a better point estimator relative to the maximum likelihood estimator as evidenced by empirical estimates of closeness to the true parameter. Confidence intervals based on the unbiased estimator tended to be shorter than two competitors because of its relatively small variance estimator but at a slight cost in terms of coverage probability.