Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.