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Avoiding Problems With Normal Approximation Confidence Intervals for Probabilities

Although it is well known that modern methods of computing confidence intervals (CIs) based on likelihood or simulation have important advantages, normal approximation confidence interval procedures (NACPs) are still widely used, especially in the analysis of censored data. This is because CIs from NACPs are easy to compute and easy to explain. But when the sample size is not large or when there is heavy censoring, the performance of NACPs can be poor. A transformation can be applied to keep CI endpoints from falling outside the parameter space and improve performance, but the degree of improvement (if any) depends on the chosen function. To obtain CIs for distribution probabilities, some seemingly useful transformation functions will cause the estimated variance to blow up in the tails of the distribution and can lead to nonsensical confidence intervals. This article compares different NACPs for distribution probabilities. Our results suggest that an NACP based on a studentized statistic, which we call the z procedure, has desirable properties compared with alternative NACPs.