On the probable error of the ratio of two gamma means

Abstract

SUMMARY The two-sample Student's t-interval can be derived by parametric bootstrap coverage calibration of the naive, plug-in interval that fails to account for the estimation of the common variance parameter. The same technique results in an essentially exact confidence interval for the ratio of two means based on independent samples from gamma distributions. The gamma problem is more difficult computationally because an exact pivot is not available. However, the computational burden of bootstrap calibration can be reduced to a routine level using the fact that the ratio of sample means is proportional to an exact F-variate that is independent of the conditional maximum likelihood estimator of the shape parameter. The density of the conditional estimator can be approximated with extreme accuracy using the p* formula, and a rejection sampler can be used to simulate from this approximate density. This greatly simplifies bootstrap calibration in this context. An alternative interval is obtained by inverting an exact test described by Jensen (1986). Simulation studies suggest that the two intervals have similar performance in terms of length and coverage. In practice, we recommend the bootstrap calibration method because it is much simpler to implement and is more versatile.