Comparisons of approximate tail probabilities for the shape parameter of the gamma distribution

This article compares the accuracy of tail probabilities obtained by various approximate inference procedures for the shape parameter of the gamma distribution which has one or more nuisance parameters. The approximate inference procedures being studied are the asymptotic normality of the maximum likelihood estimator, the asymptotic normality of the signed square root of the likelihood ratio statistic, the saddle-point approximation using two different approximations of the conditional log-likelihood function and also the exact conditional log-likelihood function, and a method discussed in Fraser and Reid (1995). The first two methods have first-order accuracy whereas the last four methods have third-order accuracy. Results indicate that all the third-order methods outperformed the first-order methods, and the saddle-point approximation using the exact conditional log-likelihood function gives the most accurate approximations. However, in terms of complexity in calculations, the method by Fraser and Reid (1995) is the simplest of the third-order methods.