Profile Likelihood–Based Confidence Intervals for the Mean of Inverse Gaussian Distribution

Abstract

The inverse Gaussian distribution is applied in a wide range of applications such as physics, engineering, medicine, and business. In this research, a confidence interval of the mean is constructed for the inverse Gaussian distribution with an unknown shape parameter. The profile likelihood is employed as a method to eliminate a shape parameter. In order to use an asymptotic distribution to calibrate the likelihood, the sample size must be large enough. Thus the optimal sample sizes need to be determined by considering the coverage probability estimated by the Monte Carlo simulation method. Likelihood function is mathematically proved that it does not converge to zero as the mean approaches infinity – in both known and unknown shape parameters, but approaches to a certain quantity depending on sample. Also, the estimator is derived for probability of success in constructing confidence interval with a given sample size. Finally, the profile likelihood function is founded to be maximum when the mean is equal to the maximum likelihood estimate. For the simulation study, based on desirable coverage probability, a larger sample size is required when the population becomes more skewed.