Simple Estimation Intervals for Poisson, Exponential, and Inverse Gaussian Means Obtained by Symmetrizing the Likelihood Function

Likelihood intervals for the Poisson, exponential, and inverse Gaussian means that have simple analytically closed expressions and good coverage frequencies for any sample size are given here explicitly. Their simplicity is striking and they should be more broadly used in applications everywhere. Their soundness is due to three statistical properties that these three distributions share as well as the fact that for all of them there exists a simple power reparameterization that symmetrizes the corresponding likelihood function. As a consequence, asymptotic maximum likelihood results are applicable even for samples of size one. Likelihood intervals of the new parameter may be easily transformed back to the original parameter of interest, the mean, by the invariance property of the likelihood function. Practical examples are given to illustrate the proposed inferential procedures.