Modified maximum likelihood estimation in one-parameter exponential family models

We propose a new pivotal quantity which is a function of the maximum likelihood estimate of a scalar parameter theta and whose distribution is standard normal excluding terms of order O(n(-3/2)) and smaller, where n is the sample size. The proposed pivot is a polynomial transformation of the standardized maximum likelihood estimate of at most third degree. We apply our main result to the one-parameter exponential family model and to a number of special distributions of this family. Some simulation results illustrate the superiority of our pivotal quantity over the usual standardized maximum likelihood estimate with regard to third-order asymptotic theory.