A Simple Method for the Adjustment of Profile Likelihoods

SUMMARY We propose a simple adjustment for profile likelihoods. The aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods, such as bias, inconsistency and overoptimistic variance estimates. The adjustment is applied to the profile log-likelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function. For cases in which explicit calculation of the adjustments is difficult, we give two methods to simplify their computation: An ‘automatic’ simulation method that requires as input only the profile log-likelihood and its first few derivatives; first-order asymptotic expressions. Some examples are provided and a comparison is made with the conditional profile log-likelihood of Cox and Reid.