Likelihood functions for inference in the presence of a nuisance parameter

Consider inference about a scalar parameter of interest 0 in the presence of a vector nuisance parameter. Inference about 0 is often based on a pseudolikelihood function. In this paper, the general problem of constructing a pseudo-loglikelihood function H(0) is considered. Conditions are given under which H has the same properties as a genuine loglikelihood function for a model without a nuisance parameter. When these conditions are satisfied to a given order of approximation, H is said to be a y th-order local loglikelihood function. The theory of local loglikelihood functions is developed and it is shown that second-order versions of these have a number of desirable properties. Several commonly used pseudolikelihood functions are studied from this point of view. One commonly used pseudolikelihood function is profile likelihood in which parameters other than 0 are replaced by their maximum likelihood estimates. A second aspect of the paper is to consider the use of other estimates in this context. Examples are given which suggest that inference about 0 may be improved if a method other than maximum likelihood is used, particularly when the number of other parameters is large relative to the sample size.