Adjusted quasi-profile likelihoods from estimating functions

Higher-order adjustments for a quasi-profile likelihood for a scalar parameter of interest in the presence of nuisance parameters are discussed. Paralleling likelihood asymptotics, these adjustments aim to alleviate some of the problems inherent to the presence of nuisance parameters. Indeed, the estimating equation for the parameter of interest, when the nuisance parameter is substituted with an appropriate estimate, is not unbiased and such a bias can lead to poor inference on the parameter of interest. Following the approach of McCullagh and Tibshirani (1990), here we propose adjustments for the estimating equation for the parameter of interest. Moreover, we discuss two methods for their computation: a bootstrap simulation method, and a first-order asymptotic expression, which can be simplified under an orthogonality assumption. Some examples, in the context of generalized linear models and of robust inference, are provided.