An approximation to the modified profile likelihood function

An approximation to the modified profile likelihood function is proposed. This approximation is invariant under interest-respecting reparameterisations, it agrees with the modified profile likelihood function to order O(n^-1) in the moderate-deviation sense and to order O(n^-1) in the large-deviation sense, and it is stable in the sense of conditional inference. For the case of a curved exponential family model this approximation agrees with the one proposed by Barndorff-Nielsen (1995). A general expression for the approximate modified profile likelihood function is given for nonlinear regression models and for generalised linear models with known dispersion parameter. Adjusted profile likelihood; Likelihood inference; Nonlinear regression; Nuisance parameter; Profile likelihood.