An explanation of generalized profile likelihoods

Let X, T, Y be random vectors such that the distribution of Y conditional on covariates partitioned into the vectors X = x and T = t is given by f(y; x, φ), where φ = (θ, η(t)). Here θ is a parameter vector and η(t) is a smooth, real-valued function of t. The joint distribution of X and T is assumed to be independent of θ and η. This semiparametric model is called conditionally parametric because the conditional distribution f(y; x, φ) of Y given X = x, T = t is parameterized by a finite dimensional parameter φ = (θ, η(t)). Severini and Wong (1992. Annals of Statistics 20: 1768-1802) show how to estimate θ and η(·) using generalized profile likelihoods, and they also provide a review of the literature on generalized profile likelihoods. Under specified regularity conditions, they derive an asymptotically efficient estimator of θ and a uniformly consistent estimator of η(·). The purpose of this paper is to provide a short tutorial for this method of estimation under a likelihood-based model, reviewing results from Stein (1956. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley, pp. 187-196), Severini (1987. Ph.D Thesis, The University of Chicago, Department of Statistics, Chicago, Illinois), and Severini and Wong (op. cit.).