Nonstandard Estimation of Inverse Conditional Density-Weighted Expectations

This paper is concerned with the semiparametric estimation of function means that are scaled by an unknown conditional density function. Parameters of this form arise naturally in the consideration of models where interest is focused on the expected value of an integral of a conditional expectation with respect to a continuously distributed “special regressor”' with unbounded support. In particular, a consistent and asymptotically normal estimator of an inverse conditional density-weighted average is proposed whose validity does not require data-dependent trimming or the subjective choice of smoothing parameters. The asymptotic normality result is also rate adaptive in the sense that it allows for the formulation of the usual Wald-type inference procedures without knowledge of the estimator's actual rate of convergence, which depends in general on the tail behaviour of the conditional density weight. The theory developed in this paper exploits recent results of Goh & Knight (2009) concerning the behaviour of estimated regression-quantile residuals. Simulation experiments illustrating the applicability of the procedure proposed here to a semiparametric binary-choice model are suggestive of good small-sample performance.