Nonparametric $n\sp {-1/2}$-consistent estimation for the general transformation models

Abstract

We propose simple estimators for the transformation function $\Lambda$ and the distribution function F of the error for the model $$\Lambda (Y) = \alpha + \mathbf{X} \mathbf{\beta} + \varepsilon.$$ It is proved that these estimators are consistent and can achieve the unusual $n^{-1/2}$ rate of convergence on any finite interval under some regularity conditions. We show that our estimators are more attractive than another class of estimators proposed by Horowitz. Interesting decompositions of the estimators are obtained. The estimator of F is independent of the unknown transformation function $\Lambda$, and the variance of the estimator for $\Lambda$ depends on $\Lambda$ only through the density function of X. Through simulations, we find that the procedure is not sensitive to the choice of bandwidth, and the computation load is very modest. In almost all cases simulated, our procedure works substantially better than median nonparametric regression.