Estimating the error distribution in semiparametric transformation models

Abstract

In this paper we consider the semiparametric transformation model $\Lambda_{\theta_{o}}(Y)=m(X)+\varepsilon$, where $\theta_{o}$ is an unknown finite dimensional parameter, the function $m(\cdot)=\mathbb{E}(\Lambda_{\theta_{o}}(Y)|X=\cdot)$ is “smooth”, but otherwise unknown, and the covariate $X$ is independent of the error $\varepsilon$. An estimator of the distribution function of $\varepsilon$ is investigated and its weak convergence is proved. The proposed estimator depends on a profile likelihood estimator of $\theta_{o}$ and a nonparametric kernel estimator of $m$. We also evaluate the practical performance of our estimator in a simulation study for several models and sample sizes. Finally, the method is applied to a data set on the scattering of sunlight in the atmosphere.