Estimation of a semiparametric transformation model: A novel approach based on least squares minimization

Abstract

Consider the following semiparametric transformation model $\Lambda_{\theta }(Y)=m(X)+\varepsilon $, where $X$ is a $d$-dimensional covariate, $Y$ is a univariate response variable and $\varepsilon $ is an error term with zero mean and independent of $X$. We assume that $m$ is an unknown regression function and that $\{\Lambda _{\theta }:\theta \in\Theta \}$ is a parametric family of strictly increasing functions. Our goal is to develop two new estimators of the transformation parameter $\theta $. The main idea of these two estimators is to minimize, with respect to $\theta $, the $L_{2}$-distance between the transformation $\Lambda _{\theta }$ and one of its fully nonparametric estimators. We consider in particular the nonparametric estimator based on the least-absolute deviation loss constructed in Colling and Van Keilegom (2019). We establish the consistency and the asymptotic normality of the two proposed estimators of $\theta $. We also carry out a simulation study to illustrate and compare the performance of our new parametric estimators to that of the profile likelihood estimator constructed in Linton et al. (2008).