Estimation of the Error Density in a Semiparametric Transformation Model

TL;DR

This paper introduces a kernel-based method to estimate the error density in a semiparametric transformation model, demonstrating its asymptotic properties and practical effectiveness through simulations.

Contribution

It proposes a novel kernel-type estimator for the error density in a semiparametric transformation model, with proven asymptotic normality.

Findings

Estimator is asymptotically normal

Simulation shows good practical performance

Method effectively estimates error density in complex models

Abstract

Consider the semiparametric transformation model $\Lambda_{\theta_o}(Y)=m(X)+\epsilon$, where $\theta_o$ is an unknown finite dimensional parameter, the functions $\Lambda_{\theta_o}$ and $m$ are smooth, $\epsilon$ is independent of $X$, and $\esp(\epsilon)=0$. We propose a kernel-type estimator of the density of the error $\epsilon$, and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of $\theta_o$ and a nonparametric kernel estimator of $m$. The practical performance of the proposed density estimator is evaluated in a simulation study.