Heteroscedastic semiparametric transformation models: estimation and testing for validity

Abstract

In this paper we consider a heteroscedastic transformation model of the form Λϑ(Y ) = m(X) + σ(X)ε, where Λϑ belongs to a parametric family of monotone transformations, m(·) and σ(·) are unknown but smooth functions, ε is independent of the d-dimensional vector of covariates X, E(ε) = 0 and Var(ε) = 1. In this model, we first consider the estimation of the unknown components of the model, namely ϑ, m(·), σ(·) and the distribution of ε, and we show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and establish the limiting distribution of the test statistics under the null hypothesis. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, we carry out a simulation study to verify the small sample behavior of the proposed estimators and tests and illustrate our method with a real dataset.