Goodness-of-fit tests in semiparametric transformation models

Abstract

Consider a semiparametric transformation model of the form \(\varLambda _{\theta }(Y)\) \(= m(X) + \varepsilon \), where \(Y\) is a univariate dependent variable, \(X\) is a \(d\)-dimensional covariate, and \(\varepsilon \) is independent of \(X\) and has mean zero. We assume that \(\{ \varLambda _{\theta } : \theta \in \varTheta \}\) is a parametric family of strictly increasing functions, while \(m\) is an unknown regression function. The goal of the paper is to develop tests for the null hypothesis that \(m(\cdot )\) belongs to a certain parametric family of regression functions. We propose a Kolmogorov–Smirnov and a Cramér–von Mises type test statistic, which measure the distance between the distribution of \(\varepsilon \) estimated under the null hypothesis and the distribution of \(\varepsilon \) without making use of this null hypothesis. The estimated distributions are based on a profile likelihood estimator of \(\theta \) and a local polynomial estimator of \(m(\cdot )\). The limiting distributions of these two test statistics are established under the null hypothesis and under a local alternative. We use a bootstrap procedure to approximate the critical values of the test statistics under the null hypothesis. Finally, a simulation study is carried out to illustrate the performance of our testing procedures, and we apply our tests to data on the scattering of sunlight in the atmosphere.