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.