Goodness-of-fit tests in semiparametric transformation models

Consider a semiparametric transformation model of the form Λθ(Y ) = m(X)+ε, where Y is a univariate dependent variable, X is a d-dimensional covariate, and ε is independent of X and has mean zero. We assume that {Λθ : θ ∈ Θ} 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(·) belongs to a certain parametric family of regression functions. We propose a Kolmogorov-Smirnov and a Cram´er-von Mises type test statistic, which measure the distance between the distribution of ε estimated under the null hypothesis and the distribution of ε without making use of this null hypothesis. The estimated distributions are based on a profile likelihood estimator of θ and a local polynomial estimator of m(·). 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.status: publishe