Goodness-of-fit tests in semiparametric transformation models using the integrated regression function

Consider the following semiparametric transformation model Λθ(Y)=m(X)+ε, where X is a d-dimensional covariate, Y is a univariate dependent variable and ε is an error term with zero mean and which is independent of X. We assume that m is an unknown regression function and that {Λθ:θ∈Θ} is a parametric family of strictly increasing functions. We use a profile likelihood estimator for the parameter θ and a local polynomial estimator for m. Our goal is to develop a new test for the parametric form of the regression function m, which has power against local directional alternatives that converge to the null model at parametric rate, and to compare its performance to that of the test proposed by Colling and Van Keilegom (2016). The idea of the new test is to compare the integrated regression function estimated in a semiparametric way to the integrated regressionfunction estimated under the null hypothesis. We consider two different test statistics, a Kolmogorov–Smirnov and a Cramér–von Mises type statistic, and establish the limiting distributions of these two test statistics 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 procedure, to compare this new test to the previous one and to see under which model conditions which test behaves the best. We also apply both methods on a real data set.