Goodness-of-fit tests for parametric regression with selection biased data

Consider the nonparametric location-scale regression model Y=m(X)+σ(X)εY=m(X)+σ(X)ε, where the error εε is independent of the covariate XX, and mm and σσ are smooth but unknown functions. The pair (X,Y)(X,Y) is allowed to be subject to selection bias. We construct tests for the hypothesis that m(·)m(·) belongs to some parametric family of regression functions. The proposed tests compare the nonparametric maximum likelihood estimator (NPMLE) based on the residuals obtained under the assumed parametric model, with the NPMLE based on the residuals obtained without using the parametric model assumption. The asymptotic distribution of the test statistics is obtained. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, the finite sample performance of the proposed tests is studied in a simulation study, and the developed tests are applied on environmental data.