Bootstrap-based goodness-of-fit test for parametric families of conditional distributions

Abstract

Many scientific studies are concerned with the relationship between some vector of covariates X and a response variable Y. In particular, scientists are often interested in finding a parametric family the conditional density function of Y given X fits in. A classical example for these types of families are parametric generalized linear models. In order to make use of such models, e.g. for predictions of Y given a new vector of covariates, it first has to be checked whether the given data fits to the assumed model. In this thesis, we propose a bootstrap-based goodness-of-fit test for this purpose. The test statistic uses a nonparametric and a semi-parametric estimator for the marginal distribution function of Y. The critical value is calculated using a parametric bootstrap method. We will verify the validity of this approach by proving that the limit distribution of the original test statistic and its bootstrap version coincide. A simulation study will show that the new method performs considerably better than other tests found in the literature in case of multivariate covariates. We will extend part of our results to parametric regression under random censorship. The methods derived in this thesis are implemented in an R-package called gofreg.