Non-parametric bootstrap tests for parametric distribution families

Durbin’s estimated empirical process is a widely used tool to testing goodness of fit for parametric distribution families. In general, statistical methods based on the process are not distribution free and the critical values can not always be calculated in a theoretical way. One can avoid these difficulties by applying the parametric or the non-parametric bootstrap procedure. Although the parametric bootstrapped estimated empirical process is well investigated, only a few papers dealt with the non-parametric version. Recently, Babu and Rao pointed out that in the latter case a bias correction is needed, and they proved the weak convergence of the bootstrapped process in continuous distribution families. Our paper presents a weak approximation theorem for the non-parametric bootstrapped estimated empirical process using similar conditions under which Durbin’s non-bootstrapped process converges. The result covers the most important continuous and discrete distribution families. Simulation studies in the Poisson and the normal distribution are also reported.