Goodness-of-fit test under length-biased sampling

In this article, we study a goodness-of-fit (GOF) test in the presence of length-biased sampling. For this purpose, we introduce a smoothed estimator of distribution function (d.f.) and we investigate its asymptotic behaviors, such as uniform consistency and asymptotic normality. Based on this estimator, we define a one-sample Kolmogorov type of GOF test for length-biased data. We conduct Monte Carlo simulations to evaluate the performance of the proposed test statistic and compare it with the one-sample Kolmogorov type of GOF test obtained by the non smoothed estimator of d.f.