A direct bootstrapping technique and its application to a novel goodness of fit test

We prove general theorems that characterize situations in which we could have asymptotic closeness between the original statistics HnHn and its bootstrap version Hn∗, without stipulating the existence of weak limits. As one possible application we introduce a novel goodness of fit test based on the modification of Total Variation metric. This new statistic is more sensitive than the Kolmogorov–Smirnov statistic, it applies to higher dimensions, and it does not converge weakly; but we show that it can be bootstrapped.