Integral distribution-free statistics of LpLp-type and their asymptotic comparison

Generalizing the Cramér–von Mises and the Kolmogorov–Smirnov test, different integral statistics based on LpLp-norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on LpLp-norms should receive more attention. It is shown that, given a distribution function F0F0 and a specific alternative, one can draw the plot of efficiency as a function of pp and determine the value of pp giving the maximum efficiency.