Integral distribution-free statistics of $L_p$-type and their asymptotic comparison

Abstract

Generalizing the Cramér–von Mises and the Kolmogorov–Smirnov test, different integral statistics based on $L_p$-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 $L_p$-norms should receive more attention. It is shown that, given a distribution function $F_0$ and a specific alternative, one can draw the plot of efficiency as a function of $p$ and determine the value of $p$ giving the maximum efficiency.