Goodness-of-fit tests for normal distribution of order p and their asymptotic effìciency

Abstract

This paper deals with the Bahadur local efficiency of the parametric score test and of the Kolmogorov, Cramér-von Mises and Chapman-Moses non parametric tests to verify the null hypothesis Ho Teta= 0 agains the alternative Hl: Teta > 0, being Teta the centrality parameter of a p normal distribution. Bahadur efficiency is calculated with respect to the theoretical upper bound for exact slopes given in terms of Kullback-Leibler information numbers. It is analytically, numerically and graphically shown that the score test is locally optimal for all p >= 1 whereas the non parametric tests behave very differently for large p and p close to 1. As a matter of fact, for large p their efficiency decreases approxi-mately as 1/p, but for p close to 1 they are serious competitors of the score test. For instance, the Kolmogorov test is for p= 1 locally optimal in the Bahadur sense.