Bahadur efficiency of nonparametric test for independence based on $L_1$-error

Abstract: We introduce new test statistic to test the independence of two multi-dimensional random variables. Based on the $L_1$-distance and the historgram density estimation method, the test is compared via Bahadur relative efficiency to several tests available in the literature. It arises that our test reaches better performances than a number of usual tests among whom we cite the Kolmogorov-Smirnov test. Beforehand, large deviation result is stated for the associated statistic. The local asymptotic optimality relative to the test is also studied.