  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.
