Test of independence for generalized Farlie–Gumbel–Morgenstern distributions

Given a pair of absolutely continuous random variables ( X , Y ) distributed as the generalized Farlie–Gumbel–Morgenstern (GFGM) distribution, we develop a test for testing the hypothesis: X and Y are independent vs. the alternative; X and Y are positively (negatively) quadrant dependent above a preassigned degree of dependence. The proposed test maximizes the minimum power over the alternative hypothesis. Also it possesses a monotone increasing power with respect to the dependence parameter of the GFGM distribution. An asymptotic distribution of the test statistic and an approximate test power are also studied.