Test of independence for generalized Farlie–Gumbel–Morgenstern distributions

Given a pair of absolutely continuous random variables (X,Y)(X,Y) distributed as the generalized Farlie–Gumbel–Morgenstern (GFGM) distribution, we develop a test for testing the hypothesis: XX and YY are independent vs. the alternative; XX and YY 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.