This paper deals with distribution-free tests for independence under the constraint that the population has a bivariate discrete distribution. The locally most powerful conditional test, given the marginal empirical distributions, is derived. The unconditional asymptotic distribution of the conditional test statistic standardized by the conditional mean and variance is also given under the hypothesis of independence and under contiguous alternatives. Furthermore, some discussions on asymptotic relative efficiency are made. Two competitive test statistics having asymptotically chi-square distributions with different degrees of freedom are compared by means of the local asymptotic relative efficiency.
"Tests for Independence in Infinite Contingency Tables." Ann. Statist. 4 (3) 542 - 553, May, 1976. https://doi.org/10.1214/aos/1176343460