Scalable kernel two-sample tests via empirical likelihood and jackknife

Abstract

Two-sample hypothesis testing is a classical problem in statistics. Given i.i.d. samples and the two-sample testing is to test whether P is significantly different from Q or not. Among the various statistical tests, the maximum mean discrepancy (MMD) has been proven to have excellent performance compared to traditional methods. In application of MMD, we are confronted with two challenges: (I) For large data set, MMD is computationally demanding with computational cost (II) The limiting distribution under the null hypothesis usually contains infinitely many unknown parameters. In this paper, we propose two testing methods that solve (I) and (II) respectively. To address (I), the data are independently and evenly divided into k groups, calculate the MMD on each group and then combine them by empirical likelihood to yield a test statistic It converges in distribution to the chi-square distribution with degree of freedom one ( ) under the null hypothesis as k goes to infinity. To address (II), we propose to compute the two-sample jackknife pseudo-values on each group and combine them by empirical likelihood to get a new test statistic Under the null hypothesis, converges in distribution to if The simulation and real data application indicates that the proposed tests performs well.