TY - JOUR

T1 - Linear Kernel Tests via Empirical Likelihood for High-Dimensional Data

AU - Ding, Lizhong

AU - Liu, Zhi

AU - Li, Yu

AU - Liao, Shizhong

AU - Liu, Yong

AU - Yang, Peng

AU - Yu, Ge

AU - Shao, Ling

AU - Gao, Xin

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): BAS/1/1624-01-01, URF/1/3007-01-01
Acknowledgements: This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No. URF/1/3007-01-01 and BAS/1/1624-01-01, National Natural Science Foundation of China (No. 61673293), National Natural Science Foundation of China (No. 61703396) and Shenzhen Government (GJHZ20180419190732022).

PY - 2019/9/7

Y1 - 2019/9/7

N2 - We propose a framework for analyzing and comparing distributions without imposing any parametric assumptions via empirical likelihood methods. Our framework is used to study two fundamental statistical test problems: the two-sample test and the goodness-of-fit test. For the two-sample test, we need to determine whether two groups of samples are from different distributions; for the goodness-of-fit test, we examine how likely it is that a set of samples is generated from a known target distribution. Specifically, we propose empirical likelihood ratio (ELR) statistics for the two-sample test and the goodness-of-fit test, both of which are of linear time complexity and show higher power (i.e., the probability of correctly rejecting the null hypothesis) than the existing linear statistics for high-dimensional data. We prove the nonparametric Wilks’ theorems for the ELR statistics, which illustrate that the limiting distributions of the proposed ELR statistics are chi-square distributions. With these limiting distributions, we can avoid bootstraps or simulations to determine the threshold for rejecting the null hypothesis, which makes the ELR statistics more efficient than the recently proposed linear statistic, finite set Stein discrepancy (FSSD). We also prove the consistency of the ELR statistics, which guarantees that the test power goes to 1 as the number of samples goes to infinity. In addition, we experimentally demonstrate and theoretically analyze that FSSD has poor performance or even fails to test for high-dimensional data. Finally, we conduct a series of experiments to evaluate the performance of our ELR statistics as compared to state-of-the-art linear statistics.

AB - We propose a framework for analyzing and comparing distributions without imposing any parametric assumptions via empirical likelihood methods. Our framework is used to study two fundamental statistical test problems: the two-sample test and the goodness-of-fit test. For the two-sample test, we need to determine whether two groups of samples are from different distributions; for the goodness-of-fit test, we examine how likely it is that a set of samples is generated from a known target distribution. Specifically, we propose empirical likelihood ratio (ELR) statistics for the two-sample test and the goodness-of-fit test, both of which are of linear time complexity and show higher power (i.e., the probability of correctly rejecting the null hypothesis) than the existing linear statistics for high-dimensional data. We prove the nonparametric Wilks’ theorems for the ELR statistics, which illustrate that the limiting distributions of the proposed ELR statistics are chi-square distributions. With these limiting distributions, we can avoid bootstraps or simulations to determine the threshold for rejecting the null hypothesis, which makes the ELR statistics more efficient than the recently proposed linear statistic, finite set Stein discrepancy (FSSD). We also prove the consistency of the ELR statistics, which guarantees that the test power goes to 1 as the number of samples goes to infinity. In addition, we experimentally demonstrate and theoretically analyze that FSSD has poor performance or even fails to test for high-dimensional data. Finally, we conduct a series of experiments to evaluate the performance of our ELR statistics as compared to state-of-the-art linear statistics.

UR - http://hdl.handle.net/10754/630864

UR - https://aaai.org/ojs/index.php/AAAI/article/view/4222

U2 - 10.1609/aaai.v33i01.33013454

DO - 10.1609/aaai.v33i01.33013454

M3 - Article

SN - 2374-3468

VL - 33

SP - 3454

EP - 3461

JO - Proceedings of the AAAI Conference on Artificial Intelligence

JF - Proceedings of the AAAI Conference on Artificial Intelligence

ER -