Copula Versions of RKHS-Based and Distance-Based Criteria

open_access
Open Access
Author:
Lin, Junli
Graduate Program:
Statistics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
June 15, 2017
Committee Members:
Michael Akritas, Dissertation Advisor/Co-Advisor
Michael Akritas, Committee Chair/Co-Chair
Matthew Reimherr, Committee Member
Bharath Sriperumbudur, Committee Member
Jesse Barlow, Outside Member
Keywords:
kernel methods
permutation method
two-sample problems
copula
kernel methods
permutation methods
V-statistic
Abstract:
Four general classes of statistics in hypothesis testing and corresponding measures are those based on reproducing kernels or distances. Among the most popular criteria for independence between two random vectors X and Y are the distance covariance (dCov) and the Hilbert-Schmidt independence criterion (HSIC). Among the most popular criteria for equal distributions of two random vectors X and Y are the Maximum Mean Discrepancy (MMD) and an energy distance (eD) criterion. Copula versions of these criteria are introduced. The estimators of the proposed criteria belong in the class of rank transform statistics and share the important property of being invariant under monotone transformations of each variable. The asymptotic theory is established under alternative hypothesis for the first two proposed statistics, and under null hypothesis for all the four proposed statistics, in which general distributions are allowed by employing mid-ranks. Dealing with the non-differentiability of the Euclidean norm, in combination with mid-ranks, presents methodological and notation challenges which are dealt with by novel arguments. Conservative tests, as well as linear time statistics for the first two proposed methods are also developed. Simulation studies suggest superior performance of the proposed statistics for certain classes of distributions.