Mathematics Ph.D. Dissertations

Title

Local Distance Correlation: An Extension of Local Gaussian Correlation

Date of Award

2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Mathematical Statistics

First Advisor

Maria Rizzo (Advisor)

Second Advisor

Jari G. Willing (Other)

Third Advisor

Wei Ning (Committee Member)

Fourth Advisor

Junfeng Shang (Committee Member)

Abstract

Distance correlation is a measure of the relationship between random vectors in arbitrary dimension. A sample distance covariance can be formulated in both an unbiased estimator and a biased estimator of distance covariance, where distance correlation is defined as the normalized coefficient of distance covariance.

The jackknife empirical likelihood for a U-statistic by Jing, Yuan, and Zhou (2009) can be applied to a distance correlation since the empirical likelihood method fails in nonlinear statistics. A Wilks' theorem for jackknife empirical likelihood is shown to hold for distance correlation. This research shows how to construct a confidence interval for distance correlation based on jackknife empirical likelihood for a U-statistic, where the sample distance covariance can be represented as a U-statistic. In comparing coverage probabilities of confidence intervals for distance correlation based on jackknife empirical likelihood and bootstrap method, coverage probabilities for the jackknife empirical likelihood show more accuracy.

We propose the estimation and the visualization of local distance correlation by using a local version of the jackknife empirical likelihood. The kernel density functional estimation is used to construct the jackknife empirical likelihood locally. The bandwidth selection for kernel function should minimize the distance between the true density and estimated density.

Local distance correlation has the property that it equals zero in the neighborhood of each point if and only if the two variables are independent in that neighborhood. The estimation and visualization of local distance correlation are shown as accurate to capture the local dependence when compared with the local Gaussian correlation in simulation studies and real examples.

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