Quantile generalized measures of correlation

Abstract

In this paper, we introduce a quantile Generalized Measure of Correlation (GMC) to describe nonlinear quantile relationship between response variable and predictors. The introduced correlation takes values between zero and one. It is zero if and only if the conditional quantile function is equal to the unconditional quantile. We also introduce a quantile partial Generalized Measure of Correlation. Estimators of these correlations are developed. Notably by adopting machine learning methods, our estimation procedures allow the dimension of predictors very large. Under mild conditions, we establish the estimators’ consistency. For construction of confidence interval, we adopt sample splitting and show that the corresponding estimators are asymptotic normal. We also consider composite quantile GMC by integrating information from different quantile levels. Numerical studies are conducted to illustrate our methods. Moreover, we apply our methods to analyze genome-wide association study data from Carworth Farms White mice.