An Extended Approach to Test of Independence between Error and Covariates under Nonparametric Regression Model

Abstract

In 2014, Bergsma et al. (2014) proposed a generalized measure of association τ∗ as an extension of widely used Kendall’s τ. Later, in testing of independence between error and covariate, under nonparametric regression model Y = m(X) + ϵ, with unknown regression function m and observation error ϵ, test statistic tailored on τ ∗ was suggested by Dhar et al. (2018). In this article, we develop a test, constructed on further extension of τ ∗, considering the ordered X and the third order difference of Y with an motive to address the same issue of independence. We deduce the asymptotic distributions of test statistics using the theory of degenerate U-statistics. Moreover, we unravel the power of the proposed tests using Le Cam’s concept of contiguous alternatives. A couple of simulated examples on normal and non normal distribution are furnished. Also, the performance of the test statistics is honed through a real data analysis.