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

Sthitadhi  Das; Soutik  Halder; Saran  Ishika Maiti

Authors

Sthitadhi Das Department of Statistics, Institute of Sciences, Visva Bharati, Santiniketan, India
Soutik Halder Department of Statistics, Institute of Sciences, Visva Bharati, Santiniketan, India
Saran Ishika Maiti Department of Statistics, Institute of Sciences, Visva Bharati, Santiniketan, India

Keywords:

Kendall’s tau, measures of association, asymptotic power, contiguous alternative, nonparametric regression model

Abstract

In 2014, Bergsma et al. (2014) proposed a generalized measure of association $gif.latex?\tau&space;^*$ as an extension
of widely used Kendall’s $gif.latex?\tau$ . Later, in testing of independence between error and covariate, under
nonparametric regression model $gif.latex?Y=m(x)+\varepsilon,$ with unknown regression function $gif.latex?m$ and observation error $gif.latex?\varepsilon,$ test statistic tailored on $gif.latex?\tau&space;^*$ was suggested by Dhar et al. (2018). In this article, we develop a test, constructed on further extension of $gif.latex?\tau&space;^*,$ considering the ordered $gif.latex?X$ and the third order difference of $gif.latex?Y$ 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.