Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model

Under the setup of a generalized partially linear model <I>Y</I> = <i>β</i><sub>1</sub><I>X</I><sub>1</sub>+ ... + <i>β<sub>p</sub>X<sub>p</sub></i>+ <i>m</i>(<I>W</I><sub>1</sub>,...,<i>W<sub>q</sub></i>) + ϵ with p parametric regressors <I>X</I><sub>1</sub>,..., <i>X<sub>p</sub></i> and <i>q</i> nonparametric regressors <I>W</I><sub>1</sub>,..., <i>W<sub>q</sub></i>, we are motivated to test on the independence between all the (<i>p</i> + <i>q</i>) regressors and the random error <i>ϵ</i>. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, <i>m</i>(<I>W</I><sub>1</sub>,..., <i>W<sub>q</sub></i>) is a Lipschitz continuous function defined on ℝ<sup>q</sup>→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (<i>p</i> + <i>q</i>) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the <i>r</i>-th order difference of estimated response and the <i>r</i>-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the <i>r</i>-th differences of the estimated and observed <I>Y</I>. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.