Dependence testing in generalized partially linear regression: asymptotic power of Kendall’s <i>τ</i>

In this study, we are motivated to study the asymptotic power of Kendall's within the framework of a generalized partially linear regression model, given by Ye where (X 1 , . . ., X p ) represents parametric regressors p, (V 1 , . . ., V q ) are nonparametric regressors q and e denotes the error term.The model incorporates p unknown parameters b 1 , . . ., b p and an unknown smooth function (, . . ., ).Our main objective is to test for dependence between the combined set of regressors S = (X 1 , . . ., X p , V 1 , . . ., V q ) and the error term e.Under the null hypothesis of independence, the general-order differences between observed responses and their estimated counterparts remain independent.To construct an appropriate test, we introduce statistics based on V-statistics, derived from bivariate observations formed using these differences.These statistics act as empirical analogs of Kendall's , as originally proposed in Kendall (1938).Additionally, we consider the test statistics based on the nonparametric measures * and dCov in analogous manner to carry out a comparative power analysis under a sequence of local alternatives.By specifying different conditional distributions for the error term e, we assess the asymptotic power through the limiting distribution of a non-degenerate V-statistic.For a real dataset, we compare their effectiveness by evaluating p-values and finite sample powers.