On the Test of Association Between Nonparametric Covariate and Error in Semiparametric Regression Model

Consider a semiparametric regression model Y = Z β + m ( X ) + ϵ , with Y being the response variable , X and Z being the covariates , β the unknown parameter, m ( · ) an unknown function preferably a non-linear one, and ϵ the random error . In this article, our objective is to test the independence between X and ϵ only, given the assumption of no relationship between Z and ϵ . Using the concept of Robinson’s (Econometrica 56:931–954, 1988) technique of β estimation at the first stage and then considering a transformed nonparametric model, test statistic is formed on the function of induced order statistics of Y . Thereafter constructing Le Cam’s contiguous alternatives , the local powers of the proposed rank-based test statistic as well as power performances of some other relevant statistics are discussed. Further, in reference to the finite sample simulation study, the power performance of newly introduced test is investigated. Finally, for a real biological data the practicability of the proposed test technique under the setting of semiparametric regression model is judged.