A class of consistent tests of independence between covariate and error in nonparametric regression with missing at random responses

Abstract

In nonparametric regression statistical relationship between covariate and random error is a matter of interest. For a traditional nonparametric regression model, Y = m(X) + ϵ with Y the response, X the covariate, ϵ the random error and m(·) a suitably chosen smooth function the null hypothesis of interest is X and ϵ are independent against all possible alternatives citing dependence between X and ϵ. A valid extension is if for an incomplete data set with several missing responses, similar testing of independence is applicable. In the present article, some observations on Y are missing at random whereas X possesses complete set of observations. Thereafter, in this missing at random (MAR) type data, test of independence between X and ϵ is investigated. Test statistics, based on some availableor modified measures of association are constructed to develop consistent test procedures against a sequence of contiguous alternatives. The asymptotic powers of the test statistics are further determined through a finite sample simulation study. Finally, real data analysis is executed to examine the usefulness of the proposed test statistics.