Checking Heteroscedasticity in Partially Linear Single-Index Models Using Pairwise Distance

Abstract

In this article, a new test is proposed for partially linear single-index models (PLSIM) based on the pairwise distances of the sample points, to test heteroscedasticity. The statistic can be formulated as a U statistic and does not have to estimate the conditional variance function by using nonparametric methods, such as kernel, local polynomial, or spline. We derive a computationally feasible approximation to deal with the complexity of the limit zero distribution under the null hypothesis. We prove that the proposed bootstrap procedure is valid approximation to the null distribution of the test. It shows that this statistic has an asymptotically normal distribution. The algorithmic program of this test method is easy to implement and has faster convergence than some existing methods. In addition, convergence rate of the statistic does not depend on the dimensions of the covariates, which greatly reduces the impact of the dimensional curse. Finally, we give the numerical simulations and a real data example. INDEX TERMS Dimension reduction, heteroscedasticity, partially linear single-index models. I. INTRODUCTION H 0 : ∃ σ 2 > 0, E(ε 2 |X , Z) = σ 2 (X , Z) = σ 2 , H 1 : ∀ σ 2 > 0, E(ε 2 |X , Z) = σ 2. (2) Under H 0 , the constant σ 2 is an unconditional variance E(ε 2). Consequently, the heteroscedasticity test in (2) is equivalent to determining whether the conditional variance function E(ε 2 |X , Z) is equal to the unconditional variance E(ε 2). Many authors have studied the heteroscedasticity test of common regression model, such as the literature [4