A new nonparametric lack-of-fit test of nonlinear regression in presence of heteroscedastic variances

In this article, a nonparametric lack-of-fit test of nonlinear regression in presence of heteroscedastic variances is proposed. We consider regression models with a discrete or continuous response variable without distributional assumptions so that the test is widely applicable. The test statistic is developed using a k-nearest neighbor augmentation defined through the ranks of the predictor variable. The asymptotic distribution of the test statistic is derived under the null and local alternatives for the case of using fixed number of nearest neighbors. The parametric standardizing rate is achieved for the asymptotic distribution of the proposed test statistic. This allows the proposed test to have faster convergence rate than most of nonparametric methods. Numerical studies show that the proposed test has good power to detect both low and high frequency alternatives even for moderate sample size. The proposed test is applied to an engineering data example.