Nonparametric lack-of-fit tests in presence of heteroscedastic variances

Abstract

It is essential to test the adequacy of a specified regression model in order to have cor- rect statistical inferences. In addition, ignoring the presence of heteroscedastic errors of regression models will lead to unreliable and misleading inferences. In this dissertation, we consider nonparametric lack-of-fit tests in presence of heteroscedastic variances. First, we consider testing the constant regression null hypothesis based on a test statistic constructed using a k-nearest neighbor augmentation. Then a lack-of-fit test of nonlinear regression null hypothesis is proposed. For both cases, 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. Numerical studies and real data analyses are presented to evaluate the perfor- mance of the proposed tests. Advantages of our tests compared to classical methods include: (1) The response variable can be discrete or continuous and can have variations depend on the predictor. This allows our tests to have broad applicability to data from many practi- cal fields. (2) Using fixed number of k-nearest neighbors avoids slow convergence problem which is a common drawback of nonparametric methods that often leads to low power for moderate sample sizes. (3) We obtained the parametric standardizing rate for our test statis- tics, which give more power than smoothing based nonparametric methods for intermediate sample sizes. The numerical simulation studies show that our tests are powerful and have noticeably better performance than some well known tests when the data were generated from high frequency alternatives. Based on the idea of the Least Squares Cross-Validation (LSCV) procedure of Hardle and Mammen (1993), we also proposed a method to estimate the number of nearest neighbors for data augmentation that works with both continuous and discrete response variable.