Testing for goodness of fit using nonparametric techniques

Abstract

When one fits a parametric model to data it is advisable to test the adequacy of the model through goodness-of-fit techniques. A nuisance of many existing nonparametric tests is that they depend on a smoothing parameter whose choice can be arbitrary. Two nonparametric tests are proposed that overcome this problem by using data-driven smoothing parameters derived from risk estimation procedures. In the first test the regression function is estimated using a Rogosinski-type Fourier series estimator, and the test statistic is a data-driven smoothing parameter that minimizes an unbiased estimator of the risk in estimating the regression. The asymptotic distribution of the test statistic is derived and the consistency of the test under fixed and local alternatives is obtained. The second test is one for checking the adequacy of a parametric regression model. The test statistic is the L2 norm of a Fourier series that is fitted to the residuals from the parametric model. The smoothing parameter is obtained by minimizing a risk criterion. When the null model is linear the distribution of the test statistic does not depend on the regression coefficients; while if the null model is nonlinear the distribution of the test statistic m ay depend on the regression parameters. A bootstrap procedure is recommended for this case. The consistency of the test under fixed alternatives is obtained. A simulation study compares the power of the two proposed tests with some existing tests. The power of the L2--based test is studied when different variance estimators are used in the risk criterion and in the test statistic. The two tests are applied to a data set.