Natalie Neumeyer

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^-^1^/^2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.

A Note on Nonparametric Estimation of the Effective Dose in Quantal Bioassay

For the common binary response model, we propose a direct method for the nonparametric estimation of the effective dose level. The estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate. The new method yields a simple and reliable monotone estimate of the effective dose-level curve α → EDα and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators, or smoothing splines. Moreover, it is computationally very efficient, because it does not require a numerical inversion of a monotonized estimate of the quantile dose-response curve. We prove asymptotic normality of the new estimate and compare it with an available alternative estimate (based on a monotonized nonparametric estimate of the dose-response curve and calculation of the inverse function) by means of a simulation study.

Testing for Symmetric Error Distribution in Nonparametric Regression Models

For the problem of testing symmetry of the error distribution in a nonparametric regression model we propose as a test statistic the difference between the two empirical distribution functions of estimated residuals and their counterparts with opposite signs. The weak convergence of the difference process to a Gaussian process is shown. The covariance structure of this process depends heavily on the density of the error distribution, and for this reason the performance of a symmetric wild bootstrap procedure is discussed in asymptotic theory and by means of a simulation study. In contrast to the available procedures the new test is also applicable under heteroscedasticity.

Testing symmetry in nonparametric regression models

In a recent paper Ahmad and Li (1997) proposed a new test for symmetry of the error distribution in linear regression models and proved asymptotic normality for the distribution of the corresponding test statistic under the null hypothesis and consistency under fixed alternatives. The present paper has three purposes. On the one hand we derive the asymptotic distribution of the statistic considered by Ahmad and Li (1997) under fixed alternatives and demonstrate that asymptotic normality is still valid but with a different rate of convergence. On the other hand we generalize Ahmad and Lis (1997) test of a symmetric error distribution to general nonparametric regression models. Moreover, it is also demonstrated that a bootstrap version of the new test for symmetry has good finite sample properties.

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

Summary In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non-parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution-free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness-of-fit testing of assumptions regarding the error distribution in linear and non-parametric regression models.

A note on testing symmetry of the error distribution in linear regression models

In the classical linear regression model, the problem of testing for symmetry of the error distribution is considered. The test statistic is a functional of the difference between the two empirical distribution functions of the estimated residuals and their counterparts with opposite signs. The weak convergence of the difference process to a Gaussian process is established. The covariance structure of this process depends heavily on the density of the error distribution, and for this reason, the performance of a symmetric wild bootstrap procedure is discussed in asymptotic theory and by means of a simulation study.

A Simple Nonparametric Estimator of a Monotone Regression Function

In this paper a new method for monotone estimation of a regression function is proposed. The estimator is obtained by the combination of a density and a regression estimate and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators or smoothing splines. The main idea of the new approach is to construct a density estimate from the estimated values ˆm(i/N) (i = 1, . . . ,N) of the regression function to use these “data” for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the conventially used techniques for monotone estimation the new method is computationally more efficient, because it does not require constrained optimization techniques for the calculation of the estimate. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular it is shown that for kernel estimates or local polynomials the monotone estimate is first order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study.

Empirical likelihood estimators for the error distribution in nonparametric regression models

The aim of this paper is to show that existing estimators for the error distribution in non-parametric regression models can be improved when additional information about the distribution is included by the empirical likelihood method. The weak convergence of the resulting new estimator to a Gaussian process is shown and the performance is investigated by comparison of asymptotic mean squared errors and by means of a simulation study.

A note on one-sided nonparametric analysis of covariance by ranking residuals

In a recent paper Speckman et al. (2002) introduced a technique for accounting covariates when their e ects are nonlinear. They proposed a test for a one-sided analysis of covariance which is based on a rank test for the residuals obtained by smoothing the dependent variable on the covariate. In this paper we study some of the asymptotic properties of this test and a modi cations of the test which try to take into account di erent sizes of the variances in both samples.

Significance testing in quantile regression

We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that in contrast to the nonparametric approach based on estimation of L2-distances the new test is able to detect local alternatives which converge to the null hypothesis with any rate an → 0 such that an √ n → ∞ (here n denotes the sample size). We also present a small simulation study illustrating the finite sample properties of a bootstrap version of the the corresponding Kolmogorov-Smirnov test. AMS Classification: 62G10, 62G08, 62G30