A note on nonparametric testing for Gaussian innovations in AR-ARCH models

In this paper we consider autoregressive models with conditional autoregressive variance, including the case of homoscedastic AR-models and the case of ARCH models. Our aim is to test the hypothesis of normality for the innovations in a completely nonparametric way, i. e. without imposing parametric assumptions on the conditional mean and volatility functions. To this end the Cram\'er-von Mises test based on the empirical distribution function of nonparametrically estimated residuals is shown to be asymptotically distribution-free. We demonstrate its good performance for finite sample sizes in 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. --empirical process of residuals,testing for symmetry,nonparametric regression

The Two-Sample Problem with Regression Errors : An Empirical Process Approach

We describe how to test the null hypothesis that errors from two parametrically specified regression models have the same distribution versus a general alternative. First we obtain the asymptotic properties of teststatistics derived from the difference between the two residual-based empirical distribution functions. Under the null distribution they are not asymptotically distribution free and, hence, a consistent bootstrap procedure is proposed to compute critical values. As an alternative, we describe how to perform the test with statistics based on martingale-transformed empirical processes, which are asymptotically distribution free. Some Monte Carlo experiments are performed to compare the behaviour of all statistics with moderate sample sizes. --

A note on uniform consistency of monotone function estimators

Recently, Dette, Neumeyer and Pilz (2005a) proposed a new monotone estimator for strictly increasing nonparametric regression functions and proved asymptotic normality. We explain two modifications of their method that can be used to obtain monotone versions of any nonparametric function estimators, for instance estimators of densities, variance functions or hazard rates. The method is appealing to practitioners because they can use their favorite method of function estimation (kernel smoothing, wavelets, orthogonal series,...) and obtain a monotone estimator that inherits desirable properties of the original estimator. In particular, we show that both monotone estimators share the same rates of uniform convergence (almost sure or in probability) as the original estimator

Heteroscedastic semiparametric transformation models: estimation and testing for validity

In this paper we consider a heteroscedastic transformation model, where the transformation belongs to a parametric family of monotone transformations, the regression and variance function are modelled nonparametrically and the error is independent of the multidimensional covariates. In this model, we first consider the estimation of the unknown components of the model, namely the transformation parameter, regression and variance function and the distribution of the error. We show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and establish the limiting distribution of the test statistics under the null hypothesis. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, we carry out a simulation study to verify the small sample behavior of the proposed estimators and tests.Comment: 33 pages, 1 figur