Bootstrap tests for the error distribution in linear and nonparametric regression models

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and nonparametric 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 nonparametric regression models. --goodness-of-fit,residual process,parametric bootstrap,linear model,analysis of variance,M-estimation,nonparametric regression

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. --M-estimation,goodness-of-fit tests,testing for symmetry,empirical process of residuals,linear model

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 nonparametric 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. As a by-product of our proofs we obtain stochastic expansions for smooth linear estimators based on residuals from the nonparametric regression model. --empirical distribution function,empirical likelihood,error distribution,estimating function,nonparametric regression,Owen estimator

Bootstrap Tests for the Error Distribution in Linear and Nonparametric Regression Models

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and nonparametric 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 nonparametric 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

Tests in a Case-Control Design Including Relatives

We present a new approach to handle dependencies within the general framework of case-control designs, illustrating our approach by a particular application from the field of genetic epidemiology. The method is derived for parent-offspring trios, which will later be relaxed to more general family structures. For applications in genetic epidemiology we consider tests on equality of allele frequencies among cases and controls utilizing well-known risk measures to test for independence of phenotype and genotype at the observed locus. These test statistics are derived as functions of the entries in the associated contingency table containing the numbers of the alleles under consideration in the case and the control group. We find the joint asymptotic distribution of these entries, which enables us to derive critical values for any test constructed on this basis. A simulation study reveals the finite sample behavior of our test statistics. --association tests,contingency tables,dependent data,risk measures

Empirical-Likelihood-Schätzung der Fehlerverteilung im nichtparametrischen Regressionsmodell

In der vorliegenden Arbeit wird ein neuer Schätzer für dieVerteilungsfunktion der Fehlervariablen im nichtparametrischen Regressionsmodell unter Verwendung von Zusatzinformationen hinsichtlich der Fehlerverteilung mithilfe des Empirical Likelihood-Ansatzes konstruiert. Für verschiedene Typen von Zusatzinformationen und im homo- und heteroskedastischen Fall wird dieser Schätzer asymptotisch untersucht, indem die schwache Konvergenz des zugehörigen stochastischen Prozesses gezeigt wird. Das Verhalten des Schätzers wird außerdem an Beispielen und für endliche Stichprobengröße dargestellt. Darüber hinaus werden Tests auf die Gültigkeit der Zusatzinformation vorgestellt und untersucht