Testing heteroscedasticity by wavelets in a nonparametric regression model

In the nonparametric regression models, a homoscedastic structure is usually assumed. However, the homoscedasticity cannot be guaranteed a priori. Hence, testing the heteroscedasticity is needed. In this paper we propose a consistent nonparametric test for heteroscedasticity, based on wavelets. The empirical wavelet coefficients of the conditional variance in a regression model are defined first. Then they are shown to be asymptotically normal, based on which a test statistic for the heteroscedasticity is constructed by using Fan's wavelet thresholding idea. Simulations show that our test is superior to the traditional nonparametric test.     

Testing independence in nonparametric regression

We propose a new test for independence of error and covariate in a nonparametric regression model. The test statistic is based on a kernel estimator for the L₂-distance between the conditional distribution and the unconditional distribution of the covariates. In contrast to tests so far available in literature, the test can be applied in the important case of multivariate covariates. It can also be adjusted for models with heteroscedastic variance. Asymptotic normality of the test statistic is shown. Simulation results and a real data example are presented.     

Testing strict monotonicity in nonparametric regression

A new test for strict monotonicity of the regression function is proposed which is based on a composition of an estimate of the inverse of the regression function with a common regression estimate. This composition is equal to the identity if and only if the “true” regression function is strictly monotone, and a test based on an L ²-distance is investigated. The asymptotic normality of the corresponding test statistic is established under the null hypothesis of strict monotonicity.      

Testing for no effect in nonparametric regression via spline smoothing techniques

We propose three statistics for testing that a predictor variable has no effect on the response variable in regression analysis. The test statistics are integrals of squared derivatives of various orders of a periodic smoothing spline fit to the data. The large sample properties of the test statistics are investigated under the null hypothesis and sequences of local alternatives and a Monte Carlo study is conducted to assess finite sample power properties.     

Testing nonparametric and semiparametric hypotheses in vector stationary processes

We propose a general nonparametric approach for testing hypotheses about the spectral density matrix of multivariate stationary time series based on estimating the integrated deviation from the null hypothesis. This approach covers many important examples from interrelation analysis such as tests for noncorrelation or partial noncorrelation. Based on a central limit theorem for integrated quadratic functionals of the spectral matrix, we derive asymptotic normality of a suitably standardized version of the test statistic under the null hypothesis and under fixed as well as under sequences of local alternatives. The results are extended to cover also parametric and semiparametric hypotheses about spectral density matrices, which includes as examples goodness-of-fit tests and tests for separability.     

Testing heteroscedasticity in nonparametric regression models based on residual analysis

The importance of detecting heteroscedasticity in regression analysis is widely recognized because efficient inference for the regression function requires that heteroscedasticity should be taken into account. In this paper, a simple test for heteroscedasticity is proposed in nonparametric regression based on residual analysis. Furthermore, some simulations with a comparison with Dette and Munk＇s method are conducted to evaluate the performance of the proposed test. The results demonstrate that the method in this paper performs quite satisfactorily and is much more powerful than Dette and Munk＇s method in some cases.     

Plug-in method for nonparametric regression

The problem of bandwidth selection for non-parametric kernel regression is considered. We will follow the Nadaraya–Watson and local linear estimator especially. The circular design is assumed in this work to avoid the difficulties caused by boundary effects. Most of bandwidth selectors are based on the residual sum of squares (RSS). It is often observed in simulation studies that these selectors are biased toward undersmoothing. This leads to consideration of a procedure which stabilizes the RSS by modifying the periodogram of the observations. As a result of this procedure, we obtain an estimation of unknown parameters of average mean square error function (AMSE). This process is known as a plug-in method. Simulation studies suggest that the plug-in method could have preferable properties to the classical one. Supported by the MSMT: LC 06024.     

Error inference for nonparametric regression

This study examines means for inferring the distribution of the error in nonparametric regression. The central objective is to develop confidence intervals for nonparametric regression. Our computational study would seem to affirm that our methods are potentially useful in cases of small sample size or heterogeneously distributed error. Theoretical developments offer sufficient conditions for asymptotic normality.This work was undertaken while Dr. Rutherford was with the University of Arizona. It was supported in part by NSF grant DPP 82-19439.     

Two tests for heteroscedasticity in nonparametric regression

In this paper, two new tests for heteroscedasticity in nonparametric regression are presented and compared. The first of these tests consists in first estimating nonparametrically the unknown conditional variance function and then using a classical least-squares test for a general linear model to test whether this function is a constant. The second test is based on using an overall distance between a nonparametric estimator of the conditional variance function and a parametric estimator of the variance of the model under the assumption of homoscedasticity. A bootstrap algorithm is used to approximate the distribution of this test statistic. Extended versions of both procedures in two directions, first, in the context of dependent data, and second, in the case of testing if the variance function is a polynomial of a certain degree, are also described. A broad simulation study is carried out to illustrate the finite sample performance of both tests when the observations are independent and when they are dependent.     

Bandwidth selection for a data sharpening estimator in nonparametric regression

This paper is concerned with data-based selection of the bandwidth for a data sharpening estimator in nonparametric regression. Two kinds of bandwidths are considered: a bandwidth vector which has a different bandwidth for each covariate, and a scalar bandwidth that is common for all covariates. A plug-in method is developed and its theoretical performance is fully investigated. The proposed plug-in method works efficiently in our simulation study.     

Optimal designs for nonparametric kernel regression

We consider the fixed design regression model Y_i = g(t_i) + ξ_i, i = 1, …, n, where ξ_i are (not necessarily i.i.d.) no variables, t_i constitute the design points where nonrepeatable measurements are to be taken and Y_i are the observations from which g and its derivatives are to be estimated. The dependency of the Integrated Mean Squared Error of two different types of kernel estimates on the design {t₁, …, t_n} is established. This allows the derivation of asymptotically optimal designs.     

Robust estimation for nonparametric generalized regression

This paper focuses on nonparametric regression estimation for the parameters of a discrete or continuous distribution, such as the Poisson or Gamma distributions, when anomalous data are present. The proposal is a natural extension of robust methods developed in the setting of parametric generalized linear models. Robust estimators bounding either large values of the deviance or of the Pearson residuals are introduced and their asymptotic behaviour is derived. Through a Monte Carlo study, for the Poisson and Gamma distributions, the finite properties of the proposed procedures are investigated and their performance is compared with that of the classical ones. A resistant cross-validation method to choose the smoothing parameter is also considered.     

Cardinal splines in nonparametric regression

We discuss a nonparametric regression model on an equidistant grid of the real line. A class of kernel type estimates based on the so-called fundamental cardinal splines will be introduced. Asymptotic optimality of these estimates will be established for certain functional classes. This model explains the often mentioned heuristic fact that cubic splines are adequate for most practical applications.      

Optimal testing for additivity in multiple nonparametric regression

We consider the problem of testing for additivity in the standard multiple nonparametric regression model. We derive optimal (in the minimax sense) non- adaptive and adaptive hypothesis testing procedures for additivity against the composite nonparametric alternative that the response function involves interactions of second or higher orders separated away from zero in L ²([0, 1] ^d )-norm and also possesses some smoothness properties. In order to shed some light on the theoretical results obtained, we carry out a wide simulation study to examine the finite sample performance of the proposed hypothesis testing procedures and compare them with a series of other tests for additivity available in the literature.     

Adaptive nonparametric estimation of a multivariate regression function

We consider the kernel estimation of a multivariate regression function at a point. Theoretical choices of the bandwidth are possible for attaining minimum mean squared error or for local scaling, in the sense of asymptotic distribution. However, these choices are not available in practice. We follow the approach of Krieger and Pickands (Ann. Statist.9 (1981) 1066–1078) and Abramson (J. Multivariate Anal.12 (1982), 562–567) in constructing adaptive estimates after demonstrating the weak convergence of some error process. As consequences, efficient data-driven consistent estimation is feasible, and data-driven local scaling is also feasible. In the latter instance, nearest-neighbor-type estimates and variance-stabilizing estimates are obtained as special cases.     

On confidence bands for multivariate nonparametric regression

In a multivariate nonparametric regression problem with fixed, deterministic design asymptotic, uniform confidence bands for the regression function are constructed. The construction of the bands is based on the asymptotic distribution of the maximal deviation between a suitable nonparametric estimator and the true regression function which is derived by multivariate strong approximation methods and a limit theorem for the supremum of a stationary Gaussian field over an increasing system of sets. The results are derived for a general class of estimators which includes local polynomial estimators as a special case. The finite sample properties of the proposed asymptotic bands are investigated by means of a small simulation study.     

A nonparametric lack-of-fit test for heteroscedastic regression models

A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residual plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can cope with heteroscedastic errors and no replicates. Recursive formulae allowing one to calculate the exact distribution of the test statistic under the null hypothesis and under a class of alternative hypotheses are given.     

Asymptotic equivalence for nonparametric regression with non-regular errors

Asymptotic equivalence in Le Cam’s sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size n and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.