Error variance function estimation in nonparametric regression models

In this article, a new class of variance function estimators is proposed in the setting of heteroscedastic nonparametric regression models. To obtain a variance function estimator, the main proposal is to smooth the product of the response variable and residuals as opposed to the squared residuals. The asymptotic properties of the proposed methodology are investigated in order to compare its asymptotic behavior with that of the existing methods. The finite sample performance of the proposed estimator is studied through simulation studies. The effect of the curvature of the mean function on its finite sample behavior is also discussed.     

Robust nonparametric estimation with missing data

In this paper, under a nonparametric regression model, we introduce two families of robust procedures to estimate the regression function when missing data occur in the response. The first proposal is based on a local M$M$-functional applied to the conditional distribution function estimate adapted to the presence of missing data. The second proposal imputes the missing responses using the local M$M$-smoother based on the observed sample and then estimates the regression function with the completed sample. We show that the robust procedures considered are consistent and asymptotically normally distributed. A robust procedure to select the smoothing parameter is also discussed.     

Robust nonparametric estimation for spatial regression

In this paper, we investigate a nonparametric robust estimation for spatial regression. More precisely, given a strictly stationary random field _Zi=(_Xi,_{Yi)i∈NNN≥1}$Z_{\mathbf{i}}=(X_{\mathbf{i}},Y_{\mathbf{i}}{)}_{\mathbf{i}\in {\mathbb{N}}^{N}N\ge 1}$, we consider a family of robust nonparametric estimators for a regression function based on the kernel method. Under some general mixing assumptions, the almost complete consistency and the asymptotic normality of these estimators are obtained. A robust procedure to select the smoothing parameter adapted to the spatial data is also discussed.     

Adaptive lifting for nonparametric regression

Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2^J for some J. These methods require serious modification or preprocessed data to cope with irregularly spaced data. The lifting scheme is a recent mathematical innovation that obtains a multiscale analysis for irregularly spaced data. A key lifting component is the “predict” step where a prediction of a data point is made. The residual from the prediction is stored and can be thought of as a wavelet coefficient. This article exploits the flexibility of lifting by adaptively choosing the kind of prediction according to a criterion. In this way the smoothness of the underlying ‘wavelet’ can be adapted to the local properties of the function. Multiple observations at a point can readily be handled by lifting through a suitable choice of prediction. We adapt existing shrinkage rules to work with our adaptive lifting methods. We use simulation to demonstrate the improved sparsity of our techniques and improved regression performance when compared to both wavelet and non-wavelet methods suitable for irregular data. We also exhibit the benefits of our adaptive lifting on the real inductance plethysmography and motorcycle data.     

Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

MOSUM monitoring for variance change in nonparametric regression models

ABSTRACT

This article considers the monitoring for variance change in nonparametric regression models. First, the local linear estimator of the regression function is given. A moving square cumulative sum procedure is proposed based on residuals of the estimator. And the asymptotic results of the statistic under the null hypothesis and the alternative hypothesis are obtained. Simulations and Application support our procedure.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

Strong uniform consistency results of the weighted average of conditional artificial data points

In this paper, we study strong uniform consistency of a weighted average of artificial data points. This is especially useful when information is incomplete (censored data, missing data …). In this case, reconstruction of the information is often achieved nonparametrically by using a local preservation of mean criterion for which the corresponding mean is estimated by a weighted average of new data points. The present approach enlarges the possible scope for applications beyond just the incomplete data context and can also be useful to treat the estimation of the conditional mean of specific functions of complete data points. As a consequence, we establish the strong uniform consistency of the Nadaraya–Watson [Nadaraya, E.A., 1964. On estimating regression. Theory Probab. Appl. 9, 141–142; Watson, G.S., 1964. Smooth regression analysis. Sankhyā Ser. A 26, 359–372] estimator for general transformations of the data points. This result generalizes the one of Härdle et al. [Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16, 1428–1449]. In addition, the strong uniform consistency of a modulus of continuity will be obtained for this estimator. Applications of those two results are detailed for some popular estimators.     

A goodness-of-fit test for logistic-normal models using nonparametric smoothing method

Logistic-normal models can be applied for analysis of longitudinal binary data. The aim of this article is to propose a goodness-of-fit test using nonparametric smoothing techniques for checking the adequacy of logistic-normal models. Moreover, the leave-one-out cross-validation method for selecting the suitable bandwidth is developed. The quadratic form of the proposed test statistic based on smoothing residuals provides a global measure for checking the model with categorical and continuous covariates. The formulae of expectation and variance of the proposed statistics are derived, and their asymptotic distribution is approximated by a scaled chi-squared distribution. The power performance of the proposed test for detecting the interaction term or the squared term of continuous covariates is examined by simulation studies. A longitudinal dataset is utilized to illustrate the application of the proposed test.     

More efficient local polynomial regression with random-effects panel data models

We propose a modification on the local polynomial estimation procedure to account for the “within-subject” correlation presented in panel data. The proposed procedure is rather simple to compute and has a closed-form expression. We study the asymptotic bias and variance of the proposed procedure and show that it outperforms the working independence estimator uniformly up to the first order. Simulation study shows that the gains in efficiency with the proposed method in the presence of “within-subject” correlation can be significant in small samples. For illustration purposes, the procedure is applied to explore the impact of market concentration on airfare.     

Estimating additive models with missing responses

ABSTRACT

For multivariate regressors, the Nadaraya–Watson regression estimator suffers from the well-known curse of dimensionality. Additive models overcome this drawback. To estimate the additive components, it is usually assumed that we observe all the data. However, in many applied statistical analysis missing data occur. In this paper, we study the effect of missing responses on the additive components estimation. The estimators are based on marginal integration adapted to the missing situation. The proposed estimators turn out to be consistent under mild assumptions. A simulation study allows to compare the behavior of our procedures, under different scenarios.     

Local linear kernel estimation for discontinuous nonparametric regression functions

In this paper, we consider using a local linear (LL) smoothing method to estimate a class of discontinuous regression functions. We establish the asymptotic normality of the integrated square error (ISE) of a LL-type estimator and show that the ISE has an asymptotic rate of convergence as good as for smooth functions, and the asymptotic rate of convergence of the ISE of the LL estimator is better than that of the Nadaraya-Watson (NW) and the Gasser-Miiller (GM) estimators.     

Geoadditive models

Summary. A study into geographical variability of reproductive health outcomes (e.g. birth weight) in Upper Cape Cod, Massachusetts, USA, benefits from geostatistical mapping or kriging . However, also observed are some continuous covariates (e.g. maternal age) that exhibit pronounced non-linear relationships with the response variable. To account for such effects properly we merge kriging with additive models to obtain what we call geoadditive models . The merging becomes effortless by expressing both as linear mixed models. The resulting mixed model representation for the geoadditive model allows for fitting and diagnosis using standard methodology and software.     

Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.     

Component selection in additive quantile regression models

Nonparametric additive models are powerful techniques for multivariate data analysis. Although many procedures have been developed for estimating additive components both in mean regression and quantile regression, the problem of selecting relevant components has not been addressed much especially in quantile regression. We present a doubly-penalized estimation procedure for component selection in additive quantile regression models that combines basis function approximation with a ridge-type penalty and a variant of the smoothly clipped absolute deviation penalty. We show that the proposed estimator identifies relevant and irrelevant components consistently and achieves the nonparametric optimal rate of convergence for the relevant components. We also provide an accurate and efficient computation algorithm to implement the estimator and demonstrate its performance through simulation studies. Finally, we illustrate our method via a real data example to identify important body measurements to predict percentage of body fat of an individual.     

Rate-optimal nonparametric estimation in classical and Berkson errors-in-variables problems

We consider nonparametric estimation of a regression curve when the data are observed with Berkson errors or with a mixture of classical and Berkson errors. In this context, other existing nonparametric procedures can either estimate the regression curve consistently on a very small interval or require complicated inversion of an estimator of the Fourier transform of a nonparametric regression estimator. We introduce a new estimation procedure which is simpler to implement, and study its asymptotic properties. We derive convergence rates which are faster than those previously obtained in the literature, and we prove that these rates are optimal. We suggest a data-driven bandwidth selector and apply our method to some simulated examples.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.