A plug-in technique in nonparametric regression with dependence

The problem addressed is that of smoothing parameter selection in kernel nonparametric regression in the fixed design regression model with dependent noise. An asymptotic expression of the optimum bandwidth parameter has been obtained in recent studies, where this takes the form h = C ₀ n ^?1/5. This paper proposes to use a plug-in methodology, in order to obtain an optimum estimation of the bandwidth parameter, through preliminary estimation of the unknown value of C ₀.     

Discrete Nonparametric Kernel and Parametric Methods for the Modeling of Pavement Deterioration

This article is concerned with one discrete nonparametric kernel and two parametric regression approaches for providing the evolution law of pavement deterioration. The first parametric approach is a survival data analysis method; and the second is a nonlinear mixed-effects model. The nonparametric approach consists of a regression estimator using the discrete associated kernels. Some asymptotic properties of the discrete nonparametric kernel estimator are shown as, in particular, its almost sure consistency. Moreover, two data-driven bandwidth selection methods are also given, with a new theoretical explicit expression of optimal bandwidth provided for this nonparametric estimator. A comparative simulation study is realized with an application of bootstrap methods to a measure of statistical accuracy.     

Robust plug-in bandwidth estimators in nonparametric regression

In this paper, we propose a robust bandwidth selection method for local M-estimates used in nonparametric regression. We study the asymptotic behavior of the resulting estimates. We use the results of a Monte Carlo study to compare the performance of various competitors for moderate samples sizes. It appears that the robust plug-in bandwidth selector we propose compares favorably to its competitors, despite the need to select a pilot bandwidth. The Monte Carlo study shows that the robust plug-in bandwidth selector is very stable and relatively insensitive to the choice of the pilot.     

On inference for a semiparametric partially linear regression model with serially correlated errors

The authors consider a semiparametric partially linear regression model with serially correlated errors. They propose a new way of estimating the error structure which has the advantage that it does not involve any nonparametric estimation. This allows them to develop an inference procedure consisting of a bandwidth selection method, an efficient semiparametric generalized least squares estimator of the parametric component, a goodness‐of‐fit test based on the bootstrap, and a technique for selecting significant covariates in the parametric component. They assess their approach through simulation studies and illustrate it with a concrete application.     

Local maximum likelihood estimation and inference

Local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique. It extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces. Associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment. This paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation. The approach is then applied to least squares nonparametric regression and to nonparametric logistic regression. Our experiences in these two settings show that the general idea outlined here is powerful and encouraging.     

Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoothing spline estimators. Each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set. In this paper an improved version of a criterion based on the Akaike information criterion (AIC), termed AIC_C, is derived and examined as a way to choose the smoothing parameter. Unlike plug-in methods, AIC_C can be used to choose smoothing parameters for any linear smoother, including local quadratic and smoothing spline estimators. The use of AIC_C avoids the large variability and tendency to undersmooth (compared with the actual minimizer of average squared error) seen when other 'classical' approaches (such as generalized cross-validation (GCV) or the AIC) are used to choose the smoothing parameter. Monte Carlo simulations demonstrate that the AIC_C-based smoothing parameter is competitive with a plug-in method (assuming that one exists) when the plug-in method works well but also performs well when the plug-in approach fails or is unavailable.     

A plug-in the number of knots selector for polynomial spline regression

A plug-in the number of interior knots (NIKs) selector is proposed for polynomial spline estimation in nonparametric regression. The existence and properties of the optimal NIKs for spline regression are established by minimising the weighted mean integrated squared error. We obtain plug-in formulae for the optimal NIKs based on the theoretical results of asymptotic optimality, and develop strategies for choosing the NIKs of the spline estimator. The proposed NIKs selection method is tested on our simulated data with quite satisfactory performance, and is illustrated by analysing a fossil data set.     

On the nonparametric smooth estimation of the reversed Hazard Rate function

The Reversed Hazard Rate (RHR) function is an important measure as a tool in the analysis of the reliability of both natural and man-made systems. In this paper, we present several new estimators of the RHR function using nonparametric techniques. These estimators are obtained by incorporating different binning techniques with fixed design local polynomial regression. We show that these estimators are asymptotically unbiased and consistent and, to determine the bandwidth, we propose two simple yet efficient plug-in bandwidth selection methods for even and odd order local polynomial estimators. Simulated and real life data are subsequently used to evaluate the performances of these estimators.     

Nonparametric estimation of a switching regression model

High-dimensional data often exhibit multi-collinearity, leading to unstable regression coefficients. To address sample selection bias and problems associated with high dimensionality, principal components were extracted and used as predictors in a switching regression model. Since principal component regression often results to decline in predictive ability due to the selection of few principal components, we formulate the model with nonparametric function of principal components in lieu of individual predictors. Simulation studies indicated better predictive ability for nonparametric principal component switching regression over the parametric counterpart while mitigating the adverse effects of multi-collinearity and high dimensionality.     

Bandwidth selection for local linear regression smoothers

Summary. The paper presents a general strategy for selecting the bandwidth of nonparametric regression estimators and specializes it to local linear regression smoothers. The procedure requires the sample to be divided into a training sample and a testing sample. Using the training sample we first compute a family of regression smoothers indexed by their bandwidths. Next we select the bandwidth by minimizing the empirical quadratic prediction error on the testing sample. The resulting bandwidth satisfies a finite sample oracle inequality which holds for all bounded regression functions. This permits asymptotically optimal estimation for nearly any regression function. The practical performance of the method is illustrated by a simulation study which shows good finite sample behaviour of our method compared with other bandwidth selection procedures.     

A new information criterion-based bandwidth selection method for non-parametric regressions

ABSTRACT

Local linear estimator is a popularly used method to estimate the non-parametric regression functions, and many methods have been derived to estimate the smoothing parameter, or the bandwidth in this case. In this article, we propose an information criterion-based bandwidth selection method, with the degrees of freedom originally derived for non-parametric inferences. Unlike the plug-in method, the new method does not require preliminary parameters to be chosen in advance, and is computationally efficient compared to the cross-validation (CV) method. Numerical study shows that the new method performs better or comparable to existing plug-in method or CV method in terms of the estimation of the mean functions, and has lower variability than CV selectors. Real data applications are also provided to illustrate the effectiveness of the new method.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

Multivariate bandwidth selection for local linear regression

The existence and properties of optimal bandwidths for multivariate local linear regression are established, using either a scalar bandwidth for all regressors or a diagonal bandwidth vector that has a different bandwidth for each regressor. Both involve functionals of the derivatives of the unknown multivariate regression function. Estimating these functionals is difficult primarily because they contain multivariate derivatives. In this paper, an estimator of the multivariate second derivative is obtained via local cubic regression with most cross-terms left out. This estimator has the optimal rate of convergence but is simpler and uses much less computing time than the full local estimator. Using this as a pilot estimator, we obtain plug-in formulae for the optimal bandwidth, both scalar and diagonal, for multivariate local linear regression. As a simpler alternative, we also provide rule-of-thumb bandwidth selectors. All these bandwidths have satisfactory performance in our simulation study.     

Bias-corrected confidence bands in nonparametric regression

Bias-corrected confidence bands for general nonparametric regression models are considered. We use local polynomial fitting to construct the confidence bands and combine the cross-validation method and the plug-in method to select the bandwidths. Related asymptotic results are obtained. Our simulations show that confidence bands constructed by local polynomial fitting have much better coverage than those constructed by using the Nadaraya–Watson estimator. The results are also applicable to nonparametric autoregressive time series models.     

A comparative simulation study of data-driven methods for estimating density level sets

Density level sets are mainly estimated using one of three methodologies: plug-in, excess mass, or a hybrid approach. The plug-in methods are based on replacing the unknown density by some nonparametric estimator, usually the kernel one. Thus, the bandwidth selection is a fundamental problem from an applied perspective. Recently, specific selectors for level sets have been proposed. However, if some a priori information about the geometry of the level set is available, then excess mass algorithms can be useful. In this case, the problem of bandwidth selection can be avoided. The third methodology is a hybrid of the others. It assumes a mild geometric restriction on the level set and it requires a pilot nonparametric estimator of the density. One interesting open question concerns the performance of these methods. In this work, existing methods are reviewed, and two new hybrid algorithms are proposed. Their practical behaviour is compared through extensive simulation study.     

Empirical Likelihood Confidence Intervals for Response Mean with Data Missing at Random

Abstract.  A kernel regression imputation method for missing response data is developed. A class of bias-corrected empirical log-likelihood ratios for the response mean is defined. It is shown that any member of our class of ratios is asymptotically chi-squared, and the corresponding empirical likelihood confidence interval for the response mean is constructed. Our ratios share some of the desired features of the existing methods: they are self-scale invariant and no plug-in estimators for the adjustment factor and asymptotic variance are needed; when estimating the non-parametric function in the model, undersmoothing to ensure root- n consistency of the estimator for the parameter is avoided. Since the range of bandwidths contains the optimal bandwidth for estimating the regression function, the existing data-driven algorithm is valid for selecting an optimal bandwidth. We also study the normal approximation-based method. A simulation study is undertaken to compare the empirical likelihood with the normal approximation method in terms of coverage accuracies and average lengths of confidence intervals.     

Smoothing for small samples with model misspecification: nonparametric and semiparametric concerns

Our goal is to find a regression technique that can be used in a small-sample situation with possible model misspecification. The development of a new bandwidth selector allows nonparametric regression (in conjunction with least squares) to be used in this small-sample problem, where nonparametric procedures have previously proven to be inadequate. Considered here are two new semiparametric (model-robust) regression techniques that combine parametric and nonparametric techniques when there is partial information present about the underlying model. A general overview is given of how typical concerns for bandwidth selection in nonparametric regression extend to the model-robust procedures. A new penalized PRESS criterion (with a graphical selection strategy for applications) is developed that overcomes these concerns and is able to maintain the beneficial mean squared error properties of the new model-robust methods. It is shown that this new selector outperforms standard and recently improved bandwidth selectors. Comparisons of the selectors are made via numerous generated data examples and a small simulation study.     

Local Adaptivity of Kernel Estimates with Plug-in Local Bandwidth Selectors

In non-parametric function estimation selection of a smoothing parameter is one of the most important issues. The performance of smoothing techniques depends highly on the choice of this parameter. Preferably the bandwidth should be determined via a data-driven procedure. In this paper we consider kernel estimators in a white noise model, and investigate whether locally adaptive plug-in bandwidths can achieve optimal global rates of convergence. We consider various classes of functions: Sobolev classes, bounded variation function classes, classes of convex functions and classes of monotone functions. We study the situations of pilot estimation with oversmoothing and without oversmoothing. Our main finding is that simple local plug-in bandwidth selectors can adapt to spatial inhomogeneity of the regression function as long as there are no local oscillations of high frequency. We establish the pointwise asymptotic distribution of the regression estimator with local plug-in bandwidth.     

Model selection in spline nonparametric regression

A Bayesian approach is presented for model selection in nonparametric regression with Gaussian errors and in binary nonparametric regression. A smoothness prior is assumed for each component of the model and the posterior probabilities of the candidate models are approximated using the Bayesian information criterion. We study the model selection method by simulation and show that it has excellent frequentist properties and gives improved estimates of the regression surface. All the computations are carried out efficiently using the Gibbs sampler.     

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.