Local and Variable Bandwidths and Local Linear Regression

Two types of non-global bandwidth, which may be called local and variable, have been defined in attempts to improve the performance of kernel density estimators. In nonparametric regression, local linear fitting has become a method of much popularity. It is natural, therefore, to consider the use of non-global bandwidths in the local linear context, and indeed local bandwidths are often used. In this paper, it is observed that a natural proposal in the literature for combining variable bandwidths with local linear fitting fails in the sense that the resulting mean squared error properties are those normally associated with local rather than variable bandwidths. We are able to understand why this happens in terms of weightings that are involved. We also attempt to investigate how the bias reduction expected of well-chosen variable bandwidths might be achieved in conjunction with local linear fitting.     

Local linear fitting and improved estimation near peaks

The authors look into the problem of estimating regression functions that exhibit jump irregularities in the first derivative. They investigate the behaviour of the bias in the local linear fit and show the superior performance of appropriate one‐sided versions of the local linear fit near such irregularities. They then propose an improved estimation procedure based on data‐driven selection of a conventional or one‐sided local linear fit according to a residual sum of squares type of criterion. The authors provide theoretical results and illustrate the method both on simulated and real‐life data examples. The Canadian Journal of Statistics 37: 453–475; 2009 © 2009 Statistical Society of Canada     

Local Linear Additive Quantile Regression

Abstract.  We consider non-parametric additive quantile regression estimation by kernel-weighted local linear fitting. The estimator is based on localizing the characterization of quantile regression as the minimizer of the appropriate 'check function'. A backfitting algorithm and a heuristic rule for selecting the smoothing parameter are explored. We also study the estimation of average-derivative quantile regression under the additive model. The techniques are illustrated by a simulated example and a real data set.     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

Local Influence Analysis for The Ridge Regression Under Stochastic Linear Restrictions

In this article, we assess the local influence for the ridge regression of linear models with stochastic linear restrictions in the spirit of Cook by using the log-likelihood of the stochastic restricted ridge regression estimator. The diagnostics under the perturbations of constant variance, responses and individual explanatory variables are derived. We also assess the local influence of the stochastic restricted ridge regression estimator under the approach suggested by Billor and Loynes. At the end, a numerical example on the Longley data is given to illustrate the theoretic results.     

Local Linear Regression for Non Grid Spatiotemporal Models with Autoregressive Errors

In various environmental studies, spatiotemporal correlated data are involved, so there has been increasing demand of proposing spatiotemporal estimation methods that capture spatiotemporal correlation so as to improve the accuracy of estimation. In this article, we construct estimators for non grid spatiotemporal models with autoregressive errors. It is proved that the estimators are asymptotic normality. Simulation results also show the estimators perform well.     

Local Linear Estimation for Time-Dependent Coefficients in Cox's Regression Models

This article develops a local partial likelihood technique to estimate the time-dependent coefficients in Cox's regression model. The basic idea is a simple extension of the local linear fitting technique used in the scatterplot smoothing. The coefficients are estimated locally based on the partial likelihood in a window around each time point. Multiple time-dependent covariates are incorporated in the local partial likelihood procedure. The procedure is useful as a diagnostic tool and can be used in uncovering time-dependencies or departure from the proportional hazards model. The programming involved in the local partial likelihood estimation is relatively simple and it can be modified with few efforts from the existing programs for the proportional hazards model. The asymptotic properties of the resulting estimator are established and compared with those from the local constant fitting. A consistent estimator of the asymptotic variance is also proposed. The approach is illustrated by a real data set from the study of gastric cancer patients and a simulation study is also presented.     

On the Uniform Strong Consistency of Local Polynomial Regression Under Dependence Conditions

Abstract

In this article, nonparametric estimators of the regression function, and its derivatives, obtained by means of weighted local polynomial fitting are studied. Consider the fixed regression model where the error random variables are coming from a stationary stochastic process satisfying a mixing condition. Uniform strong consistency, along with rates, are established for these estimators. Furthermore, when the errors follow an AR(1) correlation structure, strong consistency properties are also derived for a modified version of the local polynomial estimators proposed by Vilar-Fernández and Francisco-Fernández (Vilar-Fernández, J. M., Francisco-Fernández, M. (2002 Vilar-Fernández, J. M. and Francisco-Fernández, M. 2002. Local polynomial regression smoothers with AR-error structure. TEST, 11(2): 439–464.  [Google Scholar]). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439–464).     

Local Linear Regression in Proportional Hazards Model with Censored Data

In this article we study the method of nonparametric regression based on a transformation model, under which an unknown transformation of the survival time is nonlinearly, even more, nonparametrically, related to the covariates with various error distributions, which are parametrically specified with unknown parameters. Local linear approximations and locally weighted least squares are applied to obtain estimators for the effects of covariates with censored observations. We show that the estimators are consistent and asymptotically normal. This transformation model, coupled with local linear approximation techniques, provides many alternatives to the more general proportional hazards models with nonparametric covariates.     

Local Linear Least Squares Kernel Regression for Linear and Circular Predictors

We extend nonparametric regression models with local linear least squares fitting using kernel weights to the case of linear and circular predictors. We derive the asymptotic properties of the conditional bias and variance of bivariate local linear least squares kernel estimators. A small simulation study and a real experiment are given.     

Comparisons Between Local Linear Estimator and Kernel Smooth Estimator for a Smooth Distribution Based on MSE Under Right Censoring

We describe a method for estimating the coefficients in a logistic regression model when the predictors are subject to measurement error and an instrumental variable is present. The proposed method is based upon the theory of factor scores taken from factor analysis. Two versions of the proposed method, a simple one and an extended one, are compared to the methods referred to by Carrol, Ruppert and Stefanski (1995) through simulation studies. Our conclusion is that the simple version performs as well as the methods from Carrol et al. (1995), and the extended version performs betterwith respect to MSE, due to a reduction of bias.     

Detecting discontinuities in nonparametric regression curves and surfaces

The existence of a discontinuity in a regression function can be inferred by comparing regression estimates based on the data lying on different sides of a point of interest. This idea has been used in earlier research by Hall and Titterington (1992), Müller (1992) and later authors. The use of nonparametric regression allows this to be done without assuming linear or other parametric forms for the continuous part of the underlying regression function. The focus of the present paper is on assessing the evidence for the presence of a discontinuity within a regression function through examination of the standardised differences of ‘left’ and ‘right’ estimators at a variety of covariate values. The calculations for the test are carried out through distributional results on quadratic forms. A graphical method in the form of a reference band to highlight the sources of the evidence for discontinuities is proposed. The methods are also developed for the two covariate case where there are additional issues associated with the presence of a jump location curve. Methods for estimating this curve are also developed. All the techniques, for the one and two covariate situations, are illustrated through applications.     

D-optimal Designs for a Combined Simple Linear and Haar-Wavelet Regression Model

A combined simple linear and Haar-wavelet regression model is the combination of a simple linear model and a Haar-wavelet regression model. In this article we show how to construct D-optimal designs for a combined simple linear and Haar-wavelet regression model. An example is also given.     

Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.