A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines

An unbiased stochastic estimator of tr(I–A), where A is the influence matrix associated with the calculation of Laplacian smoothing splines, is described. The estimator is similar to one recently developed by Girard but satisfies a minimum variance criterion and does not require the simulation of a standard normal variable. It uses instead simulations of the discrete random variable which takes the values 1, -1 each with probability 1/2. Bounds on the variance of the estimator, similar to those established by Girard, are obtained using elementary methods. The estimator can be used to approximately minimize generalised cross validation (GCV) when using discretized iterative methods for fitting Laplacian smoothing splines to very large data sets. Simulated examples show that the estimated trace values, using either the estimator presented here or the estimator of Girard, perform almost as well as the exact values when applied to the minimization of GCV for n as small as a few hundred, where n is the number of data points.     

Kernel spline regression

The authors propose «kernel spline regression,» a method of combining spline regression and kernel smoothing by replacing the polynomial approximation for local polynomial kernel regression with the spline basis. The new approach retains the local weighting scheme and the use of a bandwidth to control the size of local neighborhood. The authors compute the bias and variance of the kernel linear spline estimator, which they compare with local linear regression. They show that kernel spline estimators can succeed in capturing the main features of the underlying curve more effectively than local polynomial regression when the curvature changes rapidly. They also show through simulation that kernel spline regression often performs better than ordinary spline regression and local polynomial regression.     

Confidence intervals for nonparametric quantile regression: an emphasis on smoothing splines approach

In this paper we consider the problem of constructing confidence intervals for nonparametric quantile regression with an emphasis on smoothing splines. The mean‐based approaches for smoothing splines of Wahba (1983) and Nychka (1988) may not be efficient for constructing confidence intervals for the underlying function when the observed data are non‐Gaussian distributed, for instance if they are skewed or heavy‐tailed. This paper proposes a method of constructing confidence intervals for the unknown τth quantile function (0<τ<1) based on smoothing splines. In this paper we investigate the extent to which the proposed estimator provides the desired coverage probability. In addition, an improvement based on a local smoothing parameter that provides more uniform pointwise coverage is developed. The results from numerical studies including a simulation study and real data analysis demonstrate the promising empirical properties of the proposed approach.     

Réduction de la variance dans les sondages en présence d'information auxiliarie: Une approache non paramétrique par splines de régression

The author considers the use of auxiliary information available at population level to improve the estimation of finite population totals. She introduces a new type of model‐assisted estimator based on nonparametric regression splines. The estimator is a weighted linear combination of the study variable with weights calibrated to the B‐splines known population totals. The author shows that the estimator is asymptotically design‐unbiased and consistent under conditions which do not require the superpopulation model to be correct. She proposes a design‐based variance approximation and shows that the anticipated variance is asymptotically equivalent to the Godambe‐Joshi lower bound. She also shows through simulations that the estimator has good properties.     

Local M-estimation for Conditional Variance in Heteroscedastic Regression Models

In this article, we develop a local M-estimation for the conditional variance in heteroscedastic regression models. The estimator is based on the local linear smoothing technique and the M-estimation technique, and it is shown to be not only asymptotically equivalent to the local linear estimator but also robust. The consistency and asymptotic normality of the local M-estimator for the conditional variance in heteroscedastic regression models are obtained under mild conditions. The simulation studies demonstrate that the proposed estimators perform well in robustness.     

Smooth supersaturated models

In areas such as kernel smoothing and non-parametric regression, there is emphasis on smooth interpolation. We concentrate on pure interpolation and build smooth polynomial interpolators by first extending the monomial (polynomial) basis and then minimizing a measure of roughness with respect to the extra parameters in the extended basis. Algebraic methods can help in choosing the extended basis. We get arbitrarily close to optimal smoothing for any dimension over an arbitrary region, giving simple models close to splines. We show in examples that smooth interpolators perform much better than straight polynomial fits and for small sample size, better than kriging-type methods, used, for example in computer experiments.     

Bayes and Empirical Bayes Inference in Changepoint Problems

ABSTRACT

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.     

A Flexible Method for Estimating the ROC Curve

In this paper we propose a flexible method for estimating a receiver operating characteristic (ROC) curve that is based on a continuous-scale test. The approach is easily understood and efficiently computed, and robust to the smooth parameter selection, which needs intensive computation when using local polynomial and smoothing spline techniques. The results from our simulation experiment indicate that the moderate-sample numerical performance of our estimator is better than the empirical ROC curve estimator and comparable to the local linear estimator. The availability of easy implementation is also illustrated by our simulation. We apply the proposed method to two real data sets.     

Efficient estimation for time-varying coefficient longitudinal models

For estimation of time-varying coefficient longitudinal models, the widely used local least-squares (LS) or covariance-weighted local LS smoothing uses information from the local sample average. Motivated by the fact that a combination of multiple quantiles provides a more complete picture of the distribution, we investigate quantile regression-based methods to improve efficiency by optimally combining information across quantiles. Under the working independence scenario, the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound. In the presence of dependence among within-subject measurements, we adopt a prewhitening technique to transform regression errors into independent innovations and show that the prewhitened optimally weighted quantile average estimator asymptotically achieves the Cramér–Rao bound for the independent innovations. Fully data-driven bandwidth selection and optimal weights estimation are implemented through a two-step procedure. Monte Carlo studies show that the proposed method delivers more robust and superior overall performance than that of the existing methods.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Empirical Likelihood Inference for the Cox Model with Time-dependent Coefficients via Local Partial Likelihood

Abstract.  The Cox model with time-dependent coefficients has been studied by a number of authors recently. In this paper, we develop empirical likelihood (EL) pointwise confidence regions for the time-dependent regression coefficients via local partial likelihood smoothing. The EL simultaneous confidence bands for a linear combination of the coefficients are also derived based on the strong approximation methods. The EL ratio is formulated through the local partial log-likelihood for the regression coefficient functions. Our numerical studies indicate that the EL pointwise/simultaneous confidence regions/bands have satisfactory finite sample performances. Compared with the confidence regions derived directly based on the asymptotic normal distribution of the local constant estimator, the EL confidence regions are overall tighter and can better capture the curvature of the underlying regression coefficient functions. Two data sets, the gastric cancer data and the Mayo Clinic primary biliary cirrhosis data, are analysed using the proposed method.     

Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

Robust non-parametric smoothing of non-stationary time series

Motivated by the need of extracting local trends and low frequency components in non-stationary time series, this paper discusses methods of robust non-parametric smoothing. Basic approach is the combination of the parametric M-estimation with kernel and local polynomial regression methods. The result is an iterative estimator that retains a linear structure, but has kernel weights also in the direction of the prediction errors. The design of smoothing coefficients is carried out with robust cross-validation criteria and rules of thumb. The method works well both to remove the influence of patches of outliers and to detect the local breaks and persistent structural change in time series.     

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.     

A Unified View of Nonparametric Trend-Cycle Predictors Via Reproducing Kernel Hilbert Spaces

We provide a common approach for studying several nonparametric estimators used for smoothing functional time series data. Linear filters based on different building assumptions are transformed into kernel functions via reproducing kernel Hilbert spaces. For each estimator, we identify a density function or second order kernel, from which a hierarchy of higher order estimators is derived. These are shown to give excellent representations for the currently applied symmetric filters. In particular, we derive equivalent kernels of smoothing splines in Sobolev and polynomial spaces. The asymmetric weights are obtained by adapting the kernel functions to the length of the various filters, and a theoretical and empirical comparison is made with the classical estimators used in real time analysis. The former are shown to be superior in terms of signal passing, noise suppression and speed of convergence to the symmetric filter.     

Boundary corrected cubic smoothing splines

Smoothing splines are known to exhibit a type of boundary bias that can reduce their estimation efficiency. In this paper, a boundary corrected cubic smoothing spline is developed in a way that produces a uniformly fourth order estimator. The resulting estimator can be calculated efficiently using an O(n) algorithm that is designed for the computation of fitted values and associated smoothing parameter selection criteria. A simulation study shows that use of the boundary corrected estimator can improve estimation efficiency in finite samples. Applications to the construction of asymptotically valid pointwise confidence intervals are also investigated .     

Free-knot polynomial splines with confidence intervals

Summary.  We construct approximate confidence intervals for a nonparametric regression function, using polynomial splines with free-knot locations. The number of knots is determined by generalized cross-validation. The estimates of knot locations and coefficients are obtained through a non-linear least squares solution that corresponds to the maximum likelihood estimate. Confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood estimator. Average coverage probabilities and the accuracy of the estimate are examined via simulation. This includes comparisons between our method and some existing methods such as smoothing spline and variable knots selection as well as a Bayesian version of the variable knots method. Simulation results indicate that our method works well for smooth underlying functions and also reasonably well for discontinuous functions. It also performs well for fairly small sample sizes.     

Testing the adequacy of a linear model VIA critical smoothing

An important problem for fitting local linear regression is the choice of the smoothing parameter. As the smoothing parameter becomes large, the estimator tends to a straight line, which is the least squares fit in the ordinary linear regression setting. This property may be used to assess the adequacy of a simple linear model. Motivated by Silverman's (1981) work in kernel density estimation, a suitable test statistic is the critical smoothing parameter where the estimate changes from nonlinear to linear, while linearity or non- linearity requires a more precise judgment. We define the critical smoothing parameter through the approximate F-tests by Hastie and Tibshirani (1990). To assess the significance, the “wild bootstrap” procedure is used to replicate the data and the proportion of bootstrap samples which give a nonlinear estimate when using the critical bandwidth is obtained as the p-value. Simulation results show that the critical smoothing test is useful in detecting a wide range of alternatives.     

Simple Transformation Techniques for Improved Non-parametric Regression

We propose and investigate two new methods for achieving less bias in non- parametric regression. We show that the new methods have bias of order h ⁴, where h is a smoothing parameter, in contrast to the basic kernel estimator's order h ². The methods are conceptually very simple. At the first stage, perform an ordinary non-parametric regression on { x_i , Y_i } to obtain m^ ( x_i ) (we use local linear fitting). In the first method, at the second stage, repeat the non-parametric regression but on the transformed dataset { m^ ( x_i , Y_i )}, taking the estimator at x to be this second stage estimator at m^ ( x ). In the second, and more appealing, method, again perform non-parametric regression on { m^ ( x_i , Y_i )}, but this time make the kernel weights depend on the original x scale rather than using the m^ ( x ) scale. We concentrate more of our effort in this paper on the latter because of its advantages over the former. Our emphasis is largely theoretical, but we also show that the latter method has practical potential through some simulated examples.     

Smoothing Splines with Boundary Correction

Regular smoothing splines are known to have a type of boundary bias problem that can reduce their estimation efficiency. In this paper, a boundary corrected smoothing spline with general order is designed in a way that the risk will decay at an optimal rate. An O(n) algorithm is also developed to compute the resultant estimator efficiently.