Smoothed Bootstrap Method Research Articles

SMOOTHED EMPIRICAL LIKELIHOOD METHODS FOR QUANTILE REGRESSION MODELS

This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve the higher order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order n−1 and may be Bartlett corrected to produce regions with errors of order n−2, where n denotes the sample size. Our result is an extension of the previous result of Chen and Hall (1993, Annals of Statistics 21, 1166–1181) to the regression context. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998, Econometrica 66, 1327–1351).I thank the co-editor Bruce Hansen and anonymous referees for valuable suggestions and comments. I also thank Song X. Chen, Joel Horowitz, Yuichi Kitamura, Oliver Linton, Whitney Newey, Peter Phillips, and Richard Smith for helpful comments. Parts of this paper were written while I was visiting the Cowles Foundation at Yale University, whose hospitality is gratefully acknowledged. Young-Hyun Cho has provided excellent research assistance. This work was supported by the Korea Research Foundation grant KRF 2003-041-B00072.

Comparing distribution functions of errors in linear models: A nonparametric approach

We describe how to test whether the distribution functions of errors from two linear regression models are the same, with statistics based on empirical distribution functions constructed with residuals. A smooth bootstrap method is used to approximate critical values. Simulations show that the procedure works well in practice.

Calibrated interpolated confidence intervals for population quantiles

Beran & Hall's (1993) simple linear interpolation provides a very convenient approach for constructing nonparametric confidence intervals for population quantiles based on a random sample of size n. We show that the coverage error of the interpolated interval, which is of order O(n -1 ), can be improved upon by calibrating the nominal coverage level. Three distinct methods of calibration are considered. The analytical and Monte Carlo methods succeed in reducing the order of coverage error to O(n -3/2 ), while the smoothed bootstrap method reduces it further to O(n -25/14 ). We provide guidelines for practical implementation of the calibration methods. Their performance is compared with the simple linear interpolated interval in a simulation study which confirms superiority of the calibrated intervals.

SMOOTHED BOOTSTRAP BANDWIDTH SELECTION IN NONPARAMETRIC DENSITY ESTIMATION FOR MOVING AVERAGE PROCESSES

This article is concerned with the problem of selecting a suitable bandwidth when estimating the marginal density function of a moving average process. The smoothed bootstrap method is used to implement a bandwidth selector for a convolution-type density estimator, based on the kernel method. The relative rate of convergence of this selector with respect to the MISE bandwidth is proved to be O P (n −1/2). The finite sample size performance of the selector is investigated in a simulation study.

On the minimisation of $L\sp p$ error in mode estimation

We show that, for $L^p$ convergence of the mode of a nonparametric density estimator to the mode of an unknown probability density, finiteness of the pth moment of the underlying distribution is both necessary and sufficient. The basic requirement of existence of finite variance has been overlooked by statisticians, who have earlier considered mean square convergence of nonparametric mode estimators; they have focussed on mean squared error of the asymptotic distribution, rather than on asymptotic mean squared error. The effect of bandwidth choice on the rate of $L^p$ convergence is analysed, and smoothed bootstrap methods are used to develop an empirical approximation to the $L^p$ measure of error. The resulting bootstrap estimator of $L^p$ error may be minimised with respect to the bandwidth of the nonparametric density estimator, and in this way an empirical rule may be developed for selecting the bandwidth for mode estimation. Particular attention is devoted to the problem of selecting the appropriate amount of smoothing in the bootstrap algorithm.

Optimizing the smoothed bootstrap

In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.

Bootstrapping the Mean Integrated Squared Error

A smooth bootstrap method is used to find an estimator of the mean integrated squared error in density estimation. This provides a natural bootstrap selector for the bandwidth. Some rates of convergence and limit distributions are presented for this new selector as well as for some modification of it.

Bootstrap Choice of Bandwidth for Density Estimation

Abstract A bootstrap-based choice of bandwidth for kernel density estimation is introduced. The method works by estimating the integrated mean squared error (IMSE) for any given bandwidth and then minimizing over all bandwidths. A straightforward application of the bootstrap method to estimate the IMSE fails because it does not capture the bias component. A smoothed bootstrap method based on an initial density estimate is described that solves this problem. It is possible to construct pointwise and simultaneous confidence intervals for the density. The simulation study compares cross-validation and the bootstrap method over a wide range of densities—a long-tailed, a short-tailed, an asymmetric, and a bimodal, among others. The bootstrap method uniformly outperforms cross-validation. The accuracy of the constructed confidence bands improves as the sample size increases.