Kernel Smoothing to Improve Bootstrap Confidence Intervals

Some studies of the bootstrap have assessed the effect of smoothing the estimated distribution that is resampled, a process usually known as the smoothed bootstrap. Generally, the smoothed distribution for resampling is a kernel estimate and is often rescaled to retain certain characteristics of the empirical distribution. Typically the effect of such smoothing has been measured in terms of the mean-squared error of bootstrap point estimates. The reports of these previous investigations have not been encouraging about the efficacy of smoothing. In this paper the effect of resampling a kernel-smoothed distribution is evaluated through expansions for the coverage of bootstrap percentile confidence intervals. It is shown that, under the smooth function model, proper bandwidth selection can accomplish a first-order correction for the one-sided percentile method. With the objective of reducing the coverage error the appropriate bandwidth for one-sided intervals converges at a rate of n ^−1/4, rather than the familiar n ^−1/5 for kernel density estimation. Applications of this same approach to bootstrap t and two-sided intervals yield optimal bandwidths of order n ^−1/2. These bandwidths depend on moments of the smooth function model and not on derivatives of the underlying density of the data. The relationship of this smoothing method to both the accelerated bias correction and the bootstrap t methods provides some insight into the connections between three quite distinct approximate confidence intervals.     

The estimating function bootstrap

The authors propose a bootstrap procedure which estimates the distribution of an estimating function by resampling its terms using bootstrap techniques. Studentized versions of this so‐called estimating function (EF) bootstrap yield methods which are invariant under reparametrizations. This approach often has substantial advantage, both in computation and accuracy, over more traditional bootstrap methods and it applies to a wide class of practical problems where the data are independent but not necessarily identically distributed. The methods allow for simultaneous estimation of vector parameters and their components. The authors use simulations to compare the EF bootstrap with competing methods in several examples including the common means problem and nonlinear regression. They also prove symptotic results showing that the studentized EF bootstrap yields higher order approximations for the whole vector parameter in a wide class of problems.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

A simple bootstrap bandwidth selector for local polynomial fitting

A new, fully data-driven bandwidth selector with a double smoothing (DS) bias term and a data-driven variance estimator is developed following the bootstrap idea. The data-driven variance estimation does not involve any additional bandwidth selection. The proposed bandwidth selector convergences faster than a plug-in one due to the DS bias estimate, whereas the data-driven variance improves its finite sample performance clearly and makes it stable. Asymptotic results of the proposals are obtained. A comparative simulation study was done to show the overall gains and the gains obtained by improving either the bias term or the variance estimate, respectively. It is shown that the use of a good variance estimator is more important when the sample size is relatively small.     

Local Smoothing for Kernel Distribution Function Estimation

The problem of bandwidth selection for kernel-based estimation of the distribution function (cdf) at a given point is considered. With appropriate bandwidth, a kernel-based estimator (kdf) is known to outperform the empirical distribution function. However, such a bandwidth is unknown in practice. In pointwise estimation, the appropriate bandwidth depends on the point where the function is estimated. The existing smoothing methods use one common bandwidth to estimate the cdf. The accuracy of the resulting estimates varies substantially depending on the cdf and the point where it is estimated. We propose to select bandwidth by minimizing a bootstrap estimator of the MSE of the kdf. The resulting estimator performs reliably, irrespective of where the cdf is estimated. It is shown to be consistent under i.i.d. as well as strongly mixing dependence assumption. Two applications of the proposed estimator are shown in finance and seismology. We report a dataset on the S & P Nifty index values.     

Nonparametric particle filtering and smoothing with quasi-Monte Carlo sampling

Sequential Monte Carlo methods (also known as particle filters and smoothers) are used for filtering and smoothing in general state-space models. These methods are based on importance sampling. In practice, it is often difficult to find a suitable proposal which allows effective importance sampling. This article develops an original particle filter and an original particle smoother which employ nonparametric importance sampling. The basic idea is to use a nonparametric estimate of the marginally optimal proposal. The proposed algorithms provide a better approximation of the filtering and smoothing distributions than standard methods. The methods’ advantage is most distinct in severely nonlinear situations. In contrast to most existing methods, they allow the use of quasi-Monte Carlo (QMC) sampling. In addition, they do not suffer from weight degeneration rendering a resampling step unnecessary. For the estimation of model parameters, an efficient on-line maximum-likelihood (ML) estimation technique is proposed which is also based on nonparametric approximations. All suggested algorithms have almost linear complexity for low-dimensional state-spaces. This is an advantage over standard smoothing and ML procedures. Particularly, all existing sequential Monte Carlo methods that incorporate QMC sampling have quadratic complexity. As an application, stochastic volatility estimation for high-frequency financial data is considered, which is of great importance in practice. The computer code is partly available as supplemental material.     

Estimating kurtosis and confidence intervals for the variance under nonnormality

Exact confidence intervals for variances rely on normal distribution assumptions. Alternatively, large-sample confidence intervals for the variance can be attained if one estimates the kurtosis of the underlying distribution. The method used to estimate the kurtosis has a direct impact on the performance of the interval and thus the quality of statistical inferences. In this paper the author considers a number of kurtosis estimators combined with large-sample theory to construct approximate confidence intervals for the variance. In addition, a nonparametric bootstrap resampling procedure is used to build bootstrap confidence intervals for the variance. Simulated coverage probabilities using different confidence interval methods are computed for a variety of sample sizes and distributions. A modification to a conventional estimator of the kurtosis, in conjunction with adjustments to the mean and variance of the asymptotic distribution of a function of the sample variance, improves the resulting coverage values for leptokurtically distributed populations.     

Consistent Smooth Bootstrap Kernel Intensity Estimation for Inhomogeneous Spatial Poisson Point Processes

Non‐parametric estimation and bootstrap techniques play an important role in many areas of Statistics. In the point process context, kernel intensity estimation has been limited to exploratory analysis because of its inconsistency, and some consistent alternatives have been proposed. Furthermore, most authors have considered kernel intensity estimators with scalar bandwidths, which can be very restrictive. This work focuses on a consistent kernel intensity estimator with unconstrained bandwidth matrix. We propose a smooth bootstrap for inhomogeneous spatial point processes. The consistency of the bootstrap mean integrated squared error (MISE) as an estimator of the MISE of the consistent kernel intensity estimator proves the validity of the resampling procedure. Finally, we propose a plug‐in bandwidth selection procedure based on the bootstrap MISE and compare its performance with several methods currently used through both as a simulation study and an application to the spatial pattern of wildfires registered in Galicia (Spain) during 2006.     

On the bootstrap and smoothed bootstrap

The standard bootstrap and two commonly used types of smoothed bootstrap are investigated. The saddlepoint approximations are used to evaluate the accuracy of the three bootstrap estimates of the density of a sample mean. The optimal choice for the smoothing parameter is obtained when smoothing is useful in reducing the mean squared error.     

On multivariate smoothed bootstrap consistency

This paper deals with the convergence in Mallows metric for classical multivariate kernel distribution function estimators. We prove the convergence in Mallows metric of a locally orientated kernel smooth estimator belonging to the class of sample smoothing estimators. The consistency follows for the smoothed bootstrap for regular functions of the marginal means. Two simple simulation studies show how the smoothed versions of the bootstrap give better results than the classical technique.     

An optimal local bandwidth selector for kernel density estimation

The problem of selecting the bandwidth for optimal kernel density estimation at a point is considered. A class of local bandwidth selectors which minimize smoothed bootstrap estimates of mean-squared error in density estimation is introduced. It is proved that the bandwidth selectors in the class achieve optimal relative rates of convergence, dependent upon the local smoothness of the target density. Practical implementation of the bandwidth selection methodology is discussed. The use of Gaussian-based kernels to facilitate computation of the smoothed bootstrap estimate of mean-squared error is proposed. The performance of the bandwidth selectors is investigated empirically.     

Smoothed bootstrap bandwidth selection for nonparametric hazard rate estimation

A smoothed bootstrap method is presented for the purpose of bandwidth selection in nonparametric hazard rate estimation for iid data. In this context, two new bootstrap bandwidth selectors are established based on the exact expression of the bootstrap version of the mean integrated squared error of some approximations of the kernel hazard rate estimator. This is very useful since Monte Carlo approximation is no longer needed for the implementation of the two bootstrap selectors. A simulation study is carried out in order to show the empirical performance of the new bootstrap bandwidths and to compare them with other existing selectors. The methods are illustrated by applying them to a diabetes data set.     

A simulation study of balanced and antithetic bootstrap resampling methods

An extensive simulation study is conducted to compare the performance between balanced and antithetic resampling for the bootstrap in estimation of bias, variance, and percentiles when the statistic of interest is the median, the square root of the absolute value of the mean, or the median absolute deviations from the median. Simulation results reveal that balanced resampling provide better efficiencies in most cases; however, antithetic resampling is superior in estimating bias of the median. We also investigate the possibility of combining an existing efficient bootstrap computation of Efron (1990) with balanced or antithetic resampling for percentile estimation. Results indicate that the combination method does indeed offer gains in performance though the gains are much more dramatic for the bootstrap t statistic than for any of the three statistics of interest as described above.     

The exact bootstrap mean and variance of an L-estimator

Exact analytic expressions for the bootstrap mean and variance of any L -estimator are obtained, thus eliminating the error due to bootstrap resampling. The expressions follow from the direct calculation of the bootstrap mean vector and covariance matrix of the whole set of order statistics. By using these expressions, recommendations can be made about the appropriateness of bootstrap estimation under given conditions.     

Asymptotic confidence interval of power spectrum of a continuous time process through progressively faster sampling

If the power spectral density of a continuous time stationary stochastic process is not limited to a finite bandwidth, data sampled from that process at any uniform sampling rate leads to biased and inconsistent spectrum estimators, which are unsuitable for constructing confidence intervals. In this paper, we use the smoothed periodogram estimator to construct asymptotic confidence intervals shrinking to the true spectra, by allowing the sampling rate to go to infinity suitably fast as the sample size goes to infinity. The proposed method requires minimal computation, as it does not involve bootstrap or other resampling. The method is illustrated through a Monte-Carlo simulation study, and its performance is compared with that of the corresponding method based on uniform sampling at a fixed rate.     

Bootstrapping Regression Parameters in Multivariate Survival Analysis

Bootstrap methods are proposed for estimating sampling distributions and associated statistics for regression parameters in multivariate survival data. We use an Independence Working Model (IWM) approach, fitting margins independently, to obtain consistent estimates of the parameters in the marginal models. Resampling procedures, however, are applied to an appropriate joint distribution to estimate covariance matrices, make bias corrections, and construct confidence intervals. The proposed methods allow for fixed or random explanatory variables, the latter case using extensions of existing resampling schemes (Loughin,1995), and they permit the possibility of random censoring. An application is shown for the viral positivity time data previously analyzed by Wei, Lin, and Weissfeld (1989). A simulation study of small-sample properties shows that the proposed bootstrap procedures provide substantial improvements in variance estimation over the robust variance estimator commonly used with the IWM. This revised version was published online in July 2006 with corrections to the Cover Date.     

Resampling estimation when observations are m–dependent

For m–dependent, identically distributed random observation, the bootstrap method provides inconsistent estimators of the distribution and variance of the sample mean. This paper proposes an alternative resampling procedure. For estimating the distribution and variance of a function of the sample mean, the proposed resampling estimators are shown to be strongly consistent.     

Edgeworth expansions for nonparametric distribution estimation with applications

In this paper, we will investigate the nonparametric estimation of the distribution function F of an absolutely continuous random variable. Two methods are analyzed: the first one based on the empirical distribution function, expressed in terms of i.i.d. lattice random variables and, secondly, the kernel method, which involves nonlattice random vectors dependent on the sample size n; this latter procedure produces a smooth distribution estimator that will be explicitly corrected to reduce the effect of bias or variance. For both methods, the non-Studentized and Studentized statistics are considered as well as their bootstrap counterparts and asymptotic expansions are constructed to approximate their distribution functions via the Edgeworth expansion techniques. On this basis, we will obtain confidence intervals for F(x) and state the coverage error order achieved in each case.     

Partial Saddlepoint Approximations for Transformed Means

The full saddlepoint approximation for real valued smooth functions of means requires the existence of the joint cumulant generating function for the entire vector of random variables which are being transformed. We propose a mixed saddlepoint-Edgeworth approximation requiring the existence of a cumulant generating function for only part of the random vector considered, while retaining partially the relative nature of the errors. Tail probability approximations are obtained under conditions which enable the approximations to be used in resampling situations and hence to obtain a result on the relative error of coverage in the case of the bootstrap approximation to the confidence interval for the Studentized mean.     

Statistical inference on transformation models: a self-induced smoothing approach

This paper deals with a general class of transformation models that contains many important semiparametric regression models as special cases. It develops a self-induced smoothing for the maximum rank correlation estimator, resulting in simultaneous point and variance estimation. The self-induced smoothing does not require bandwidth selection, yet provides the right amount of smoothness so that the estimator is asymptotically normal with mean zero (unbiased) and variance–covariance matrix consistently estimated by the usual sandwich-type estimator. An iterative algorithm is given for the variance estimation and shown to numerically converge to a consistent limiting variance estimator. The approach is applied to a data set involving survival times of primary biliary cirrhosis patients. Simulation results are reported, showing that the new method performs well under a variety of scenarios.