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.     

Kernel Estimation of Rate Function for Recurrent Event Data

Abstract.  Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth will produce a curve with nicks occurring at the censoring times, whereas there is no such problem with the least squares method. Furthermore, the asymptotic variance of the least squares estimator is shown to be smaller under regularity conditions. However, in the implementation of the bootstrap procedures, the moment method is computationally more efficient than the least squares method because the former approach uses condensed bootstrap data. The performance of the proposed procedures is studied through Monte Carlo simulations and an epidemiological example on intravenous drug users.     

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.     

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.     

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.     

Local Smoothing Using the Bootstrap

We consider the problem of data-based choice of the bandwidth of a kernel density estimator, with an aim to estimate the density optimally at a given design point. The existing local bandwidth selectors seem to be quite sensitive to the underlying density and location of the design point. For instance, some bandwidth selectors perform poorly while estimating a density, with bounded support, at the median. Others struggle to estimate a density in the tail region or at the trough between the two modes of a multimodal density. We propose a scale invariant bandwidth selection method such that the resulting density estimator performs reliably irrespective of the density or the design point. We choose bandwidth by minimizing a bootstrap estimate of the mean squared error (MSE) of a density estimator. Our bootstrap MSE estimator is different in the sense that we estimate the variance and squared bias components separately. We provide insight into the asymptotic accuracy of the proposed density estimator.     

A Generalization of Turnbull’s Estimator for Interval-Censored and Doubly Truncated Data

Nonparametric estimates of the conditional distribution of a response variable given a covariate are important for data exploration purposes. In this article, we propose a nonparametric estimator of the conditional distribution function in the case where the response variable is subject to interval censoring and double truncation. Using the approach of Dehghan and Duchesne (2011), the proposed method consists in adding weights that depend on the covariate value in the self-consistency equation of Turnbull (1976), which results in a nonparametric estimator. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection all perform well in finite samples.     

Bootstrap bandwidth selection method for local linear estimator in exponential family models

Many biological experiments involve data whose distribution belongs to the exponential family. Such data are often analysed using generalised linear models but this method requires specification of the link function which can have strong influence on the resulting estimate. Instead a local method based on quasi-likelihood can be used, but the choice of the smoothing parameter is crucial for its performance. A bootstrap bandwidth selection method is proposed and shown to be consistent. Examples of application to data from biological and psychometric experiments are given.     

Importance resampling for the smoothed bootstrap

Alternative methods of estimating properties of unknown distributions include the bootstrap and the smoothed bootstrap. In the standard bootstrap setting, Johns (1988) introduced an importance resam¬pling procedure that results in more accurate approximation to the bootstrap estimate of a distribution function or a quantile. With a suitable “exponential tilting” similar to that used by Johns, we derived a smoothed version of importance resampling in the framework of the smoothed bootstrap. Smoothed importance resampling procedures were developed for the estimation of distribution functions of the Studentized mean, the Studentized variance, and the correlation coefficient. Implementation of these procedures are presented via simulation results which concentrate on the problem of estimation of distribution functions of the Studentized mean and Studentized variance for different sample sizes and various pre-specified smoothing bandwidths for the normal data; additional simulations were conducted for the estimation of quantiles of the distribution of the Studentized mean under an optimal smoothing bandwidth when the original data were simulated from three different parent populations: lognormal, t(3) and t(10). These results suggest that in cases where it is advantageous to use the smoothed bootstrap rather than the standard bootstrap, the amount of resampling necessary might be substantially reduced by the use of importance resampling methods and the efficiency gains depend on the bandwidth used in the kernel density estimation.     

Random Weighted Bootstrap Method for Recurrent Events with Informative Censoring

Using the data from the AIDS Link to Intravenous Experiences cohort study as an example, an informative censoring model was used to characterize the repeated hospitalization process of a group of patients. Under the informative censoring assumption, the estimators of the baseline rate function and the regression parameters were shown to be related to a latent variable. Hence, it becomes impractical to directly estimate the unknown quantities in the moments of the estimators for the bandwidth selection of a smoothing estimator and the construction of confidence intervals, which are respectively based on the asymptotic mean squared errors and the asymptotic distributions of the estimators. To overcome these difficulties, we develop a random weighted bootstrap procedure to select appropriate bandwidths and to construct approximated confidence intervals. One can see that our method is simple and faster to implement from a practical point of view, and is at least as accurate as other bootstrap methods. In this article, it is shown that the proposed method is useful through the performance of a Monte Carlo simulation. An application of our procedure is also illustrated by a recurrent event sample of intravenous drug users for inpatient cares over time.     

Accuracy Analysis of the Percentile Method for Estimating Non Normal Manufacturing Quality

We consider asymmetric kernel estimates based on grouped data. We propose an iterated scheme for constructing such an estimator and apply an iterated smoothed bootstrap approach for bandwidth selection. We compare our approach with competing methods in estimating actuarial loss models using both simulations and data studies. The simulation results show that with this new method, the estimated density from grouped data matches the true density more closely than with competing approaches.     

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.     

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.     

Variable Selection Methods in High-dimensional Regression—A Simulation Study

A challenging problem in the analysis of high-dimensional data is variable selection. In this study, we describe a bootstrap based technique for selecting predictors in partial least-squares regression (PLSR) and principle component regression (PCR) in high-dimensional data. Using a bootstrap-based technique for significance tests of the regression coefficients, a subset of the original variables can be selected to be included in the regression, thus obtaining a more parsimonious model with smaller prediction errors. We compare the bootstrap approach with several variable selection approaches (jack-knife and sparse formulation-based methods) on PCR and PLSR in simulation and real data.     

Parallel Bootstrap and Optimal Subsample Lengths in Smooth Function Models

Parallel bootstrap is an extremely useful statistical method with good performance. In the present study, we introduce a working correlation matrix on the method, which is called parallel bootstrap matrix. We consider some properties of it and the optimal size of the subsample in smooth function models. We also present some performance results of parallel bootstrap estimators, the subsample length selection on the finance time series data.     

Nonparametric localized bandwidth selection for Kernel density estimation

As conventional cross-validation bandwidth selection methods do not work properly in the situation where the data are serially dependent time series, alternative bandwidth selection methods are necessary. In recent years, Bayesian-based methods for global bandwidth selection have been studied. Our experience shows that a global bandwidth is however less suitable than a localized bandwidth in kernel density estimation based on serially dependent time series data. Nonetheless, a di?cult issue is how we can consistently estimate a localized bandwidth. This paper presents a nonparametric localized bandwidth estimator, for which we establish a completely new asymptotic theory. Applications of this new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate and the S&P 500 daily return demonstrate the effectiveness and competitiveness of the proposed localized bandwidth.     

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.     

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.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.     

Smooth nonparametric quantile estimation under censoring: simulations and bootstrap methods

Based on right-censored data from a lifetime distribution F , a smooth nonparametric estimator of the quantile function Q (p) is given by Qn(p)=h 1jjQn(t)K((t-p)/h)dt, where QR(p) denotes the product-limit quantile function. Extensive Monte Carlo simulations indicate that at fixed p this kernel-type quantile estimator has smaller mean squared error than (L(p) for a range of values of the bandwidth h. A method of selecting an "optimal" bandwidth (in the sense of small estimated mean squared error) based on the bootstrap is investigated yielding results consistent with the simulation study. The bootstrap is also used to obtain interval estimates for Q (p) after determining the optimal value of h.