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 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.     

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.     

Boundary Aware Estimators of Integrated Density Derivative Products

Integrated squared density derivatives are important to the plug-in type of bandwidth selector for kernel density estimation. Conventional estimators of these quantities are inefficient when there is a non-smooth boundary in the support of the density. We introduce estimators that utilize density derivative estimators obtained from local polynomial fitting. They retain the rates of convergence in mean-squared error that are familiar from non-boundary cases, and the constant coefficients have similar forms. The estimators and the formula for their asymptotically optimal bandwidths, which depend on integrated products of density derivatives, are applied to automatic bandwidth selection for local linear density estimation. Simulation studies show that the constructed bandwidth rule and the Sheather–Jones bandwidth are competitive in non-boundary cases, but the former overcomes boundary problems whereas the latter does not.     

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.     

The influence function of the optimal bandwidth for kernel density estimation

Theories about the bandwidth of kernel density estimation have been well established by many statisticians. However, the influence function of the bandwidth has not been well investigated. The influence function of the optimal bandwidth that minimizes the mean integrated square error is derived and the asymptotic property of the bandwidth selectors based on the influence function is provided.     

Local bandwidth selectors for deconvolution kernel density estimation

We consider kernel density estimation when the observations are contaminated by measurement errors. It is well-known that the success of kernel estimators depends heavily on the choice of a smoothing parameter called the bandwidth. A number of data-driven bandwidth selectors exist, but they are all global. Such techniques are appropriate when the density is relatively simple, but local bandwidth selectors can be more attractive in more complex settings. We suggest several data-driven local bandwidth selectors and illustrate via simulations the significant improvement they can bring over a global bandwidth.     

Bandwidth selection for kernel density estimation with length-biased data

Length-biased data are a particular case of weighted data, which arise in many situations: biomedicine, quality control or epidemiology among others. In this paper we study the theoretical properties of kernel density estimation in the context of length-biased data, proposing two consistent bootstrap methods that we use for bandwidth selection. Apart from the bootstrap bandwidth selectors we suggest a rule-of-thumb. These bandwidth selection proposals are compared with a least-squares cross-validation method. A simulation study is accomplished to understand the behaviour of the procedures in finite samples.     

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.     

Iterated Bootstrap-t Confidence Intervals for Density Functions

Abstract.  Conventional bootstrap- t intervals for density functions based on kernel density estimators exhibit poor coverages due to failure of the bootstrap to estimate the bias correctly. The problem can be resolved by either estimating the bias explicitly or undersmoothing the kernel density estimate to undermine its bias asymptotically. The resulting bias-corrected intervals have an optimal coverage error of order arbitrarily close to second order for a sufficiently smooth density function. We investigated the effects on coverage error of both bias-corrected intervals when the nominal coverage level is calibrated by the iterated bootstrap. In either case, an asymptotic reduction of coverage error is possible provided that the bias terms are handled using an extra round of smoothed bootstrapping. Under appropriate smoothness conditions, the optimal coverage error of the iterated bootstrap- t intervals has order arbitrarily close to third order. Examples of both simulated and real data are reported to illustrate the iterated bootstrap procedures.     

Absolute error criteria for bandwidth selection in density estimation from censored data

In Kernel density estimation, a criticism of bandwidth selection techniques which minimize squared error expressions is that they perform poorly when estimating tails of probability density functions. Techniques minimizing absolute error expressions are thought to result in more uniform performance and be potentially superior. An asympotic mean absolute error expression for nonparametric kernel density estimators from right-censored data is developed here. This expression is used to obtain local and global bandwidths that are optimal in the sense that they minimize asymptotic mean absolute error and integrated asymptotic mean absolute error, respectively. These estimators are illustrated fro eight data sets from known distributions. Computer simulation results are discussed, comparing the estimation methods with squared-error-based bandwidth selection for right-censored data.     

Local Adaptivity of Kernel Estimates with Plug-in Local Bandwidth Selectors

In non-parametric function estimation selection of a smoothing parameter is one of the most important issues. The performance of smoothing techniques depends highly on the choice of this parameter. Preferably the bandwidth should be determined via a data-driven procedure. In this paper we consider kernel estimators in a white noise model, and investigate whether locally adaptive plug-in bandwidths can achieve optimal global rates of convergence. We consider various classes of functions: Sobolev classes, bounded variation function classes, classes of convex functions and classes of monotone functions. We study the situations of pilot estimation with oversmoothing and without oversmoothing. Our main finding is that simple local plug-in bandwidth selectors can adapt to spatial inhomogeneity of the regression function as long as there are no local oscillations of high frequency. We establish the pointwise asymptotic distribution of the regression estimator with local plug-in bandwidth.     

Adaptive normal reference bandwidth based on quantile for kernel density estimation

Bandwidth selection is an important problem of kernel density estimation. Traditional simple and quick bandwidth selectors usually oversmooth the density estimate. Existing sophisticated selectors usually have computational difficulties and occasionally do not exist. Besides, they may not be robust against outliers in the sample data, and some are highly variable, tending to undersmooth the density. In this paper, a highly robust simple and quick bandwidth selector is proposed, which adapts to different types of densities.     

UNCONSTRAINED PILOT SELECTORS FOR SMOOTHED CROSS‐VALIDATION

Two of the most useful multivariate bandwidth selection techniques are the plug‐in and cross‐validation methods. The smoothed version of the cross‐validation method is known to reduce the variability of its non‐smoothed counterpart; however, it shares with the plug‐in choice the need for a pilot bandwidth matrix. Owing to the mathematical difficulties encountered in the optimal pilot choice, it is common to restrict this pilot matrix to be a scalar multiple of the identity matrix, at the expense of losing the flexibility afforded by the unconstrained approach. Here we show how to overcome these difficulties and propose a smoothed cross‐validation selector using an unconstrained pilot matrix. Our numerical results indicate that the unconstrained selector outperforms the constrained one in practice, and is a viable competitor to unconstrained plug‐in selectors.     

Data-Based Choice of the Number of Pilot Stages for Plug-in Bandwidth Selection

The choice of the bandwidth is a crucial issue for kernel density estimation. Among all the data-dependent methods for choosing the bandwidth, the direct plug-in method has shown a particularly good performance in practice. This procedure is based on estimating an asymptotic approximation of the optimal bandwidth, using two “pilot” kernel estimation stages. Although two pilot stages seem to be enough for most densities, for a long time the problem of how to choose an appropriate number of stages has remained open. Here we propose an automatic (i.e., data-based) method for choosing the number of stages to be employed in the plug-in bandwidth selector. Asymptotic properties of the method are presented and an extensive simulation study is carried out to compare its small-sample performance with that of the most recommended bandwidth selectors in the literature.     

An evaluation of likelihood-based bandwidth selectors for spatial and spatiotemporal kernel estimates

Spatial point pattern data sets are commonplace in a variety of different research disciplines. The use of kernel methods to smooth such data is a flexible way to explore spatial trends and make inference about underlying processes without, or perhaps prior to, the design and fitting of more intricate semiparametric or parametric models to quantify specific effects. The long-standing issue of ‘optimal’ data-driven bandwidth selection is complicated in these settings by issues such as high heterogeneity in observed patterns and the need to consider edge correction factors. We scrutinize bandwidth selectors built on leave-one-out cross-validation approximation to likelihood functions. A key outcome relates to previously unconsidered adaptive smoothing regimens for spatiotemporal density and multitype conditional probability surface estimation, whereby we propose a novel simultaneous pilot-global selection strategy. Motivated by applications in epidemiology, the results of both simulated and real-world analyses suggest this strategy to be largely preferable to classical fixed-bandwidth estimation for such data.     

Multivariate bandwidth selection for local linear regression

The existence and properties of optimal bandwidths for multivariate local linear regression are established, using either a scalar bandwidth for all regressors or a diagonal bandwidth vector that has a different bandwidth for each regressor. Both involve functionals of the derivatives of the unknown multivariate regression function. Estimating these functionals is difficult primarily because they contain multivariate derivatives. In this paper, an estimator of the multivariate second derivative is obtained via local cubic regression with most cross-terms left out. This estimator has the optimal rate of convergence but is simpler and uses much less computing time than the full local estimator. Using this as a pilot estimator, we obtain plug-in formulae for the optimal bandwidth, both scalar and diagonal, for multivariate local linear regression. As a simpler alternative, we also provide rule-of-thumb bandwidth selectors. All these bandwidths have satisfactory performance in our simulation study.     

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.     

An Investigation of Quantile Function Estimators Relative to Quantile Confidence Interval Coverage

In this article, we investigate the limitations of traditional quantile function estimators and introduce a new class of quantile function estimators, namely, the semi-parametric tail-extrapolated quantile estimators, which has excellent performance for estimating the extreme tails with finite sample sizes. The smoothed bootstrap and direct density estimation via the characteristic function methods are developed for the estimation of confidence intervals. Through a comprehensive simulation study to compare the confidence interval estimations of various quantile estimators, we discuss the preferred quantile estimator in conjunction with the confidence interval estimation method to use under different circumstances. Data examples are given to illustrate the superiority of the semi-parametric tail-extrapolated quantile estimators. The new class of quantile estimators is obtained by slight modification of traditional quantile estimators, and therefore, should be specifically appealing to researchers in estimating the extreme tails.     

Variable kernel density estimation

This paper considers the problem of selecting optimal bandwidths for variable (sample‐point adaptive) kernel density estimation. A data‐driven variable bandwidth selector is proposed, based on the idea of approximating the log‐bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross‐validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions.