A New Characterization and Estimation of the Zero–bias Bandwidth

It is well established that bandwidths exist that can yield an unbiased non–parametric kernel density estimate at points in particular regions (e.g. convex regions) of the underlying density. These zero–bias bandwidths have superior theoretical properties, including a 1/n convergence rate of the mean squared error. However, the explicit functional form of the zero–bias bandwidth has remained elusive. It is difficult to estimate these bandwidths and virtually impossible to achieve the higher–order rate in practice. This paper addresses these issues by taking a fundamentally different approach to the asymptotics of the kernel density estimator to derive a functional approximation to the zero–bias bandwidth. It develops a simple approximation algorithm that focuses on estimating these zero–bias bandwidths in the tails of densities where the convexity conditions favourable to the existence of the zerobias bandwidths are more natural. The estimated bandwidths yield density estimates with mean squared error that is O(n^–4/5), the same rate as the mean squared error of density estimates with other choices of local bandwidths. Simulation studies and an illustrative example with air pollution data show that these estimated zero–bias bandwidths outperform other global and local bandwidth estimators in estimating points in the tails of densities.     

样本选择模型截距项的半参数估计及应用———户籍工资差异分解研究

样本选择模型是解决样本选择问题的主要工具,广泛应用于工资差异分解、平均处理效应测算等实证研究。截距项的估计是样本选择模型半参数估计中相对独立且重要的一部分,现有的以无穷处识别为代表的半参数估计方法存在窗宽参数难以选取的问题。为此,本文把无穷处识别等价转化为边界处识别,并基于新的识别关系给出样本选择模型截距项的核估计方法。这种新方法的好处在于将样本选择模型截距项的估计纳入核估计框架中,从而可以采用经验法则解决现有方法的窗宽选取难题。数值模拟结果表明,本文所提出的估计方法在不同设定下均有良好的有限样本表现。把这种新的半参数估计方法应用于户籍工资差异分解后发现,我国劳动力市场目前不存在明显的户籍差别待遇。     

Local orthogonal polynomial expansion for density estimation

A local orthogonal polynomial expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localised expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalises straightforwardly to multivariate settings. By manner of construction, it is similar to local likelihood density estimation (LLDE). In the limit of small bandwidths, LOrPE functions as kernel density estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Consistency and faster asymptotic convergence rates follow. In the limit of large bandwidths LOrPE is equivalent to orthogonal series density estimation (OSDE) with Legendre polynomials, thereby inheriting its consistency. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.     

Optimal smoothing in adaptive location estimation

In this paper, we consider higher order performance of kernel based adaptive location estimates. We show how much one loses in efficiency without knowing the underlying translation density, and derive the optimal order of the bandwidths involved in kernel estimation of the efficient score function. The optimal order is obtained by minimizing the loss of efficiency in terms of estimating the location parameter. The main lesson here is that the optimal order of the bandwidths are different from those for optimal estimation of the score function. This implies that optimal estimation of the score function does not lead to second order optimal location estimation.     

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.     

Bandwidth selection for nonparametric modal regression

In the context of estimating local modes of a conditional density based on kernel density estimators, we show that existing bandwidth selection methods developed for kernel density estimation are unsuitable for mode estimation. We propose two methods to select bandwidths tailored for mode estimation in the regression setting . Numerical studies using synthetic data and a real-life dataset are carried out to demonstrate the performance of the proposed methods in comparison with several well-received bandwidth selection methods for density estimation.     

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.     

Analytic bias correction for maximum likelihood estimators when the bias function is non-constant

Recently, many articles have obtained analytical expressions for the biases of various maximum likelihood estimators, despite their lack of closed-form solution. These bias expressions have provided an attractive alternative to the bootstrap. Unless the bias function is “flat,” however, the expressions are being evaluated at the wrong point(s). We propose an “improved” analytical bias-adjusted estimator, in which the bias expression is evaluated at a more appropriate point (at the bias adjusted estimator itself). Simulations illustrate that the improved analytical bias-adjusted estimator can eliminate significantly more bias than the simple estimator, which has been well established in the literature.     

On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives

Abstract. We consider the properties of the local polynomial estimators of a counting process intensity function and its derivatives. By expressing the local polynomial estimators in a kernel smoothing form via effective kernels, we show that the bias and variance of the estimators at boundary points are of the same magnitude as at interior points and therefore the local polynomial estimators in the context of intensity estimation also enjoy the automatic boundary correction property as they do in other contexts such as regression. The asymptotically optimal bandwidths and optimal kernel functions are obtained through the asymptotic expressions of the mean square error of the estimators. For practical purpose, we suggest an effective and easy‐to‐calculate data‐driven bandwidth selector. Simulation studies are carried out to assess the performance of the local polynomial estimators and the proposed bandwidth selector. The estimators and the bandwidth selector are applied to estimate the rate of aftershocks of the Sichuan earthquake and the rate of the Personal Emergency Link calls in Hong Kong.     

Zero-Bias Locally Adaptive Density Estimators

Strategies for improving fixed non-negative kernel estimators have focused on reducing the bias, either by employing higher-order kernels or by adjusting the bandwidth locally. Intuitively, bandwidths in the tails should be relatively larger in order to reduce wiggles since there is less data available in the tails. We show that in regions where the density function is convex, it is theoretically possible to find local bandwidths such that the pointwise bias is exactly zero. The corresponding pointwise mean squared error converges at the parametric rate of O ( n ⁻¹ ) rather than the slower O ( n ^−4/5). These so-called zero-bias bandwidths are constant and are usually orders of magnitude larger than the optimal locally adaptive bandwidths predicted by asymptotic mean squared error analysis. We describe data-based algorithms for estimating zero-bias bandwidths over intervals where the density is convex. We find that our particular density estimator attains the usual O ( n ^−4/5) rate. However, we demonstrate that the algorithms can provide significant improvement in mean squared error, often clearly visually superior curves, and a new operating point in the usual bias-variance tradeoff.     

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.     

Local and Variable Bandwidths and Local Linear Regression

Two types of non-global bandwidth, which may be called local and variable, have been defined in attempts to improve the performance of kernel density estimators. In nonparametric regression, local linear fitting has become a method of much popularity. It is natural, therefore, to consider the use of non-global bandwidths in the local linear context, and indeed local bandwidths are often used. In this paper, it is observed that a natural proposal in the literature for combining variable bandwidths with local linear fitting fails in the sense that the resulting mean squared error properties are those normally associated with local rather than variable bandwidths. We are able to understand why this happens in terms of weightings that are involved. We also attempt to investigate how the bias reduction expected of well-chosen variable bandwidths might be achieved in conjunction with local linear fitting.     

Central limit theorem for the variable bandwidth kernel density estimators

In this paper we study the ideal variable bandwidth kernel density estimator introduced by McKay (1993a, b) and Jones et al. (1994) and the plug-in practical version of the variable bandwidth kernel estimator with two sequences of bandwidths as in Giné and Sang (2013). Based on the bias and variance analysis of the ideal and plug-in variable bandwidth kernel density estimators, we study the central limit theorems for each of them. The simulation study confirms the central limit theorem and demonstrates the advantage of the plug-in variable bandwidth kernel method over the classical kernel method.     

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.     

Secondary design considerations for minimum bias estimation

Several authors have suggested the method of minimum bias estimation for estimating response surfaces. The minimum bias estimation procedure achieves minimum average squared bias of the fitted model without depending on the values of the unknown parameters of the true surface. The only requirement is that the design satisfies a simple estimability condition. Subject to providing minimum average squared bias, the minimum bias estimator also provides minimum average variance of ?(x) where ?(x) is the estimate of the response at the point x.

To support the estimation of the parameters in the fitted model, very little has been suggested in the way of experimental designs except to say that a full rank matrix X of independent variables should be used. This paper presents a closer look at the estimability conditions that are required for minimum bias estimation, and from the form of the matrix X, a formula is derived which measures the amount of design flexibility available. The design flexibility is termed “the degrees of freedom” of the X matrix and it is shown how the degrees of freedom can be used to decide if other design optimality criteria might be considered along with minimum bias estimation. Several examples are provided.     

CORRECTING FOR KURTOSIS IN DENSITY ESTIMATION

Using a global window width kernel estimator to estimate an approximately symmetric probability density with high kurtosis usually leads to poor estimation because good estimation of the peak of the distribution leads to unsatisfactory estimation of the tails and vice versa. The technique proposed corrects for kurtosis via a transformation of the data before using a global window width kernel estimator. The transformation depends on a “generalised smoothing parameter” consisting of two real-valued parameters and a window width parameter which can be selected either by a simple graphical method or, for a completely data-driven implementation, by minimising an estimate of mean integrated squared error. Examples of real and simulated data demonstrate the effectiveness of this approach, which appears suitable for a wide range of symmetric, unimodal densities. Its performance is similar to ordinary kernel estimation in situations where the latter is effective, e.g. Gaussian densities. For densities like the Cauchy where ordinary kernel estimation is not satisfactory, our methodology offers a substantial improvement.     

Hazard function estimation from homogeneous right censored data with missing censoring indicators

The kernel smoothed Nelson–Aalen estimator has been well investigated, but is unsuitable when some of the censoring indicators are missing. A representation introduced by Dikta, however, facilitates hazard estimation when there are missing censoring indicators. In this article, we investigate (i) a kernel smoothed semiparametric hazard estimator and (ii) a kernel smoothed “pre-smoothed” Nelson–Aalen estimator. We derive the asymptotic normality of the proposed estimators and compare their asymptotic variances.     

A note on bias reduction in variable-kernel density estimates

It is well known that the inverse-square-root rule of Abramson (1982) for the bandwidth h of a variable-kernel density estimator achieves a reduction in bias from the fixed-bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O(h⁴) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O(h⁴) bias expression holds uniformly on any set where the target density is bounded away from zero.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.