Kernel Smoothing Density Estimation when Group Membership is Subject to Missing

Density function is a fundamental concept in data analysis. Non-parametric methods including kernel smoothing estimate are available if the data is completely observed. However, in studies such as diagnostic studies following a two-stage design the membership of some of the subjects may be missing. Simply ignoring those subjects with unknown membership is valid only in the MCAR situation. In this paper, we consider kernel smoothing estimate of the density functions, using the inverse probability approaches to address the missing values. We illustrate the approaches with simulation studies and real study data in mental health.     

The Minimum Density Power Divergence Estimation for the Lognormal Density

In this article, we implement the minimum density power divergence estimation for estimating the parameters of the lognormal density. We compare the minimum density power divergence estimator (MDPDE) and the maximum likelihood estimator (MLE) in terms of robustness and asymptotic distribution. The simulations and an example indicate that the MDPDE is less biased than MLE and is as good as MLE in terms of the mean square error under various distributional situations.     

人口密度的统计价值

自有人类社会以来 ,人口和土地面积的统计是产生最早、历史最悠久的统计 ,人口密度也是最基本的反映国情国力的统计指标。如何发现并挖掘出人口密度指标应有的价值和使用价值 ,是本文讨论的一个主要目标。　　 一、人口密度与经济社会发展的关系引起人口密度变化的因素 ,除人口自然增长率各地略有不同以外 ,主要是因为人口迁移引起人口密度在空间分布短期内的较大变化。而引起人口迁移的因素较多 ,按性质来分 ,分为自然和社会经济两大因素。历史上一国自然环境条件直接导致大规模的移民 ,如旱灾、水灾、瘟疫等自然灾害的发生 ,都会发生大批…     

Kernel Density Estimation with Generalized Binning

We propose kernel density estimators based on prebinned data. We use generalized binning schemes based on the quantiles points of a certain auxiliary distribution function. Therein the uniform distribution corresponds to usual binning. The statistical accuracy of the resulting kernel estimators is studied, i.e. we derive mean squared error results for the closeness of these estimators to both the true function and the kernel estimator based on the original data set. Our results show the influence of the choice of the auxiliary density on the binned kernel estimators and they reveal that non-uniform binning can be worthwhile.     

Wavelet-Based Density Estimation in a Heteroscedastic Convolution Model

We consider a heteroscedastic convolution density model under the “ordinary smooth assumption.” We introduce a new adaptive wavelet estimator based on term-by-term hard thresholding rule. Its asymptotic properties are explored via the minimax approach under the mean integrated squared error over Besov balls. We prove that our estimator attains near optimal rates of convergence (lower bounds are determined). Simulation results are reported to support our theoretical findings.     

Skewing and Generalized Jackknifing in Kernel Density Estimation

Abstract

Kernel methods are very popular in nonparametric density estimation. In this article we suggest a simple estimator which reduces the bias to the fourth power of the bandwidth, while the variance of the estimator increases only by at most a moderate constant factor. Our proposal turns out to be a fourth order kernel estimator and may be regarded as a new version of the generalized jackknifing approach (Schucany W. R., Sommers, J. P. (1977 Schucany, W. R. and Sommers, J. P. 1977. Improvement of kernel type estimators. Journal of the American Statistical Association, 72: 420–423. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Improvement of Kernal type estimators. Journal of the American Statistical Association 72:420–423.) applied to kernel density estimation.     

Best Invariant and Minimax Estimation of Quantiles in Finite Populations

The theoretical literature on quantile and distribution function estimation in infinite populations is very rich, and invariance plays an important role in these studies. This is not the case for the commonly occurring problem of estimation of quantiles in finite populations. The latter is more complicated and interesting because an optimal strategy consists not only of an estimator, but also of a sampling design, and the estimator may depend on the design and on the labels of sampled individuals, whereas in iid sampling, design issues and labels do not exist.We study the estimation of finite population quantiles, with emphasis on estimators that are invariant under the group of monotone transformations of the data, and suitable invariant loss functions. Invariance under the finite group of permutation of the sample is also considered. We discuss nonrandomized and randomized estimators, best invariant and minimax estimators, and sampling strategies relative to different classes. Invariant loss functions and estimators in finite population sampling have a nonparametric flavor, and various natural combinatorial questions and tools arise as a result.     

Estimation of Several Intraclass Correlation Coefficients

An intraclass correlation coefficient observed in several populations is estimated. The basis is a variance-stabilizing transformation. It is shown that the intraclass correlation coefficient from any elliptical distribution should be transformed in the same way. Four estimators are compared. An estimator where the components in a vector consisting of the transformed intraclass correlation coefficients are estimated separately, an estimator based on a weighted average of these components, a pretest estimator where the equality of the components is tested and then the outcome of the test is used in the estimation procedure, and a James-Stein estimator which shrinks toward the mean.     

Automatic Local Smoothing for Spectral Density Estimation

This article uses local polynomial techniques to fit Whittle's likelihood for spectral density estimation. Asymptotic sampling properties of the proposed estimators are derived, and adaptation of the proposed estimator to the boundary effect is demonstrated. We show that the Whittle likelihood-based estimator has advantages over the least-squares based log-periodogram. The bandwidth for the Whittle likelihood-based method is chosen by a simple adjustment of a bandwidth selector proposed in Fan & Gijbels (1995). The effectiveness of the proposed procedure is demonstrated by a few simulated and real numerical examples. Our simulation results support the asymptotic theory that the likelihood based spectral density and log-spectral density estimators are the most appealing among their peers     

Logspline Density Estimation under Censoring and Truncation

ABSTRACT. In this paper we consider logspline density estimation for data that may be left-truncated or right-censored. For randomly left-truncated and right-censored data the product-limit estimator is known to be a consistent estimator of the survivor function, having a faster rate of convergence than many density estimators. The product-limit estimator and B-splines are used to construct the logspline density estimate for possibly censored or truncated data. Rates of convergence are established when the log-density function is assumed to be in a Besov space. An algorithm involving a procedure similar to maximum likelihood, stepwise knot addition, and stepwise knot deletion is proposed for the estimation of the density function based upon sample data. Numerical examples are used to show the finite-sample performance of inference based on the logspline density estimation.     

Robust Likelihood Cross-Validation for Kernel Density Estimation

Likelihood cross-validation for kernel density estimation is known to be sensitive to extreme observations and heavy-tailed distributions. We propose a robust likelihood-based cross-validation method to select bandwidths in multivariate density estimations. We derive this bandwidth selector within the framework of robust maximum likelihood estimation. This method establishes a smooth transition from likelihood cross-validation for nonextreme observations to least squares cross-validation for extreme observations, thereby combining the efficiency of likelihood cross-validation and the robustness of least-squares cross-validation. We also suggest a simple rule to select the transition threshold. We demonstrate the finite sample performance and practical usefulness of the proposed method via Monte Carlo simulations and a real data application on Chinese air pollution.     

我国人口控制综合效果核算

人口控制效果既表现为采取节育措施的人次数,如晚婚,避孕人数和终止妊娠人次数;亦表现为采取节育措施使育龄人口晚育、延长间隔生育和减少多胎生育;还表现为孩子出生数的减少,出生率和生育率的下降。而少生孩子必将导致孩子抚育成本的节约。因此,人口控制效果的表现形式是多种多样且各具特色的。但其中最能综合反映人口控制效果的指标是少生孩子数和孩子抚育成本节约额。然而,在我国人口统计实践中,初婚年龄、晚婚晚育     

Wavelet Estimation of a Density in a GARCH-type Model

We consider the GARCH-type model: S = σ² Z, where σ² and Z are independent random variables. The density of σ² is unknown whereas the one of Z is known. We want to estimate the density of σ² from n observations of S under some dependence assumption (the exponentially strongly mixing dependence). Adopting the wavelet methodology, we construct a nonadaptive estimator based on projections and an adaptive estimator based on the hard thresholding rule. Taking the mean integrated squared error over Besov balls, we prove that the adaptive one attains a sharp rate of convergence.     

Polynomial Histograms for Multivariate Density and Mode Estimation

Abstract. We consider the problem of efficiently estimating multivariate densities and their modes for moderate dimensions and an abundance of data. We propose polynomial histograms to solve this estimation problem. We present first‐ and second‐order polynomial histogram estimators for a general d‐dimensional setting. Our theoretical results include pointwise bias and variance of these estimators, their asymptotic mean integrated square error (AMISE), and optimal binwidth. The asymptotic performance of the first‐order estimator matches that of the kernel density estimator, while the second order has the faster rate of O(n^?6/(d+6)). For a bivariate normal setting, we present explicit expressions for the AMISE constants which show the much larger binwidths of the second order estimator and hence also more efficient computations of multivariate densities. We apply polynomial histogram estimators to real data from biotechnology and find the number and location of modes in such data.     

Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation

Abstract.  The performance of multivariate kernel density estimates depends crucially on the choice of bandwidth matrix, but progress towards developing good bandwidth matrix selectors has been relatively slow. In particular, previous studies of cross-validation (CV) methods have been restricted to biased and unbiased CV selection of diagonal bandwidth matrices. However, for certain types of target density the use of full (i.e. unconstrained) bandwidth matrices offers the potential for significantly improved density estimation. In this paper, we generalize earlier work from diagonal to full bandwidth matrices, and develop a smooth cross-validation (SCV) methodology for multivariate data. We consider optimization of the SCV technique with respect to a pilot bandwidth matrix. All the CV methods are studied using asymptotic analysis, simulation experiments and real data analysis. The results suggest that SCV for full bandwidth matrices is the most reliable of the CV methods. We also observe that experience from the univariate setting can sometimes be a misleading guide for understanding bandwidth selection in the multivariate case.     

Density Estimation for the Metropolis–Hastings Algorithm

Abstract. Kernel density estimation is an important tool in visualizing posterior densities from Markov chain Monte Carlo output. It is well known that when smooth transition densities exist, the asymptotic properties of the estimator agree with those for independent data. In this paper, we show that because of the rejection step of the Metropolis–Hastings algorithm, this is no longer true and the asymptotic variance will depend on the probability of accepting a proposed move. We find an expression for this variance and apply the result to algorithms for automatic bandwidth selection.     

Estimation of the Spectral Density with Assigned Risk

It is well known that adaptive sequential nonparametric estimation of differentiable functions with assigned mean integrated squared error and minimax expected stopping time is impossible. In other words, no sequential estimator can compete with an oracle estimator that knows how many derivatives an estimated curve has. Differentiable functions are typical in probability density and regression models but not in spectral density models, where considered functions are typically smoother. This paper shows that for a large class of spectral densities, which includes spectral densities of classical autoregressive moving average processes, an adaptive minimax sequential estimation with assigned mean integrated squared error is possible. Furthermore, a two‐stage sequential procedure is proposed, which is minimax and adaptive to smoothness of an underlying spectral density.     

On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.