Nonlinear wavelet density estimation with censored dependent data

In this paper, we provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet estimator of survival density for a censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary and α‐mixing sequence. This asymptotic MISE expansion, when the density is only piecewise smooth, is same. However, for the kernel estimators, the MISE expansion fails if the additional smoothness assumption is absent. Also, we establish the asymptotic normality of the nonlinear wavelet estimator. Copyright © 2011 John Wiley & Sons, Ltd.     

Convergence rates in density estimation for data from infinite-order moving average processes

Summary The effect of long-range dependence in nonparametric probability density estimation is investigated under the assumption that the observed data are a sample from a stationary, infinite-order moving average process. It is shown that to first order, the mean integrated squared error (MISE) of a kernel estimator for moving average data may be expanded as the sum of MISE of the kernel estimator for a same-sizerandom sample, plus a term proportional to the variance of the moving average sample mean. The latter term does not depend on bandwidth, and so imposes a ceiling on the convergence rate of a kernel estimator regardless of how bandwidth is chosen. This ceiling can be quite significant in the case of long-range dependence. We show thatall density estimators have the convergence rate ceiling possessed by kernel estimators.The research of Dr. Hart was done while he was visiting the Australian National University, and was supported in part by ONR Contract N00014-85-K-0723     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

Asymptotically optimal choice of the smoothing parameter in a nonparametric kernel estimator for a probability density

This paper presents a method of estimation of an “optimal” smoothing parameter (window width) in kernel estimators for a probability density. The obtained estimator is calculated directly from observations. By “optimal” smoothing parameters we mean those parameters which minimize the mean integral square error (MISE) or the integral square error (ISE) of approximation of an unknown density by the kernel estimator. It is shown that the asymptotic “optimality” properties of the proposed estimator correspond (with respect to the order) to those of the well-known cross-validation procedure [1, 2]. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 67–80, Perm, 1990.     

On Kernel-Based Intensity Estimation of Spatial Point Patterns on Linear Networks

We propose an extension of Diggle’s nonparametric edge-corrected kernel-based intensity estimator to the case of events coming from an inhomogenous point pattern on a linear network. We analyze its statistical properties, showing that it is an unbiased estimator of the first-order intensity; we also provide an expression for the variance, and comment on the appropriate bandwidth selection. Our estimator is compared with the current existing equal-split discontinuous kernel density estimator in terms of the mean integrated squared error (MISE). We then use our estimator on two real datasets. We first revisit street crimes in an area of Chicago, obtaining similar results to previously published ones based on a parametric intensity function. Then, we study network-based spatial events consisting of calls to the Police department reporting anti-social behavior in the city of Castellon (Spain).     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Nonparametric estimation of multivariate density with direct and auxiliary data and application

We consider the problem of multivariate density estimation, using samples from the distribution of interest as well as auxiliary samples from a related distribution. We assume that the data from the target distribution and the related distribution may occur individually as well as in pairs. Using nonparametric maximum likelihood estimator of the joint distribution, we derive a kernel density estimator of the marginal density. We show theoretically, in a simple special case, that the implied estimator of the marginal density has smaller integrated mean squared error than that of a similar estimator obtained by ignoring dependence of the paired observations. We establish consistency of the marginal density estimator under suitable conditions. We demonstrate small sample superiority of the proposed estimator over the estimator that ignores dependence of the samples, through a simulation study with dependent and non-normal populations. The application of the density estimator in nonparametric classification is also discussed. It is shown that the misclassification probability of the resulting classifier is asymptotically equivalent to that of the Bayes classifier. We also include a data analytic illustration.     

Mean Integrated Squared Error of Nonlinear Wavelet-based Estimators with Long Memory Data

We consider the nonparametric regression model with long memory data that are not necessarily Gaussian and provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet-based mean regression function estimators. We show this MISE expansion, when the underlying mean regression function is only piecewise smooth, is the same as analogous expansion for the kernel estimators. However, for the kernel estimators, this MISE expansion generally fails if an additional smoothness assumption is absent. Research supported in part by the NSF grant DMS-0103939.     

Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Summary. Consider the stationary linear process , , where is an i.i.d. finite variance sequence. The spectral density of may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form , where denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels and for to converge to a Gaussian distribution. We show that this convergence holds if and are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain. Received: 28 February 1996 / In revised form: 10 July 1996     

Asymptotic normality of Powell’s kernel estimator

We establish asymptotic normality of Powell’s kernel estimator for the asymptotic covariance matrix of the quantile regression estimator for both i.i.d. and weakly dependent data. As an application, we derive the optimal bandwidth that minimizes the approximate mean squared error of the kernel estimator. We also derive the corresponding results to censored quantile regression.     

Almost sure convergence of the Bartlett estimator

Summary We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a.\ s.\ behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants and assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator.     

方向数据密度核估计的对数律

设X为取值于k维单位球面上的单位随机向量,具有概率密度函数f(x),X_1,…,X_n为X的n个i.i.d.的观察,讨论f(x)具有形式的核估计,其中K为定义于[0,+∞]上的非负核函数,ω_k为Ω_k上的Lebesque测度,本文建立了fn(x)的对数律,并给出了fn(x)的一致强相合速度。     

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.     

强相关数据回归函数的小波估计

讨论了在强相关数据情形下对回归函数的小波估计,并且给出了估计量的均方误差的一个渐近展开表示式． 对研究估计量的优劣,所推导的近似表示式显得非常重要．对一般的回归函数核估计,如果回归函数不是充分光滑,这个均方误差表示式并不成立A·D2但对小波估计,即使回归函数间断连续,这个均方误差表示式仍然成立．因此,小波估计的收敛速度要比核估计来得快,从而小波估计在某种程度上改进了现有的核估计．     

Some statistical properties of the kernel-diffeomorphism estimator

The kernel density estimation method is not so attractive when the density has its support confined to a bounded space U of R^d. In a recent paper, we suggested a new nonparametric probability density function (p.d.f.) estimator called the ‘kernel-diffeomorphism estimator’, which suppressed border convergence difficulties by using an appropriate regular change of variable. The present paper gives more asymptotic theory (uniform consistency, normality). An invariance criterion for p.d.f. estimators is discussed. The invariance of the kernel diffeomorphism estimator under special affine motion (a translation followed by any member of the special linear group SL(d, R) is proved. © 1997 by John Wiley & Sons, Ltd.     

Lower bounds for bandwidth selection in density estimation

Summary This paper establishes asymptotic lower bounds which specify, in a variety of contexts, how well (in terms of relative rate of convergence) one may select the bandwidth of a kernel density estimator. These results provide important new insights concerning how the bandwidth selection problem should be considered. In particular it is shown that if the error criterion is Integrated Squared Error (ISE) then, even under very strong assumptions on the underlying density, relative error of the selected bandwidth cannot be reduced below ordern ^–1/10 (as the sample size grows). This very large error indicates that any technique which aims specifically to minimize ISE will be subject to serious practical difficulties arising from sampling fluctuations. Cross-validation exhibits this very slow convergence rate, and does suffer from unacceptably large sampling variation. On the other hand, if the error criterion is Mean Integrated Squared Error (MISE) then relative error of bandwidth selection can be reduced to ordern ^–1/2, when enough smoothness is assumed. Therefore bandwidth selection techniques which aim to minimize MISE can be much more stable, and less sensitive to small sampling fluctuations, than those which try to minimize ISE. We feel this indicates that performance in minimizing MISE, rather than ISE, should become the benchmark for measuring performance of bandwidth selection methods.Research partially supported by National Science Foundation Grants DMS-8701201 and DMS-8902973Research of the first author was done while on leave from the Australian National University     

Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter

In this contribution, the statistical performance of the wavelet-based estimation procedure for the Hurst parameter is studied for non-Gaussian long-range dependent processes obtained from point transformations of Gaussian processes. The statistical properties of the wavelet coefficients and the estimation performance are compared both for processes having the same covariance but different marginal distributions and for processes having the same covariance and same marginal distributions but obtained from different point transformations, analyzed using mother wavelets with different number of vanishing moments. It is shown that the reduction of the dependence range from long to short by increasing the number of vanishing moments, observed for Gaussian processes, and at the origin of the popularity of the wavelet-based estimator, does not hold in general for non-Gaussian processes. Crucially, it is also observed that the Hermite rank of the point transformation impacts significantly the statistical properties of the wavelet coefficients and the estimation performance and also that processes having identical marginal distributions and covariance function can yet yield significantly different estimation performance. These results are interpreted in the light of central and noncentral limit theorems that are fundamental when dealing with long-range dependent processes. Moreover, it will be shown that, on condition that estimation is performed using a range of scales restricted to the coarsest practically available, an approximate, yet analytical and simple to use in practice, formula can be proposed for the evaluation of the variance of the wavelet-based estimator of the Hurst parameter.     

Polygonal smoothing of the empirical distribution function

We present two families of polygonal estimators of the distribution function: the first family is based on the knowledge of the support while the second addresses the case of an unknown support. Polygonal smoothing is a simple and natural method for regularizing the empirical distribution function \(F_n\) but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of \(F_n\). Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of \(F_n\). We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.     

A note on wavelet density deconvolution for weakly dependent data

In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.     

Consistency of kernel density estimators for causal processes

Using the blocking techniques and m-dependent methods,the asymptotic behavior of kernel density estimators for a class of stationary processes,which includes some nonlinear time series models,is investigated.First,the pointwise and uniformly weak convergence rates of the deviation of kernel density estimator with respect to its mean(and the true density function)are derived.Secondly,the corresponding strong convergence rates are investigated.It is showed,under mild conditions on the kernel functions and bandwidths,that the optimal rates for the i.i.d.density models are also optimal for these processes.