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.     

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.     

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.     

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.