On the Accuracy of Binned Kernel Density Estimators

The accuracy of the binned kernel density estimator is studied for general binning rules. We derive mean squared error results for the closeness of this estimator to both the true density and the unbinned kernel estimator. The binning rule and smoothness of the kernel function are shown to influence the accuracy of the binned kernel estimators. Our results are used to compare commonly used binning rules, and to determine the minimum grid size required to obtain a given level of accuracy.     

Multivariate Hazard Rates under Random Censorship

The data consists of multivariate failure times under right random censorship. By the kernel smoothing technique, convolutions of cumulative multivariate hazard functions suggest estimators of the so-called multivariate hazard functions. We establish strong i.i.d. representations and uniform bounds of the remainder terms on some compact sets of the underlying space. Thus asymptotic normality and uniform consistency on such sets are obtained. The asymptotic mean squared error gives an optimal bandwidth by the plug-in method. Simulations assess the performance of our estimators.     

Fast Algorithms for Nonparametric Curve Estimation

Abstract

Naive implementations of local polynomial fits and kernel estimators require almost O(n ²) operations. In this article two fast O(n) algorithms for nonparametric local polynomial fitting are presented. They are based on updating normal equations. Numerical stability is guaranteed by controlling ill-conditioned situations for small bandwidths and data-tuned restarting of the updating procedure. Restarting at every output point results in a moderately fast but highly stable O(n ^7/5) algorithm. Applicability of algorithms is evaluated for estimation of regression curves and their derivatives. The idea is also applied to kernel estimators of regression curves and densities.     

Invariant tests for symmetry about an unspecified point based on the empirical characteristic function

This paper considers a flexible class of omnibus affine invariant tests for the hypothesis that a multivariate distribution is symmetric about an unspecified point. The test statistics are weighted integrals involving the imaginary part of the empirical characteristic function of suitably standardized given data, and they have an alternative representation in terms of an L²-distance of nonparametric kernel density estimators. Moreover, there is a connection with two measures of multivariate skewness. The tests are performed via a permutational procedure that conditions on the data.     

Linear and convex aggregation of density estimators

We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.      

Fast Computation of Auxiliary Quantities in Local Polynomial Regression

Abstract

We investigate the extension of binning methodology to fast computation of several auxiliary quantities that arise in local polynomial smoothing. Examples include degrees of freedom measures, cross-validation functions, variance estimates, and exact measures of error. It is shown that the computational effort required for such approximations is of the same order of magnitude as that required for a binned local polynomial smooth.     

Dichteschätzungen mit gleichverteilungsmethoden

Density estimates with methods of uniform distribution mod 1. Some nonparametric multivariate density estimators for continuous functions, based on the Fejér and Jackson kernel, are presented. Uniform strong consistency results are obtained with methods of uniform distribution mod 1.

     

Asymptotics for Kernel Estimation of Slicing Average Third-Moment Estimation

To estimate central dimension-reduction space in multivariate nonparametric rcgression, Sliced Inverse Regression （SIR）, Sliced Average Variance Estimation （SAVE） and Slicing Average Third-moment Estimation （SAT） have been developed, Since slicing estimation has very different asymptotic behavior for SIR, and SAVE, the relevant study has been madc case by case, when the kernel estimators of SIH and SAVE share similar asymptotic properties. In this paper, we also investigate kernel estimation of SAT. We. prove the asymptotic normality, and show that, compared with tile existing results, the kernel Slnoothing for SIR, SAVE and SAT has very similar asymptotic behavior,     

Minimum disparity estimation for continuous models: Efficiency,distributions and robustness

A general class of minimum distance estimators for continuous models called minimum disparity estimators are introduced. The conventional technique is to minimize a distance between a kernel density estimator and the model density. A new approach is introduced here in which the model and the data are smoothed with the same kernel. This makes the methods consistent and asymptotically normal independently of the value of the smoothing parameter; convergence properties of the kernel density estimate are no longer necessary. All the minimum distance estimators considered are shown to be first order efficient provided the kernel is chosen appropriately. Different minimum disparity estimators are compared based on their characterizing residual adjustment function (RAF); this function shows that the robustness features of the estimators can be explained by the shrinkage of certain residuals towards zero. The value of the second derivative of theRAF at zero,A ₂, provides the trade-off between efficiency and robustness. The above properties are demonstrated both by theorems and by simulations.     

Estimating and Visualizing Conditional Densities

Abstract

We consider the kernel estimator of conditional density and derive its asymptotic bias, variance, and mean-square error. Optimal bandwidths (with respect to integrated mean-square error) are found and it is shown that the convergence rate of the density estimator is order n ^–2/3. We also note that the conditional mean function obtained from the estimator is equivalent to a kernel smoother. Given the undesirable bias properties of kernel smoothers, we seek a modified conditional density estimator that has mean equivalent to some other nonparametric regression smoother with better bias properties. It is also shown that our modified estimator has smaller mean square error than the standard estimator in some commonly occurring situations. Finally, three graphical methods for visualizing conditional density estimators are discussed and applied to a data set consisting of maximum daily temperatures in Melbourne, Australia.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

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.     

A Fast and Stable Updating Algorithm for Bivariate Nonparametric Curve Estimation

Abstract

An updating algorithm for bivariate local linear regression is proposed. Thereby, we assume a rectangular design and a polynomial kernel constrained to rectangular support as weight function. Results of univariate regression estimators are extended to the bivariate setting. The updates are performed in a way that most of the well-known numerical instabilities of a naive update implementation can be avoided. Some simulation results illustrate the properties of several algorithms with respect to computing time and numerical stability.     

Kernel Density and Hazard Rate Estimation for Censored Data under α-Mixing Condition

We derive rates of uniform strong convergence for kernel density estimators and hazard rate estimators in the presence of right censoring. It is assumed that the failure times (survival times) form a stationary -mixing sequence. Moreover, we show that, by an appropriate choice of the bandwidth, both estimators attain the optimal strong convergence rate known from independent complete samples. The results represent an improvement over that of Cai's paper (cf. Cai (1998b, J. Multivariate Anal., 67, 23–34)).     

A Visualization Technique for Studying the Iterative Estimation of Mixture Densities

Abstract

This article focuses on recent work that analyzes the expectation maximization (EM) evolution of mixtures-based estimators. The goal of this research is the development of effective visualization techniques to portray the mixture model parameters as they change in time. This is an inherently high-dimensional process. Techniques are presented that portray the time evolution of univariate, bivariate, and trivariate finite and adaptive mixtures estimators. Adaptive mixtures is a recently developed variable bandwidth kernel estimator where each of the kernels is not constrained to reside at a sample location. The future role of these techniques in developing new versions of the adaptive mixtures procedure is also discussed.     

On the estimation of entropy

Motivated by recent work of Joe (1989,Ann. Inst. Statist. Math.,41, 683–697), we introduce estimators of entropy and describe their properties. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. In particular, we argue that root-n consistency of entropy estimation requires appropriate assumptions about each of these three features. Our estimators are different from Joe's, and may be computed without numerical integration, but it can be shown that the same interaction of tail behaviour, smoothness and dimensionality also determines the convergence rate of Joe's estimator. We study both histogram and kernel estimators of entropy, and in each case suggest empirical methods for choosing the smoothing parameter.     

Nonparametric estimation of the spectral measure,and associated dependence measures,for multivariate extreme values using a limiting conditional representation

The traditional approach to multivariate extreme values has been through the multivariate extreme value distribution G, characterised by its spectral measure H and associated Pickands’ dependence function A. More generally, for all asymptotically dependent variables, H determines the probability of all multivariate extreme events. When the variables are asymptotically dependent and under the assumption of unit Fréchet margins, several methods exist for the estimation of G, H and A which use variables with radial component exceeding some high threshold. For each of these characteristics, we propose new asymptotically consistent nonparametric estimators which arise from Heffernan and Tawn’s approach to multivariate extremes that conditions on variables with marginal values exceeding some high marginal threshold. The proposed estimators improve on existing estimators in three ways. First, under asymptotic dependence, they give self-consistent estimators of G, H and A; existing estimators are not self-consistent. Second, these existing estimators focus on the bivariate case, whereas our estimators extend easily to describe dependence in the multivariate case. Finally, for asymptotically independent cases, our estimators can model the level of asymptotic independence; whereas existing estimators for the spectral measure treat the variables as either being independent, or asymptotically dependent. For asymptotically dependent bivariate random variables, the new estimators are found to compare favourably with existing estimators, particularly for weak dependence. The method is illustrated with an application to finance data.     

Strong convergence in nonparametric regression with truncated dependent data

In this paper we derive rates of uniform strong convergence for the kernel estimator of the regression function in a left-truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The estimation of the covariate’s density is considered as well. Under the assumption that the lifetime observations are bounded, we show that, by an appropriate choice of the bandwidth, both estimators of the covariate’s density and regression function attain the optimal strong convergence rate known from independent complete samples.     

Hazard Rate Regression Using Ordinary Nonparametric Regression Smoothers

Abstract

This article proposes a method for nonparametric estimation of hazard rates as a function of time and possibly multiple covariates. The method is based on dividing the time axis into intervals, and calculating number of event and follow-up time contributions from the different intervals. The number of event and follow-up time data are then separately smoothed on time and the covariates, and the hazard rate estimators obtained by taking the ratio. Pointwise consistency and asymptotic normality are shown for the hazard rate estimators for a certain class of smoothers, which includes some standard approaches to locally weighted regression and kernel regression. It is shown through simulation that a variance estimator based on this asymptotic distribution is reasonably reliable in practice. The problem of how to select the smoothing parameter is considered, but a satisfactory resolution to this problem has not been identified. The method is illustrated using data from several breast cancer clinical trials.