Bayesian local bandwidth selector in multivariate associated kernel estimator for joint probability mass functions

ABSTRACT

This work treats non-parametric estimation of multivariate probability mass functions, using multivariate discrete associated kernels. We propose a Bayesian local approach to select the matrix of bandwidths considering the multivariate Dirac Discrete Uniform and the product of binomial kernels, and treating the bandwidths as a diagonal matrix of parameters with some prior distribution. The performances of this approach and the cross-validation method are compared using simulations and real count data sets. The obtained results show that the Bayes local method performs better than cross-validation in terms of integrated squared error.     

A family of kernels and their associated deconvolving kernels for normally distributed measurement errors

A family of kernels (with the sinc kernel as the simplest member) is introduced for which the associated deconvolving kernels (assuming normally distributed measurement errors) can be represented by relatively simple analytic functions. For this family, deconvolving kernel density estimation is not more sophisticated than ordinary kernel density estimation. Application examples suggest that it may be advantageous to overestimate the measurement error, because the resulting deconvolving kernels can partially compensate for the blurring inherent to the density estimation itself. A corollary of this proposition is that, even without error, it may be rational to use deconvolving rather than ordinary kernels.     

A Simple Estimator of Error Correlation in Non-parametric Regression Models

Abstract.  It is well known that major strength of non-parametric regression function estimation breaks down when correlated errors exist in the data. Positively (negatively) correlated errors tend to produce undersmoothing (oversmoothing). Several remedies have been proposed in the context of bandwidth selection problem, but they are hard to implement without prior knowledge of error correlations. In this paper we propose a simple estimator of error correlation which is ready to implement and reports a reasonably good performance.     

What happens when bootstrapping the smoothing spline

The usual smoothing spline method is modified by a bootstrap bias correction inserted. It is shown that such a modification is asymptotically equivalent to a higher order method in sense they share the same best obtainable mean square error convergence rate. Similar results about the kernel method and their relation are discussed. It turns out that all of our effort towards better mean square error convergence rates result in some equivalent higher order kernels. The Fourier analysis canied out by Rice and Rosenblatt (1983) is used as the working horse.     

Exact mean integrated squared error and bandwidth selection for kernel distribution function estimators

Abstract

An exact, closed form, and easy to compute expression for the mean integrated squared error (MISE) of a kernel estimator of a normal mixture cumulative distribution function is derived for the class of arbitrary order Gaussian-based kernels. Comparisons are made with MISE of the empirical distribution function, the infeasible minimum MISE, and the uniform kernel. A simple plug-in method of simultaneously selecting the optimal bandwidth and kernel order is proposed based on a non asymptotic approximation of the unknown distribution by a normal mixture. A simulation study shows that the method provides a viable alternative to existing bandwidth selection procedures.     

Asymptotic normality of the multiscale wavelet density estimator

A multiscale wavelet density estimator (MWDE) was recently introduced and was shown to have nice convergence properties as well as good simulation results (Wu, 1995). This paper studies the asymptotic normality of the MWDE. It is proved that, under mild conditions, the MWDE has the asymptotic normality in the support of the unknown density f.As by-products, the author establishes the asymptotic normality of the wavelet estimator and discovers several interesting statistical properties of the reproducing kernel q_m(x,t)ofV_m .     

Non Parametric Estimation of Second-Order Diffusion Equation by Using Asymmetric Kernels

In this paper, we study the non parametric estimation of drift coefficient and diffusion coefficient in the second-order diffusion equation by using the asymmetric kernel functions, based on the difference of discrete time observations. The basic idea relies upon replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the non parametric estimation for second-order diffusion equation. Under the appropriate assumptions, we prove that the proposed estimators of second-order diffusion equation are consistent and asymptotically follow normal distribution.     

Multiple Kernel Procedure: an Asymptotic Support

ABSTRACT. This paper deals with kernel non-parametric estimation. The multiple kernel method, as proposed by Berlinet (1993), consists in choosing both the smoothing parameter and the order of the kernel function. In this paper we follow this general idea, and the selection is carried out by a combination of plug-in and cross-validation techniques. In a first attempt we give an asymptotic optimality theorem which is stated in a general unifying setting that includes many curve estimation problems. Then, as an illustration, it will be seen how this behaves in both special cases of kernel density and kernel regression estimation.     

Limiting bias-reduced Amoroso kernel density estimators for non-negative data

The Amoroso kernel density estimator (Igarashi and Kakizawa 2017 Igarashi, G., and Y. Kakizawa. 2017. Amoroso kernel density estimation for nonnegative data and its bias reduction. Department of Policy and Planning Sciences Discussion Paper Series No. 1345, University of Tsukuba. [Google Scholar]) for non-negative data is boundary-bias-free and has the mean integrated squared error (MISE) of order O(n^{? 4/5}), where n is the sample size. In this paper, we construct a linear combination of the Amoroso kernel density estimator and its derivative with respect to the smoothing parameter. Also, we propose a related multiplicative estimator. We show that the MISEs of these bias-reduced estimators achieve the convergence rates n^{? 8/9}, if the underlying density is four times continuously differentiable. We illustrate the finite sample performance of the proposed estimators, through the simulations.     

Minimax properties of beta kernel estimators

In this paper, we are interested in the study of beta kernel density estimators from an asymptotic minimax point of view. These estimators allows to estimate density functions with support in [0,1]. It is well-known that beta kernel estimators are - on the contrary of classical kernel estimators - “free of boundary effect” and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular density functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses.     

Estimating a frequency distribution when the sampling is biased

A simple random sample on a random variable A allows its density to be consistently estimated, by a histogram or preferably a kernel density estimate. When the sampling is biased towards certain x-values these methods instead estimate a weighted version of the density function. This article proposes a method for estimating both the density and the sampling bias simultaneously. The technique requires two independent samples and utilises ideas from mark-recapture experiments. An estimator of the size of the sampled population also follows simply from this density estimate.     

Nonparametric boundary detection

We propose a two-step nonparametric method for detecting the boundary curve of an object in an image. First we treat boundary points as change-points on lines across the image, and identify them by the one-sided kernel smoothing method. After obtaining potential boundary points, we use the principal curve method to smooth these points in order to obtain an estimate of smooth boundary curve, Computer simulations are provided to illustrate the effectiveness of the method.     

Convergence de I estimateur a noyau de derivees de Radon-Nikodym generales dans le cas melangeant

In many situations, nonparametric inference in point-process theory consists in estimating a Radon-Nikodym derivative of a nonnegative measure p with respect to another nonnegative measure v, where p and v are intensities of point processes. We consider the case of a mixing andstrictly stationary sequence of point processes and establish convergence results for the kernel estimator.     

Asymptotic Normality of Kernel-Type Deconvolution Estimators

Abstract.  We derive asymptotic normality of kernel-type deconvolution estimators of the density, the distribution function at a fixed point, and of the probability of an interval. We consider so-called super smooth deconvolution problems where the characteristic function of the known distribution decreases exponentially, but faster than that of the Cauchy distribution. It turns out that the limit behaviour of the pointwise estimators of the density and distribution function is relatively straightforward, while the asymptotic behaviour of the estimator of the probability of an interval depends in a complicated way on the sequence of bandwidths.     

Smooth nonparametric estimation of thedistribution and density functions from record-breaking data

In some experiments, such as destructive stress testing and industrial quality control experiments, only values smaller than all previous ones are observed. Here, for such record-breaking data, kernel estimation of the cumulative distribution function and smooth density estimation is considered. For a single record-breaking sample, consistent estimation is not possible, and replication is required for global results. For m independent record-breaking samples, the proposed distribution function and density estimators are shown to be strongly consistent and asymptotically normal as m → ∞. Also, for small m, the mean squared errors and biases of the estimators and their smoothing parameters are investigated through computer simulations.     

Non-parametric Variance Estimation from Ergodic Samples

It is shown that a kernel estimate for the variance function is uniformly strongly consistent, under the ergodic condition     

Pointwise and uniform convergence of multivariate kernel density estimators using random bandwidths

We obtain the rates of pointwise and uniform convergence of multivariate kernel density estimators using a random bandwidth vector obtained by some data-based algorithm. We are able to obtain faster rate for pointwise convergence. The uniform convergence rate is obtained under some moment condition on the marginal distribution. The rates are obtained under i.i.d. and strongly mixing type dependence assumptions.