Wavelet density estimation for stratified size-biased sample

In this paper we consider the estimation of a density function on the basis of a random stratified sample from weighted distributions. We propose a linear wavelet density estimator and prove its consistency. The behavior of the proposed estimator and its smoothed versions is eventually illustrated by simulated examples and a case study involving alcohol blood level in DUI cases.     

Wavelet density estimation for negatively associated stratified size-biased sample

This paper provides upper bounds of wavelet estimations on L^p (1≤p<∞) risk for a density function in Besov spaces based on negatively associated stratified size-biased random samples. It turns out that the classical theorem of Donoho, Johnstone, Kerkyacharian and Picard is completely extended to more general cases. More precisely, we consider the model with multiplication noise and allow the sample negatively associated. Our theory is illustrated with a simulation study.     

Kernel density estimation with binned data

Continuous data are often measured or used in binned or rounded form. In this paper we follow up on Hall's work analyzing the effect of using equally-spaced binned data in a kernel density estimator. It is shown that a surprisingly large amount of binning does not adversely affect the integrated mean squared error of a kernel estimate.     

On density and regression estimation with incomplete data

We consider the problem of estimation of a density function in the presence of incomplete data and study the Hellinger distance between our proposed estimators and the true density function. Here, the presence of incomplete data is handled by utilizing a Horvitz–Thompson-type inverse weighting approach, where the weights are the estimates of the unknown selection probabilities. We also address the problem of estimation of a regression function with incomplete data.     

Nonparametric density estimation from censored data

Nonparametric estimation of the probability density function f° of a lifetime distribution based on arbitrarily right-censor-ed observations from f° has been studied extensively in recent years. In this paper the density estimators from censored data that have been obtained to date are outlined. Histogram, kernel-type, maximum likelihood, series-type, and Bayesian nonparametric estimators are included. Since estimation of the hazard rate function can be considered as giving a density estimate, all known results concerning nonparametric hazard rate estimation from censored samples are also briefly mentioned.     

Density and hazard rate estimation for right-censored data by using wavelet methods

This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.     

Minimum local distance density estimation

We present a local density estimator based on first-order statistics. To estimate the density at a point, x, the original sample is divided into subsets and the average minimum sample distance to x over all such subsets is used to define the density estimate at x. The tuning parameter is thus the number of subsets instead of the typical bandwidth of kernel or histogram-based density estimators. The proposed method is similar to nearest-neighbor density estimators but it provides smoother estimates. We derive the asymptotic distribution of this minimum sample distance statistic to study globally optimal values for the number and size of the subsets. Simulations are used to illustrate and compare the convergence properties of the estimator. The results show that the method provides good estimates of a wide variety of densities without changes of the tuning parameter, and that it offers competitive convergence performance.     

Kernel density estimation for hierarchical data

Abstract

Multistage sampling is a common sampling technique employed in many studies. In this setting, observations are identically distributed but not independent, thus many traditional kernel smoothing techniques, which assume that the data are independent and identically distributed (i.i.d.), may not produce reasonable density estimates. In this paper, we sample repeatedly with replacement from each cluster, create multiple i.i.d. samples containing one observation from each cluster, and then create a kernel density estimate from each i.i.d. sample. These estimates will then be combined to form an estimate of the marginal probability density function of the population.     

On boundary correction in kernel density estimation

It is well known now that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated. This is due to boundary effects that occur in nonparametric curve estimation problems. A number of proposals have been made in the kernel density estimation context with some success. As of yet there appears to be no single dominating solution that corrects the boundary problem for all shapes of densities. In this paper, we propose a new general method of boundary correction for univariate kernel density estimation. The proposed method generates a class of boundary corrected estimators. They all possess desirable properties such as local adaptivity and non-negativity. In simulation, it is observed that the proposed method perform quite well when compared with other existing methods available in the literature for most shapes of densities, showing a very important robustness property of the method. The theory behind the new approach and the bias and variance of the proposed estimators are given. Results of a data analysis are also given.     

Wavelet kernel penalized estimation for non-equispaced design regression

The paper considers regression problems with univariate design points. The design points are irregular and no assumptions on their distribution are imposed. The regression function is retrieved by a wavelet based reproducing kernel Hilbert space (RKHS) technique with the penalty equal to the sum of blockwise RKHS norms. In order to simplify numerical optimization, the problem is replaced by an equivalent quadratic minimization problem with an additional penalty term. The computational algorithm is described in detail and is implemented with both the sets of simulated and real data. Comparison with existing methods showed that the technique suggested in the paper does not oversmooth the function and is superior in terms of the mean squared error. It is also demonstrated that under additional assumptions on design points the method achieves asymptotic optimality in a wide range of Besov spaces.     

Wavelet estimators for the derivatives of the density function from data contaminated with heteroscedastic measurement errors

Measurement errors occur in many real data applications. In this paper, the linear and the non linear wavelet estimators of the derivatives of the density function are constructed in the case of data contaminated with heteroscedastic measurement errors. We establish L_p risk performance of the estimators and show that they achieve fast convergence rates under quite general conditions.     

Nonparametric localized bandwidth selection for Kernel density estimation

As conventional cross-validation bandwidth selection methods do not work properly in the situation where the data are serially dependent time series, alternative bandwidth selection methods are necessary. In recent years, Bayesian-based methods for global bandwidth selection have been studied. Our experience shows that a global bandwidth is however less suitable than a localized bandwidth in kernel density estimation based on serially dependent time series data. Nonetheless, a di?cult issue is how we can consistently estimate a localized bandwidth. This paper presents a nonparametric localized bandwidth estimator, for which we establish a completely new asymptotic theory. Applications of this new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate and the S&P 500 daily return demonstrate the effectiveness and competitiveness of the proposed localized bandwidth.     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.     

A generalized reflection method of boundary correction in kernel density estimation

The kernel method of estimation of curves is now popular and widely used in statistical applications. Kernel estimators suffer from boundary effects, however, when the support of the function to be estimated has finite endpoints. Several solutions to this problem have already been proposed. Here the authors develop a new method of boundary correction for kernel density estimation. Their technique is a kind of generalized reflection involving transformed data. It generates a class of boundary corrected estimators having desirable properties such as local smoothness and nonnegativity. Simulations show that the proposed method performs quite well when compared with the existing methods for almost all shapes of densities. The authors present the theory behind this new methodology, and they determine the bias and variance of their estimators.     

On posterior distribution of Bayesian wavelet thresholding

We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in Abramovich et al. (2004) where the authors studied the convergence of several Bayesian estimators.     

Recursive estimation of the mode of a multivariate density

Let f be an unknown possibly multimodal density on R^d and let X₁, X₂, … be a sequence of independent random vectors with density f. Several recursive estimates of the mode of f are proposed, and sufficient conditions ensuring their weak and strong consistency are established.     

Nonparametric density estimation for censored survival data: Regression-spline approach

A method for nonparametric estimation of density based on a randomly censored sample is presented. The density is expressed as a linear combination of cubic M -splines, and the coefficients are determined by pseudo-maximum-likelihood estimation (likelihood is maximized conditionally on data-dependent knots). By using regression splines (small number of knots) it is possible to reduce the estimation problem to a space of low dimension while preserving flexibility, thus striking a compromise between parametric approaches and ordinary nonparametric approaches based on spline smoothing. The number of knots is determined by the minimum AIC. Examples of simulated and real data are presented. Asymptotic theory and the bootstrap indicate that the precision and the accuracy of the estimates are satisfactory.     

On Wavelet Estimation of the Derivatives of a Density Based on Biased Data

Here, we consider wavelet based estimation of the derivatives of a probability density function under random sampling from a weighted distribution and extend the results regarding the asymptotic convergence rates under the i.i.d. setup studied in Prakasa Rao (1996 Rao, B. L.S. (1996). Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernat. 28:91–100. [Google Scholar]) to the biased-data setup. We compare the performance of the wavelet based estimator with that of the kernel based estimator obtained by differentiating the Efromovich (2004 Efromovich, S. (2004). Density estimation for biased data. Ann. Statist. 32:1137–1161.[Crossref], [Web of Science ®] , [Google Scholar]) kernel density estimator through a simulation study.