The strong uniform convergence of multivariate variable kernel estimates

We show that sup, completely as, where f is a uniformly continuous density on are independent random vectors with common density f, and f_n is the variable kernel estimate Here H_ni is the distance between X_i and its kth nearest neighbour, K is a given density satisfying some regularity conditions, and k is a sequence of integers with the property that log asn     

The kernel regression estimation for randomly censored functional stationary ergodic data

In this paper, we investigate the asymptotic properties of the kernel estimator for non parametric regression operator when the functional stationary ergodic data with randomly censorship are considered. More precisely, we introduce the kernel-type estimator of the non parametric regression operator with the responses randomly censored and obtain the almost surely convergence with rate as well as the asymptotic normality of the estimator. As an application, the asymptotic (1 ? ζ) confidence interval of the regression operator is also presented (0 < ζ < 1). Finally, the simulation study is carried out to show the finite-sample performances of the estimator.     

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.     

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.     

Uniform strong consistency of histogram density estimation for dependent process

Histogram density estimator is very intuitive and easy to compute and has been widely adopted. Especially in today's big data environment, people pay more attention to the computational cost and are more willing to choose estimators with less to compute. And so, many scholars have been interested in the various estimates based on the histogram technique. Under strong mixing process, this article studies the uniform strong consistency of histogram density estimator and the convergence rate. Our conditions on the mixing coefficient and the bin width are very mild.     

On sufficient conditions for the strong consistency of least-squares estimates

The strong consistency of the least-squares estimates in regression models is obtained when the errors are i.i.d. with absolute moment of order r, 0<r? 2. The assumptions presented for the random error sequence will permit us to obtain improvements of the conditions on the regressors in order to obtain the strong consistency of the least-squares estimates in linear and nonlinear regression models.     

Weak and strong uniform consistency of a kernel error density estimator in nonparametric regression

This paper considers the problem of estimating the kernel error density function in nonparametric regression models. Sufficient conditions are given under which the kernel error density estimator based on nonparametric residuals is uniformly weakly and strongly consistent.     

Local likelihood density estimation for interval censored data

The authors propose a class of procedures for local likelihood estimation from data that are either interval‐censored or that have been aggregated into bins. One such procedure relies on an algorithm that generalizes existing self‐consistency algorithms by introducing kernel smoothing at each step of the iteration. The entire class of procedures yields estimates that are obtained as solutions of fixed point equations. By discretizing and applying numerical integration, the authors use fixed point theory to study convergence of algorithms for the class. Rapid convergence is effected by the implementation of a local EM algorithm as a global Newton iteration. The latter requires an explicit solution of the local likelihood equations which can be found by using the symbolic Newton‐Raphson algorithm, if necessary.     

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.     

Location and scale parameter estimation from randomly censored data

The problem of location and scale parameter estimation from randomly censored data is analyzed through use of a regression model for the Kaplan-Meier quantlle process. Continuous time regression techniques are employed to construct estimators that are both asymptotically normal and efficient. Estimators with a particularly simple form are obtained for the Koziol-Green model for random censorship. In the event of no censoring the regression model, and resulting estimators, reduce to those proposed by Parzen (1979 a, b).     

Strong uniform consistency rates of kernel estimators of cumulative distribution functions

In this paper we consider the uniform strong consistency, along with a rate, of the cumulative distribution function (CDF) estimator. We extend the extended Glivenko–Cantelli lemma (for empirical distribution function) in Fabian and Hannan (1985 Fabian, V., Hannan, J. (1985). Introduction to Probability and Mathematical Statistics. New York: Wiley, ISBN-13:978-0471250234. [Google Scholar], pp. 80–83) to the kernel estimator of the CDF.     

Tail density estimation for exploratory data analysis using kernel methods

It is often critical to accurately model the upper tail behaviour of a random process. Nonparametric density estimation methods are commonly implemented as exploratory data analysis techniques for this purpose and can avoid model specification biases implied by using parametric estimators. In particular, kernel-based estimators place minimal assumptions on the data, and provide improved visualisation over scatterplots and histograms. However kernel density estimators can perform poorly when estimating tail behaviour above a threshold, and can over-emphasise bumps in the density for heavy tailed data. We develop a transformation kernel density estimator which is able to handle heavy tailed and bounded data, and is robust to threshold choice. We derive closed form expressions for its asymptotic bias and variance, which demonstrate its good performance in the tail region. Finite sample performance is illustrated in numerical studies, and in an expanded analysis of the performance of global climate models.     

Asymptotic normality of conditional density estimation under truncated,censored and dependent data

Abstract

In this paper, we focus on the left-truncated and right-censored model, and construct the local linear and Nadaraya-Watson type estimators of the conditional density. Under suitable conditions, we establish the asymptotic normality of the proposed estimators when the observations are assumed to be a stationary α-mixing sequence. Finite sample behavior of the estimators is investigated via simulations too.     

Strong consistency of non parametric kernel regression estimator for strong mixing samples

For α-mixing samples, we study Priestley–Chao kernel estimator for non parametric regression model. By using the moment inequality and the exponential inequality, the strong consistency and the uniformly strong consistency of the estimator are obtained for some weak conditions.     

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.     

Constrained estimation and likelihood intervals for censored data

The authors propose a reduction technique and versions of the EM algorithm and the vertex exchange method to perform constrained nonparametric maximum likelihood estimation of the cumulative distribution function given interval censored data. The constrained vertex exchange method can be used in practice to produce likelihood intervals for the cumulative distribution function. In particular, the authors show how to produce a confidence interval with known asymptotic coverage for the survival function given current status data.     

Monte carlo results for conditional survival tests under randomly censored data

Within a Monte Carlo study finite sample results are obtained for different generalized rank tests based on randomly censored life time data. It is pointed out that conditional tests should be applied in practice whenever drastic differences between the censoring distributions for the underlying groups do not appear. The tests are slight modifications of known permutation tests for censored data.