Class Of Kernel Estimators Research Articles

Sup-norm adaptive drift estimation for multivariate nonreversible diffusions

We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup-norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup-norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β-mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.

Multiplicative bias correction for inverse gamma and beta prime kernel density estimators

In this paper, we demonstrate that the multiplicative bias correction (MBC) approaches can be extended for both Inverse Gamma (IG) and Beta Prime (BP) kernel density estimators. First, some properties of the MBC-IG and MBC-BP kernel density estimators (bias, variance and mean integrated squared error) are shown. Second, the least square cross validation technique (LSCV) is adapted for the choice of bandwidth. Finally, the performances of the MBC estimators based on IG and BP kernels are illustrated by two studies, a simulation study, followed by a real application study with positive support. In general, in terms of the two criterions, integrated squared bias (ISB) and integrated squared error (ISE), the proposed estimators outperform the classical IG and BP kernel estimators.

NONPARAMETRIC KERNEL DISTRIBUTION FUNCTION ESTIMATION NEAR ENDPOINTS

In this paper, two kernel cumulative distribution function estimators are introduced and investigated in order to improve the boundary effects, we will restrict our attention to the right boundary. The first estimator uses a self-elimination between modify theoretical Bias term and the classical kernel estimator itself. The basic technique of construction the second estimator is kind of a generalized reflection method involving reflection a transformation of the observed data. The theoretical properties of our estimators turned out that the Bias has been reduced to the second power of the bandwidth, simulation studies and two real data applications were carried out to check these phenomena and are conducted that the proposed estimators are better than the existing boundary correction methods.

A New Kernel Estimator of Copulas Based on Beta Quantile Transformations

A copula is a multivariate cumulative distribution function with marginal distributions Uniform(0,1). For this reason, a classical kernel estimator does not work and this estimator needs to be corrected at boundaries, which increases the difficulty of the estimation and, in practice, the bias boundary correction might not provide the desired improvement. A quantile transformation of marginals is a way to improve the classical kernel approach. This paper shows a Beta quantile transformation to be optimal and analyses a kernel estimator based on this transformation. Furthermore, the basic properties that allow the new estimator to be used for inference on extreme value copulas are tested. The results of a simulation study show how the new nonparametric estimator improves alternative kernel estimators of copulas. We illustrate our proposal with a financial risk data analysis.

Nonparametric Estimation of Multivariate Density and its Derivative by Dependent Data Using Gamma Kernels

We consider the nonparametric estimation of the multivariate probability density function and its partial derivative with a support on nonnegative axis by dependent data. We use the class of kernel estimators with asymmetric gamma kernel functions. The gamma kernels are nonnegative; they may change their shape depending on the position on the semi-axis and possess good boundary properties for a wide class of densities. Asymptotic estimates of the multivariate density and of its partial derivatives such as biases, variances, and covariances are derived. The optimal bandwidth of both estimates is obtained as a minimum of the mean integrated squared error (MISE) by dependent data with a strong mixing. Optimal convergence rates of the MISE both for the density and its derivative are found.

A non-parametric density estimate adaptation for population abundance when the shoulder condition is violated

The non-parametric kernel density estimation is used in practice to estimate population abundance using the line transect sampling. In general, the classical kernel estimator of f(0) tends to be underestimated. In this article, a shifted logarithmic transformation of perpendicular distance is proposed for the kernel estimator when the shoulder condition is violated. Mathematically, the proposed estimator is more efficient than the classical kernel estimator. A simulation study is also carried out to compare the performance of the proposed estimators and the classical kernel estimators.

Convergence of spectral density estimators in the locally stationary framework

Asymptotic properties of classical kernel estimators for the spectral density are studied in the locally stationary framework. In particular, it is shown that for a locally stationary process standard spectral density estimators consistently estimate the time-averaged spectral density. This result is complemented by some illustrative examples and applications including HAC-inference in the multiple linear regression model, a simple visual tool for the detection of unconditional heteroskedasticity and a test for covariance stationarity.

A General Base of Power Transformation to Improve the Boundary Effect in Kernel Density without Shoulder Condition

In this paper, a general base of power transformation under the kernel method is suggested and applied in the line transect sampling to estimate abundance. The suggested estimator performs well at the boundary compared to the classical kernel estimator without using the shoulder condition assumption. The transformed estimator show smaller value of mean squared error and absolute bias from the efficiency results obtained using simulation.

An Adaptation of Kernel Density Estimation for Population Abundance using Line Transect Sampling When the Shoulder Condition is Violated

Kernel estimation is a commonly used method to estimate the population density in line transect sampling. In general, the classical kernel estimator of (0) X f , which is the probability density function at perpendicular distance x  0 , inclines to be underestimated. In this study, a power transformation of perpendicular distance is proposed for the kernel estimator when the shoulder condition is violated. The mathematical properties of the proposed estimator are derived. A simulation study is also carried out for comparing the proposed estimator with the classical kernel estimators.

Spatial local linear estimation of the L1-conditional quantiles for functional regressors

L1-norm approach is used to construct the local linear estimator of the spatial regression quantile for functional regressors. Under mixing spatial condition, we establish the almost complete convergence of the constructed approach. The applicability of the constructed estimator is examined by a Monte-Carlo study. The finite sample performance of the proposed estimator is compared to the classical kernel estimator of the functional spatial quantile regression. The result indicates that our new approach is more accurate than the classical one.

Estimation on semi-functional linear errors-in-variables models

Semi-functional linear regression models are important in practice. In this paper, their estimation is discussed when function-valued and real-valued random variables are all measured with additive error. By means of functional principal component analysis and kernel smoothing techniques, the estimators of the slope function and the non parametric component are obtained. To account for errors in variables, deconvolution is involved in the construction of a new class of kernel estimators. The convergence rates of the estimators of the unknown slope function and non parametric component are established under suitable norm and conditions. Simulation studies are conducted to illustrate the finite sample performance of our method.

Adaptive density estimation on bounded domains under mixing conditions

In this article, we propose a new adaptive estimator for multivariate density functions defined on a bounded domain in the framework of multivariate mixing processes. Several procedures have been proposed in the literature to tackle the boundary bias issue encountered using classical kernel estimators. Most of them are designed to work in dimension $d=1$ or on the unit $d$-dimensional hypercube. We extend such results to more general bounded domains such as simple polygons or regular domains that satisfy a rolling condition. We introduce a specific family of kernel-type estimators devoid of boundary bias. We then propose a data-driven Goldenshluger and Lepski type procedure to jointly select a kernel and a bandwidth. We prove the optimality of our procedure in the adaptive framework, stating an oracle-type inequality. We illustrate the good behavior of our new class of estimators on simulated data. Finally, we apply our procedure to a real dataset.

Combination of parametric and nonparametric estimators for population abundance using line transect sampling

This paper introduced a modification of the classical kernel estimator for population abundance using line transect sampling. The procedure allows the incorporation of a small set of parametric models with the usual kernel estimator. Akaike Information Criterion (AIC) is used to choose the most appropriate parametric detection function. The resultant estimators of f(0); the probability density function at perpendicular distance x = 0, are compared with the existing usual kernel estimator by using the simulation technique and a set of real data. Numerical results demonstrate the superiority of these estimators over the usual kernel estimator for almost all considered cases.

Functional data analysis: estimation of the relative error in functional regression under random left-truncation model

ABSTRACTIn this paper, we investigate the relationship between a functional random covariable and a scalar response which is subject to left-truncation by another random variable. Precisely, we use the mean squared relative error as a loss function to construct a nonparametric estimator of the regression operator of these functional truncated data. Under some standard assumptions in functional data analysis, we establish the almost sure consistency, with rates, of the constructed estimator as well as its asymptotic normality. Then, a simulation study, on finite-sized samples, was carried out in order to show the efficiency of our estimation procedure and to highlight its superiority over the classical kernel estimation, for different levels of simulated truncated data.

Conditional kernel density estimation for some incomplete data models

A class of density estimators based on observed incomplete data are proposed. The method is to use a conditional kernel, defined as the expectation of a given kernel for the complete data conditioning on the observed data, to construct the density estimator. We study such kernel density estimators for several commonly used incomplete data models and establish their basic asymptotic properties. Some characteristics different from the classical kernel estimators are discovered. For instance, the asymptotic results of the proposed estimator do not depend on the choice of the kernel $k(\cdot )$. Simulation study is conducted to evaluate the performance of the estimator and compared with some exising methods.

Redukcja efektu brzegowego w estymacji jądrowej wybranych charakterystyk funkcyjnych zmiennej losowej

For a random variable with bounded support, the kernel estimation of functional characteristics may lead to the occurrence of the so-called boundary effect. In the case of the kernel density estimation it can mean an increase of the estimator bias in the areas near the ends of the support, and can lead to a situation where the estimator is not a density function in the support of a random variable. In the paper the procedures for reducing boundary effect for kernel estimators of density function, distribution function and regression function are analyzed. Modifications of the classical kernel estimators and examples of applications of these procedures in the analysis of the functional characteristics relating to gross national product per capita are presented. The advantages of procedures are indicated taking into account the reduction of the bias in the boundary region of the support of the random variable considered.

Adaptive kernel estimation of the baseline function in the Cox model with high-dimensional covariates

We propose a novel kernel estimator of the baseline function in a general high-dimensional Cox model, for which we derive non-asymptotic rates of convergence. To construct our estimator, we first estimate the regression parameter in the Cox model via a LASSO procedure. We then plug this estimator into the classical kernel estimator of the baseline function, obtained by smoothing the so-called Breslow estimator of the cumulative baseline function. We propose and study an adaptive procedure for selecting the bandwidth, in the spirit of Goldenshluger and Lepski (2011). We state non-asymptotic oracle inequalities for the final estimator, which leads to a reduction in the rate of convergence when the dimension of the covariates grows.

Double Kernel Method Using Line Transect Sampling

A double kernel method as an alternative to the classical kernel method is proposed to estimate the population abundance by using line transect sampling. The proposed method produces an estimator that is essentially a kernel type of estimator use the kernel estimator twice to improve the performances of the classical kernel estimator. The feasibility of using bootstrap techniques to estimate the bias and variance of the proposed estimator is also addressed. Some numerical examples based on simulated and real data are presented. The results show that the proposed estimator outperforms existingclassical kernel estimator in most considered cases.

Parametrically guided nonparametric density and hazard estimation with censored data

The parametrically guided kernel smoother is a promising nonparametric estimation approach that aims to reduce the bias of the classical kernel density estimator without increasing its variance. Theoretically, the estimator is unbiased if a correct parametric guide is used, which can never be achieved by the classical kernel estimator even with an optimal bandwidth. The estimator is generalized to the censored data case and used for density and hazard function estimation. The asymptotic properties of the proposed estimators are established and their performance is evaluated via finite sample simulations. The method is also applied to data coming from a study where the interest is in the time to return to drug use.

A note on nonparametric density deconvolution by weighted kernel estimators

Abstract Recently Hazelton and Turlach (2009) proposed a weighted kernel density estimatorfor the deconvolution problem. In the case of Gaussian kernels and measurement er-ror, they argued that the weighted kernel density estimator is a competitive estimatorover the classical deconvolution kernel estimator. In this paper we consider weightedkernel density estimators when sample observations are contaminated by double expo-nentially distributed errors. The performance of the weighted kernel density estimatorsis compared over the classical deconvolution kernel estimator and the kernel density es-timator based on the support vector regression method by means of a simulation study.The weighted density estimator with the Gaussian kernel shows numerical instabilityin practical implementation of optimization function. However the weighted densityestimates with the double exponential kernel has very similar patterns to the classicalkernel density estimates in the simulations, but the shape is less satisfactory than theclassical kernel density estimator with the Gaussian kernel.Keywords: Deconvolution, kernel density estimator, support vector regression, weightedkernel density estimator.