On the modal resolution of kernel density estimators

The ability of a kernel density estimator to resolve modes of the underlying density is investigated. For various bimodal densities and three different kernels, the smallest sample size required for the expectation of an optimally smoothed kernel estimator to be bimodal is determined. The optimality criterion employed is equivalent to asymptotic mean integrated squared error for sufficiently smooth densities.     

Large deviations and moderate deviations for kernel density estimators of directional data

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.     

线性过程误差下概率密度函数核估计的均方相合性

设{Xt,t≥1}为一单边线性平稳过程序列,具有共同的未知密度函数f(x),本文定义通常的f(x)的核估计,在适当条件下,证明了其均方相合性.     

Tests for normality based on density estimators of convolutions

Recent results show that densities of convolutions can be estimated by local U-statistics at the root-n rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L₁-distances between the standard normal density and local U-statistics based on standardized observations. We show that such test statistics converge at the root-n rate and determine their limit distributions as functionals of Gaussian processes. We also address a choice of bandwidth. Simulations show that our tests are competitive with other tests of normality.     

Recursive kernel density estimators under a weak dependence condition

Let X _t, t= ..., \s-1,0,1,... be a strietly stationary sequence of random variables (r.v.'s) defined on a probability space (,P) and taking values in R ^d.Let X ₁,...,X _nbe n consecutive observations of X _t.Let f be the density of X ₁.As an estimator of f(x), we shall consider % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqGaaO% qaaiqadAgagaqcamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGa% aiykaiabg2da9iaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcda% aeWbqaaiaadkgadaWgaaWcbaGaamOAaaqabaGcdaahaaWcbeqaaiab% gkHiTiaadsgaaaGccaWGlbGaaiikaiaacIcacaWG4bGaeyOeI0Iaam% iwamaaBaaaleaacaWGQbaabeaakiaacMcacaGGVaGaamOyamaaBaaa% leaacaWGQbaabeaakiaacMcaaSqaaiaadQgacqGH9aqpcaaIXaaaba% GaamOBaaqdcqGHris5aaaa!58A9!\[\hat f_n (x) = n^{ - 1} \sum\limits_{j = 1}^n {b_j ^{ - d} K((x - X_j )/b_j )} \]. Here K is a kernel function and b _nis a esquence of bandwidths tending to zero as n . The asymptotic distribution and uniform convergence of f n are obtained under general conditions. Appropriate bandwidths are given explicitly. The process X _tis assumed to satisfy a weak dependence condition defined in terms of joint densities. The results are applicable to a large class of time series models.     

Large and moderate deviations principles for kernel estimators of the multivariate regression

In this paper, we prove large deviations principle for the Nadaraya-Watson estimator and for the semi-recursive kernel estimator of the regression in the multidimensional case. Under suitable conditions, we show that the rate function is a good rate function. We thus generalize the results already obtained in the one-dimensional case for the Nadaraya-Watson estimator. Moreover, we give a moderate deviations principle for these two estimators. It turns out that the rate function obtained in the moderate deviations principle for the semi-recursive estimator is larger than the one obtained for the Nadaraya-Watson estimator.      

Strong consistency of kernel estimators for Markov transition densities

Let P(x, dy) = t (x, y)ν(d y) be the transition kernel of a Markov chain, where t (x, y) is a density with respect to a σ-finite measure ν on (E,ℰ), with E ⊂ R ^d . In this note, we propose a general class of estimates for t (x, y) that are strongly consistent and that extend the classical results for continuous densities on R ^d . Received: 2 June 2002     

左截断右删失下核密度估计的L1距离的界

本文在左截断右删失数据下获得了概论密度的核估计的L1距离的一个上界．     

A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of quantile estimators

A Berry-Esseen bound is established for the kernel quantile estimator under various conditions. The results improve an earlier result of Falk (1985,Ann. Statist.,13, 428–433) and rely on the local smoothness of the quantile function. This new Berry-Esseen bound is applied to studying the deficiency of the sample quantile estimator with respect to the kernel quantile estimator. A new result is obtained which is an extension of that in Falk (1985).     

Rates of strong uniform consistency for multivariate kernel density estimators

Let f_n denote the usual kernel density estimator in several dimensions. It is shown that if {a_n} is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K₁) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on R^d, then for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to . Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman [13] for univariate densities.     

NA、PA样本下密度核估计的相合性

设{Xn,n≥1}为同分布的NA或PA随机变量序列，f(x)为X1概率密度函数，基于样本X1，X2，…，Xn，本对密度函数（f(x)的核估计进行了讨论，在适当条件下证明了其强相合和r阶矩相合。     

Bootstrap bandwidth selection in kernel density estimation from a contaminated sample

In this paper we consider kernel estimation of a density when the data are contaminated by random noise. More specifically we deal with the problem of how to choose the bandwidth parameter in practice. A theoretical optimal bandwidth is defined as the minimizer of the mean integrated squared error. We propose a bootstrap procedure to estimate this optimal bandwidth, and show its consistency. These results remain valid for the case of no measurement error, and hence also summarize part of the theory of bootstrap bandwidth selection in ordinary kernel density estimation. The finite sample performance of the proposed bootstrap selection procedure is demonstrated with a simulation study. An application to a real data example illustrates the use of the method. This research was supported by ‘Projet d’Actions de Recherche Concertées’ (No. 98/03-217) from the Belgian government. Financial support from the IAP research network nr P5/24 of the Belgian State (Federal Office for Scientific, Technical and Cultural Affairs) is also gratefully acknowledged.     

球面数据核近邻估计的强相合性

该文绘出了球面数据密度函数的核近邻估计,通过对核估计与近邻估计相互关系的讨论,建立了核近邻估计的逐点强相合性及一致强相合性．     

On Consistency of the Nearest Neighbor Estimator of the Density Function and Its Applications

In this paper, we mainly study the consistency of the nearest neighbor estimator of the density function based on asymptotically almost negatively associated samples. The weak consistency,strong consistency, uniformly strong consistency and the convergence rates are established under some mild conditions. As applications, we further investigate the strong consistency and the rate of strong consistency for hazard rate function estimator.     

Consistency of change point estimators for symmetrical stable distribution with parameters shift

Assume that the characteristic indexαof stable distribution satisfies 1<α<2,and that the distribution is symmetrical about its mean.We consider the change point estimators for stable distribution withαor scale parameterβshift.For the one case that mean is a known constant,ifαorβchanges,then density function will change too.To this end,we suppose the kernel estimation for a change point.For the other case that mean is an unknown constant,we suppose to apply empirical characteristic function to estimate the change-point location.In the two cases,we consider the consistency and strong convergence rate of estimators.Furthermore,we consider the mean shift case.If mean changes,then corresponding characteristic function will change too.To this end,we also apply empirical characteristic function to estimate change point.We obtain the similar convergence rate.Finally,we consider its application on the detection of mean shift in financial market.     

A cross-validation method for data with ties in kernel density estimation

Limitation of the cross-validation method of bandwidth selection is well known when applied to data with ties. A method which resolves this problem and which is easy to understand and implement is proposed. We show that the proposed approach is viable in theory, by proving its asymptotic equivalence to the standard cross-validation method. The practical usefulness is shown in simulations and an application to a real data example.     

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

相依样本下条件密度递归双重核估计的强相合性

本文给出了条件密度的递归形式的双重核估计,并且在样本序列为平稳φ-混合的条件下讨论了它的强相合性。