UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION

Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.     

���ܶȹ��ƵĶԳƼ�����~$L_1(\mathbb{R}^d)$~�µ���ƫ��

Let f_n be a non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in \mathbb{R}^d. In this paper we prove two moderate deviation theorems in L_1(\mathbb{R}^d) for \{f_n(x)-f_n(-x),\,n\ge1\}.     

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.     

基于方向数据的核密度估计的对称检验的大偏差

设$f_n$是基于核函数$K$和取值于$d$-维单位球面${\mathbb{S}}^{d-1}$的独立同分布随机变量列的非参数核密度估计. 我们证明了若核函数是有界变差函数, 随机变量的密度函数$f$是连续的和对称的, $\{\sup_{x\in {\mathbb{SS}}^{d-1}}|f_n(x)-f_n(-x)|,n\ge 1\}$的大偏差原理成立.     

CONVERGENCE RATE OF MULTIVARIATE K-NEAREST NEIGHBOR DENSITY ESTIMATES

Let be the collection of m-times continuously differentiable probability densities fon R~d such that 丨D~af(x_1)-D~af(x_2)丨≤M‖x_1-x_2‖~β for x_1,x_2∈R~d,[a]=m,where D~adenotes the differential operator defined by D~a=([a])/(x_1~a…x_d~a_d).Under rather weak conditionson K(x),the necessary and sufficient conditions for sup丨_n(x)-f(x)丨=0(((logn/n)~λ/(d+3λ),λ=m+β,f∈ are that ∫x~aK(xi)dx=0 for 0<[a]≤m.Finally the convergenco rate at apoint is given.     

ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES

Let X_1,…,X,be a sequence of p-dimensional iid.random vectors with a commondistribution F(x).Denote the kernel estimate of the probability density of F(if it exists)by_n(x)=n~(-1)h~_n(-p)K((x-X_i)/h_n)Suppose that there exists a measurable function g(x)and h_n>0,h_n→0 such thatlim sup丨f_n(x)-g(x)丨=0 a.s.Does F(x)have a uniformly continuous density function f(x)and f(x)=g(x)?This paperdeals with the problem and gives a sufficient and necessary condition for generalp-dimensional case.     

一列连续函数的遍历优化

设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq 1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in \mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac 1 n f_n (x)=\sup\limits_{x\in X} \limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x) =\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n (x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$ 表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了 一些相关的应用.     

Moderate Deviations and Large Deviations for Kernel Density Estimators

Let f _n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ^d . It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) 0 as |x| , then the moderate deviation principle and large deviation principle for hold.     

On the uniform complete convergence of estimates for multivariate density functions and regression curves

Let (X ₁,Y ₁),...(X _n ,Y _n ) be a random sample from the (k+1)-dimensional multivariate density functionf ^*(x,y). Estimates of thek-dimensional density functionf(x)=∫f ^*(x,y)dy of the form $$\hat f_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n W \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered whereW(x) is a bounded, nonnegative weight function andb ₁ (n),...,b _k (n) and bandwidth sequences depending on the sample size and tending to 0 asn→∞. For the regression function $$m(x) = E(Y|X = x) = \frac{{h(x)}}{{f(x)}}$$ whereh(x)=∫y(f) ^* (x, y)dy , estimates of the form $$\hat h_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n {Y_i W} \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered. In particular, unform consistency of the estimates is obtained by showing that \(||\hat f_n (x) - f(x)||_\infty \) and \(||\hat m_n (x) - m(x)||_\infty \) converge completely to zero for a large class of “good” weight functions and under mild conditions on the bandwidth sequencesb _k (n)'s.     

Uniform boundedness inL p (p=p(x)) of some families of convolution operators

Suppose that a measurable 2π-periodic essentially bounded function (the kernel) κ_λ =κ_λ(x) is given for any realλ≥1. We consider the following linear convolution operator inL ^p: $$\kappa _\lambda = \kappa _\lambda f = (\kappa _\lambda f)(x) = \int_{ - \pi }^\pi {f(t)} k_\lambda (t - x) dt.$$ Uniform boundedness of the family of operators {Κ_λ}_λ≥1 is studied. Conditions on the variable exponentp=p(x) and on the kernel κ_λ that ensure the uniform boundedness of the operator family {Κ_λ}_λ≥1 inL ^p are obtained. The condition on the exponentp=p(x) is given in its final form.     

Stability Results for Ekeland's ε Variational Principle for Set-Valued Mappings

In this paper, we define the Mosco convergence and Kuratowski-Painleve (P.K.) convergence for set-valued mapping sequence F n . Under some conditions, we derive the following result If a set-valued mapping sequence F n , which are nonempty, compact valued, upper semicontinuous and uniformly bounded below, Mosco (or P.K.) converges to a set-valued mapping F , which is upper semicontinuous, nonempty, compact valued, then Q l >0, u >0, $\varepsilon / \lambda - {\rm ext}\, F := \{ \bar x \in X : (F(x) - \bar y + \varepsilon / \lambda \Vert x - \bar x \Vert e)$     

ASYMPTOTICS OF THE RESIDUALS DENSITY ESTIMATION IN NONPARAMETRIC REGRESSION UNDER m（n）-DEPENDENT SAMPLE

Let Y_i=M(X_i)+ei, where M(x)=E(Y|X=x) is an unknown realfunction on B(? R), {(X_1,Y_i)} is a stationary and m(n)-dependent sample from(X, Y), the residuals {e_i} are independent of {X_i} and have unknown common densityf(x). In [2] a nonparametric estimate f_n(x) for f(x) has been proposed on the basisof the residuals estimates. In this paper, we further obtain the asymptotic normalityand the law of the iterated logarithm of f_n(x) under some suitable conditions. Theseresults together with those in [2] bring the asymptotic theory for the residuals densityestimate in nonparametric regression under m(n)-dependent sample to completion.     

Estimates of linear combinations of near-exponential functions with positive and negative exponents

Mathematical Notes - The functions $$\begin{gathered} f_n (z) = e^{\lambda _n ^z } [1 + \alpha _n (z)], \hfill \\ \varphi _n (z) = e^{\mu _n ^z } [1 + \beta _n (z)](n = 1.2, ...), \hfill \\...     

Estimating the Density of the Residuals in Autoregressive Models

We consider a (nonlinear) autoregressive model with unknown parameters (vector θ). The aim is to estimate the density of the residuals by a kernel estimator. Since the residuals are not observed, the usual procedure for estimating the density of the residuals is the following: first, compute an estimator for θ; second, calculate the residuals by use of the estimated model; and third, calculate the kernel density estimator by use of these residuals. We show that the resulting density estimator is strong consistent at the best possible convergence rate. Moreover, we prove asymptotic normality of the estimator. This revised version was published online in June 2006 with corrections to the Cover Date.     

Differentiable Functionals and Smoothed Bootstrap

The differentiability properties of statistical functionals have several interesting applications. We are concerned with two of them. First, we prove a result on asymptotic validity for the so-called smoothed bootstrap (where the artificial samples are drawn from a density estimator instead of being resampled from the original data). Our result can be considered as a smoothed analog of that obtained by Parr (1985, Stat. Probab. Lett., 3, 97-100) for the standard, unsmoothed bootstrap. Second, we establish a result on asymptotic normality for estimators of type generated by a density functional being a density estimator. As an application, a quick and easy proof of the asymptotic normality of , (the plug-in estimator of the integrated squared density ) is given.     

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

Strong Convergence of the Kernel Estimates of Nonparametric Regression Functions

Let (X, Y), (X_1, Y_1),\cdots, (X_n, Y_n) be i. i. d. random vectors taking values in R_d\times R with E(|Y|)<\infinity, To estimate the regression function m(x)=E(Y|X=x), we use the kernel estimate $m_n(x)=[\sum\limits_{i = 1}^n {K(\frac{{{X_i} - x}}{{{h_n}}}){Y_i}/} \sum\limits_{i = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} \]$ where K(x) is a kernel function and h_n a window width. In this paper, we establish the strong consistency of m_n(x) when E(|Y|^p)<\infinity for some p>l or E{exp(t|Y|^\lambda)}<\infinity for some \lambda>0 and t>0. It is remakable that other conditions imposed here are independent of the distribution of (X, Y).     

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.     

On the uniform convergence of Cauchy principal values of quasi-interpolating splines

In this paper, quasi-interpolating splines are used to approximate the Cauchy principal value integral $$J(w_{\alpha \beta } f;\lambda ): = \smallint - _{ - 1}^1 w_{\alpha \beta } (x)\frac{{f(x)}}{{x - \lambda }}dx, \lambda \in ( - 1,1)$$ where $w_{\alpha \beta } (x): = (1 - x)^\alpha (1 + x)^\beta ,\alpha ,\beta > - 1.$ . We prove uniform convergence for the quadrature rules proposed here and give an algorithm for the numerical evaluation of these rules.     

NA样本的递归密度估计

设$\{X_n,n\geq 1\}$是一个严平稳的负相协的随机变量序列, 其概率密度函数为$f(x)$.本文讨论了$f(x)$的递归核估计量的联合渐近正态性.