关于密度估计的不变原理

设f(x)为i.i.d随机变量序列X_1,X_2,…共同的分布密度函数,它的核估计为其中h_n↓0,核函数K∈D(-∞,+∞)。 本文首先考虑了由f_n产生的一类D中的随机元的有限维分布的收敛性,然后着重讨论了由f_n产生的另一类D中的随机元在对核函数的适当限制下向正态随机元的弱收敛性。     

基于次序统计量之伴随量的非参数回归的强相合性

设(X,Y),(X_1,Y_1),(X_2,Y_2),…是 i.i.d.二维随机变量,m(x)=E(Y|X=x)是回归函数.Yang,S.S.构造了 m(x)的下述估计:记 X_(i=n) 是 X_1,…,X_n 的第 i 个次序统计量,Y_([i∶n]) 是 X_(i∶n)相应的伴随量,则m_n(x)=1/(nh_n) sum from i=1 to n K((i/n-F_n(x))/h_n)Y_([i∶n]) (1.1)是 m(x)的一个估计,其中 F_n(x)是 X_1,…,X_n 的经验分布函数,K(·)是 R 上的一个概率密度函数而{h_n}是一个正常数序列,易见 m_n(x)可应用在许多非标准情形,如 X 的观察值已自然地排好序或 X_(i∶n)比 X_i 更容易获得等.与古典强大数定律相比,一个在理论上很有兴趣的问题是假定 E|Y|<∞,能否找到 m(x)的强相合估计.成平及成平、赵林城分别用截尾的核估计和近邻估计的方法肯定地回答了这一问题.对于由(1.1)定义的 m_n(x),我们也可以讨论如下截尾形式的     

密度核估计的强一致相合性

关于密度函数f(x)的核估计,f_n(x)=(nh_n~d)~(-1)sum from i=1 to (?)(k(x-X_i/h_n)),Devroye和Wagner[1]给出一维情形下,f_n一致强相合于f的一组充分条件,本文给出多维情形下f_n的一致强相合性,而对K和h_n所加的限制是弱的,同时本文改善了徐达明,白志东[2]所得的结果。     

平稳序列密度函数及其导函数的随机窗宽核估计的一致强相合性

设{X_n}_(n=1)~∞是R~1上的平稳、强混合随机变量序列,具有公共的密度函数f(x)。我们选定一个概率密度K(x),假定f(x)与K(x)都具有r(≥0)阶导数,则可定义基于观测值X_1,…,X_n的f~(r)(x)的核估计其中窗宽h_n(x)■h_n(x;X_1,…,X_n)>0不仅依赖于X_1,…,X_n,而且也与x有关。本文是在随机序列{X_n}_(n=1)~∞是平稳、强混合的情况下,讨论f_n~(r)(x)的一致强相合性。     

最近邻密度估计强一致收敛性的必要条件问题

设x_1,…,X_n为取自具有分布密度f中的iid样本,1965年Loftsgarden等提出一个常称为“最近邻估计”的 f_n(x)=k_n/2na_n(x) x∈R~1去估计f(x),关于f_n的收敛性质已有不少的研究,到1977年Devroye等(见Ann.Statist.,5(1977),p.536)得到了最佳结果。 若i)f在R~1上一致连续,则 关于本结果之逆,即(1)成立的必要条件,柴根象,陈希孺(见中国科大学报83.4.407)分别证得:条件i)是必要的,条件及也是必要的。那末条件iii)是否也     

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.     

条件L泛函的核估计及其Bootstrap逼近

设(X,y)为取值于 R~d×R~1的随机变量,X 具有边缘分布 F(x),Y 关于 X 的条件分布为 F(y|x).对于条件 L 泛函θ_1(x)=integral from n=0 to 1 J(y)F~(-1)(y|x)dy(1)θ(x)=integral from n=0 to 1 J(y)F~(-1)(y|x)dy+sum from j=1 to k a_jF~(-1)(p_j|x)(2)在[1]中曾给出了它们的近邻估计,并讨论了估计的渐近性质(其中 F~(-1)(x)=inf{t:F(t)≥x}).在本文中,我们将用核函数方法构造它们的另一类估计,并讨论估计的一些渐近性质.设(X_1,Y_1),(X_2,Y_2),…是(X,Y)的一个样本列,取 w_n_i(x)=K((x-X_i)/h_n)/sum from i=1 to n K((x-X_i)/h_n),其中 K 为 R~d 上的概率密度函数,并有0相似文献   

ON THE STRONG CONSISTENT CONDITIONS OF THE NEAREST NEIGHBOR DENSITY ESTIMATE IN GENERAL KERNEL CASE

§1. Introduction and Main Results Let X_1, X_2, …, X_n be iid. Samples drawn from a population with distribution F and density f in R~1. The method using the kernel estimator f_n(x) to estimate, f(x) was suggested by Rosenblatt and Parzen, where     

最近邻密度估计的逐点强收敛速度

Let X_1,…,X_n be i.i.d,samples drawn from an one-dimenslonal,population withdensity f.Definef_n(x)=(na_n(x))~(1-) sum form i=1 to n K((X-X_i)/(a_n(x))).We study the strong convergence rate of f_n(x) to f(x)at a predetermined point x_o.Under some properly chosen conditions,for f(x_o) and g_n(x_o)proposed in [3],we havepointwisebywhere C_n is any sequence tending to ∞,and n approaches ∞.If f(x)is only assumed tobe continuous at x_o.Then f_n(x_o)may converges to f(x_o)arbitrarily slowly.     

具有随机窗宽核估计的一致强相合性

设 X_1,…,X_m i.i.d.是取值于 R~n 中的随机向量,X_1 有概率密度 f(x),取正随机变量 H_m(x,ω)=H_m(x,X_2(ω),…,(ω))为随机窗宽,f(x)的核估计与最近邻估计分别如下:f_m(x)=(mH_m~n(x,ω))~(-1)sum from i=1 to m K((X_i-x)/H_m(x,w))f_m(x)=(ma_m~n(x,w))~(-1) sum from i=1 to m K((X_i-x)/a_m(x,w)),m≥1,x∈R~n.假定 K 为 R~n 中有界变差函数,当 f(x)与 K(x)的条件比[1]弱时,我们讨论了 f_m(x)与 f_m(x)的一致强相合性。本文所得随机窗宽的结果与[1]中常数窗宽的结果相同,这些结果也比[2]和[5]中的要好。     

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．     

Effective methods of determining the modulus of doubly connected domains

The modulus of a doubly connected domain is determined by a quotient of certain kernel functions, namely the Bergman kernel, the reduced Bergman kernel and the square of the Szegö kernel. These methods are more efficient than methods involving the curvature of the Bergman metric.     

New derivation of the heisenberg kernel

The Heisenberg group gives rise to the simplest interesting example of a subellipticoperator. The hear kernel for this operator is known in terms of Fourier transforms. Here this hear kernel is derived in a simpleminded way from standard theroems in mathematical physics. A formula is given relating this kernel to the heat kernel of a magnetic field and ageneralization is given for similar geometries     

基于不同核函数的非参数与参数利率模型的国债定价

以上海证券交易所的国债回购利率数据为样本,本文采用两种不同核函数:高斯核和抛物线核对非参数利率期限结构模型进行估计.结果显示:短期利率的密度函数是非正态的,扩散过程的漂移函数和扩散函数都是非线性的,高斯核比抛物线核对扩散函数拟合更平滑.然后,给出了基于非参数和参数利率模型的国债定价的方法,并对非参数利率模型、Vasicek模型、CIR模型、多项式样条静态模型进行国债定价预测比较与分析.     

装备费用预测中的混合核函数支持向量机

在支持向量机预测建模中,核函数用来将低维特征空间中的非线性问题映射为高维特征空间中的线性问题.核函数的特征对于支持向量机的学习和预测都有很重要的影响.考虑到两种典型核函数—全局核(多项式核函数)和局部核(RBF核函数)在拟合与泛化方面的特性,采用了一种基于混合核函数的支持向量机方法用于预测建模.为了评价不同核函数的建模效果、得到更好的预测性能,采用遗传算法自适应进化支持向量机模型的各项参数,并将其应用于装备费用预测的实际问题中.实际计算表明采用混合核函数的支持向量机较单一核函数时有更好的预测性能,可以作为一种有效的预测建模方法在装备管理中推广应用.     

Canonical kernels for density estimation

The kernel function in density estimation is uniquely determined up to a scale factor. In this paper, we advocate one particular rescaling of a kernel function, called the canonical kernel, because it is the only version which uncouples the problems of choice of kernel and choice of scale factor. This approach is useful for both pictorial comparison of kernel density estimators and for optimal kernel theory.     

Reproducing Properties of Differentiable Mercer-Like Kernels on the Sphere

We study differentiability of functions in the reproducing kernel Hilbert space (RKHS) associated with a smooth Mercer-like kernel on the sphere. We show that differentiability up to a certain order of the kernel yields both, differentiability up to the same order of the elements in the series representation of the kernel and a series representation for the corresponding derivatives of the kernel. These facts are used to embed the RKHS into spaces of differentiable functions and to deduce reproducing properties for the derivatives of functions in the RKHS. We discuss compactness and boundedness of the embedding and some applications to Gaussian-like kernels.     

A new kernel function for SPH with applications to free surface flows

Smoothed particle hydrodynamics (SPH) is a popular meshfree Lagrangian particle method, which uses a kernel function for numerical approximations. The kernel function is closely related to the computational accuracy and stability of the SPH method. In this paper, a new kernel function is proposed, which consists of two cosine functions and is referred to as double cosine kernel function. The newly proposed double cosine kernel function is sufficiently smooth, and is associated with an adjustable support domain. It also has smaller second order momentum, and therefore it can have better accuracy in terms of kernel approximation. SPH method with this double cosine kernel function is applied to simulate a dam-break flow and water entry of a horizontal circular cylinder. The obtained SPH results agree very well with the experimental results. The double cosine kernel function is also comparatively studied with two frequently used SPH kernel functions, Gaussian and cubic spline kernel functions.     

The Construction and Properties of Boundary Kernels for Smoothing Sparse Multinomials

In recent years several authors have investigated the use of smoothing methods for sparse multinomial data. In particular, Hall and Titterington (1987) studied kernel smoothing in detail. It is pointed out here that the bias of kernel estimates of probabilities for cells near the boundaries of the multinomial vector can dominate the mean sum of squared error of the estimator for most true probability vectors. Fortunately, boundary kernels devised to correct boundary effects for kernel regression estimators can achieve the same result for these estimators. Properties of estimates based on boundary kernels are investigated and compared to unmodified kernel estimates and maximum penalized likelihood estimates. Monte Carlo evidence indicates that the boundary-corrected kernel estimates usually outperform uncorrected kernel estimates and are quite competitive with penalized likelihood estimates.     

Applications of the Bernstein-Durrmeyer operators in estimating the norm of Mercer kernel matrices

The paper is related to the norm estimate of Mercer kernel matrices.The lower and upper bound estimates of Rayleigh entropy numbers for some Mercer kernel matrices on[0,1]×[0,1]based on the Bernstein-Durrmeyer operator kernel ale obtained,with which and the approximation property of the Bernstein-Durrmeyer operator the lower and upper bounds of the Rayleigh entropy number and the l2-norm for general Mercer kernel matrices on[0,1]×[0,1]are provided.