An Exponential Inequality for U-Statistics Under Mixing Conditions

The family of U-statistics plays a fundamental role in statistics. This paper proves a novel exponential inequality for U-statistics under the time series setting. Explicit mixing conditions are given for guaranteeing fast convergence, the bound proves to be analogous to the one under independence, and extension to non-stationary time series is straightforward. The proof relies on a novel decomposition of U-statistics via exploiting the temporal correlatedness structure. Such results are of interest in many fields where high-dimensional time series data are present. In particular, applications to high-dimensional time series inference are discussed.     

On the rate of convergence and aymptotic expansions forU-statistics under alternatives

In this paper, we consider asymptotic expansions and the rate of convergence for the distribution function of asymptotically efficient U-statistics under alternatives in the one-sample problem. Section 1 is an introduction. Section 2 contains the theorem concerning the rate of convergence for U-statistics; in Sec. 3, we formulate sets of sufficient conditions under which Edgeworth-type asymptotic expansions for U-statistics under alternatives will be constructed (see Theorem 2). Finally, these theorems are proved in Sec. 4. Supported by the Russian Foundation for Basic Research (grant No. 96-01-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

关于自助 U-统计量的渐近性质

一、引言设 X_1,X_2,…,X_n 为来自分布 F 的独立随机变量,h(x_1,x_2)为关于两个变元 x_1,x_2对称的 Borel 可测函数。设 Eh(X_1,X_2)=θ,那么下面定义的 U-统计量     

关于U-统计量完全收敛性

本文讨论了满足E[f(xi,yj)xi]=E[f(xi,yj)xj]=0的U－统计量最大值完全收敛性的充分条件,降低了王岳宝1996年论文中的矩条件,进一步对一般形式的多元函数的U－统计量最大值的完全收敛性的充分条件进行了讨论,得到了较理想的结果。     

关于U-统计量最大值完全收敛的进一步讨论

本文讨论了Ｕ－统计量最大值完全收敛的充分条件，拓宽了周元■及拙文［１］中核函数的范围，降低了矩的阶数，更确切合理地阐明了Ｕ－统计量最大值与熟知的独立和最大值的完全收敛之间的内在系与区别。     

随机足标U-统计量逼近正态的阶

随机变量随机和的收敛性问题无论在理论上还是实用上都是有重要意义的。关于随机和的中心极限定理已有相当一般的结果。近十年来又有一系列讨论收敛速度的文章(如Landers和Rogge[1],Sreehari[2]和Prakasa Rao[3])。关于U-统计量,它的随机中心极限定理已在Sproule[4]中给出。近年采对U-统计量的Berry-Esseen不等式也有相当深入的结果(如赵林城[5],林正炎[6])。本文进一步讨论U-统计量的随机中心极限定理的收敛速度。     

Threshold Results for U-Statistics of Dependent Binary Variables

Large sample results for certain U-statistics, and related statistics, of binary dependent random variables are studied. The class of U-statistics include partial sums and polynomials of partial sums of a sequence of random variables. A very wide range of limit results are found. The form of the limit result can depend substantially on the magnitude of the appropriate normalizing sequence for the sum. Unexpectedly, the nature of the limit result also depends significantly on whether the degree of the U-statistic is even or odd. It is shown that dependence is a major factor contributing to this result. The limit results are illustrated with reference to a simple dynamic sequence of binary variables and a reinforced random walk.     

关于Von－Mises统计量的非一致性收敛速度

本文研究了Ｖｏｎ－Ｍｉｓｅｓ统计量的非一致性收敛速度，得到与Ｕ－统计量类似的收敛速度．     

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.     

Asymptotic Comparisons of Several Variance Estimators and their Effects for Studentizations

In this paper we obtain asymptotic representations of several variance estimators of U-statistics and study their effects for studentizations via Edgeworth expansions. Jackknife, unbiased and Sen's variance estimators are investigated up to the order o^p(n^-1). Substituting these estimators to studentized U-statistics, the Edgeworth expansions with remainder term o(n^-1) are established and inverting the expansions, the effects on confidence intervals are discussed theoretically. We also show that Hinkley's corrected jackknife variance estimator is asymptotically equivalent to the unbiased variance estimator up to the order o^p(n^-1).     

A note on the ASCLT for triangular arrays of random variables with an extension to U-statistics

In this paper, we obtain an almost sure central limit theorem for products of independent sums of positive random variables. An extension of the result gives an ASCLT for the U-statistics.     

检验的渐近展开和功效损失

本文研究统计假设检验问题中的渐近展开和功效损失,给出一阶渐近展开,二阶效率和功效损失,并且研究了建立在L－,R－,U－统计量及组合L－统计量上的检验问题。     

Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge G_Q. This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of G_Q-integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.     

Central limit theorem for Gibbsian U-statistics of facet processes

A special case of a Gibbsian facet process on a fixed window with a discrete orientation distribution and with increasing intensity of the underlying Poisson process is studied. All asymptotic joint moments for interaction U-statistics are calculated and the central limit theorem is derived using the method of moments.     

由一段样本产生的U-统计量的强极限定理

本文提出了一类不完全统计量,它们是由一段相继观察值产生的,并且研究了由这类观察值产生的 Ｕ－统计量的强极限定理．     

U-统计量的几乎处处中心极限定理

本文得到了U-统计量的几乎处处中心极限定理(ASCLT).在EX1=0,EX2=1下,Berkes等[7]在一定条件下获得了i.i.d.随机变量序列部分和的函数型ASCLT,本文在同样的条件下取得了类似的结果     

广义U—统计量的指数收敛速度

文「１」研究了一样本Ｕ－统计量的指数收敛速度，本文则研究了广义Ｕ－统计量的指数收敛速度。     

U-统计量的指数收敛速度的进一步讨论

在独立未必同分布的情形下，对U－统计量的指数收敛速度进行了讨论，减弱文[2]中的部分条件，给出了类似的结果，同时对Von-Mises统计量也给出了指数收敛速度。     

k-U统计量的指数收敛速度

研究了k-U统计量的收敛速度,在一组适当的正则条件下,获得了k-U统计量的指数收敛速度,推广了U-统计量的指数收敛速度的相应结果.     

Some Moment and Exponential Inequalities for V-Statistics with Bounded Kernels

This paper provides moment and exponential inequalities for V-statistics with uniformly bounded kernels. The main results are based on moment inequalities for sub-Bernoulli functions of independent random variables. These moment inequalities lead to extensions of the Bernstein inequality to V-statistics. U-statistics are also discussed.