Stochastic Taylor Expansions for Functionals of Diffusion Processes

In this article, a stochastic Taylor expansion of some functional applied to the solution process of an Itô or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean-square and the mean truncation errors are given.     

The Logarithmic Asymptotic Expansions for the Norms of Evaluation Functionals

Let μ be a compactly supported finite Borel measure in ℂ, and let Π_n be the space of holomorphic polynomials of degree at most n furnished with the norm of L ²(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Π_n their values at a point z ∈ ℂ. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl-Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n → ∞.Original Russian Text Copyright © 2005 Dovgoshei A. A., Abdullaev F., and Kucukaslan M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 774–785, July–August, 2005.     

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Abstract

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.     

Empirical Likelihood for a Class of Functionals of Survival Distribution with Censored Data

The empirical likelihood was introduced by Owen, although its idea originated from survival analysis in the context of estimating the survival probabilities given by Thomas and Grunkemeier. In this paper, we investigate how to apply the empirical likelihood method to a class of functionals of survival function in the presence of censoring. We define an adjusted empirical likelihood and show that it follows a chi-square distribution. Some simulation studies are presented to compare the empirical likelihood method with the Studentized-t method. These results indicate that the empirical likelihood method works better than or equally to the Studentized-t method, depending on the situations.     

Partial Sum Process for Records

Suppose the upper records from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z_n(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z_n} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal. AMS 2000 Subject Classification Primary—60F17 Secondary—60G70     

广义Pareto分布的广义有偏概率加权矩估计方法

广义Pareto分布(GPD)是统计分析中一个极为重要的分布,被广泛应用于金融、保险、水文及气象等领域.传统的参数估计方法如极大似然估计、矩估计及概率加权矩估计方法等已被广泛应用,但使用中存在一定的局限性.虽然提出很多改进方法如广义概率加权矩估计、L矩和LH矩法等,但都是研究完全样本的估计问题,而在水文及气象等应用领域常出现截尾样本.本文基于概率加权矩理论,利用截尾样本对三参数GPD提出一种应用范围广且简单易行的参数估计方法,可有效减弱异常值的影响.首先求解出具有较高精度的形状参数的参数估计,其次得出位置参数及尺度参数的参数估计.通过Monte Carlo模拟说明该方法估计精度较高.     

Limit Theorems for Weighted Functionals of Cyclical Long-Range Dependent Random Fields

The article investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.     

Bayesian Inference for the Beta-Binomial Distribution via Polynomial Expansions

A commonly used paradigm in modeling count data is to assume that individual counts are generated from a Binomial distribution, with probabilities varying between individuals according to a Beta distribution. The marginal distribution of the counts is then BetaBinomial. Bradlow, Hardie, and Fader (2002, p. 189) make use of polynomial expansions to simplify Bayesian computations with Negative-Binomial distributed data. This article exploits similar expansions to facilitate Bayesian inference with data from the Beta-Binomial model. This has great application and computational importance to many problems, as previous research has resorted to computationally intensive numerical integration or Markov chain Monte Carlo techniques.     

B-值随机变量序列加权和的收敛结果

本文讨论了双向无穷B-值随机变量序列加权和的弱大数定律、Lr收敛性、完全收敛性.并由此刻画了空间的几何性质.     

一类随机变量序列加权和的Chover型重对数律

对于具有某种尾渐近行为的独立同分布的随机变量序列,本文通过积分检验刻划了其加权部分和的极限结果,并作为推论获得了Chover型重对数律。把这些结果应用到经典的可和方式,获得了相应的结果。     

关于记录值序列部分和的渐近正态性

证明了 Burr分布和 Frechét分布的记录值序列的部分和是渐进对数正态的 ,从而解决了文[2 ]中的一个悬而未决的问题     

Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions

To date, Bayesian inferences for the negative binomial distribution (NBD) have relied on computationally intensive numerical methods (e.g., Markov chain Monte Carlo) as it is thought that the posterior densities of interest are not amenable to closed-form integration. In this article, we present a “closed-form” solution to the Bayesian inference problem for the NBD that can be written as a sum of polynomial terms. The key insight is to approximate the ratio of two gamma functions using a polynomial expansion, which then allows for the use of a conjugate prior. Given this approximation, we arrive at closed-form expressions for the moments of both the marginal posterior densities and the predictive distribution by integrating the terms of the polynomial expansion in turn (now feasible due to conjugacy). We demonstrate via a large-scale simulation that this approach is very accurate and that the corresponding gains in computing time are quite substantial. Furthermore, even in cases where the computing gains are more modest our approach provides a method for obtaining starting values for other algorithms, and a method for data exploration.     

Partial Regularity of the Minimizers of Quadratic Functionals with VMO Coefficients

The paper investigates the partial regularity of the minimizersfor quadratic functionals whose integrands have VMO coefficientsin principal part and nonlinear terms that are Carathéodoryfunctions. We use some majorizations for the functional, ratherthan the well known Euler equation associated to it.     

On the Distribution of Denominators in Sylvester Expansions

For any x (0, 1], let the series be the Sylvester expansion of x. Galambos has shown that theLebesgue measure of the set [formula] is 1 when = e, the base of the natural logarithm. This paperprovides a proof that for any 1, A() has Hausdorff dimension1 when e. 2000 Mathematics Subject Classification 11K55 (primary),28A80 (secondary).     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form s/gr - 1, where (> 0) and (= ±1) are parameters indicating the types of expansion. For = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis ( = -1) or by numerical computation ( = 1).     

强相依非平稳高斯序列超过数点过程与部分和的联合渐近分布

设{Xi}∞i=1是标准化强相依非平稳高斯序列,记Sn=∑Xi,σn=√var(Sn),Mktn为X1,X2,…,Xtn的第k个最大值,Ntn为X1,X2,…,Xtn对水平μn(x)的超过数形成的点过程,tn是-列单调增加的正整数列,在一定条件下得到Ntn与Sn/σn,Mktn与Sn/σn的联合渐近分布.     

Poisson Distribution for a Sum of Additive Functions

We consider the limit distribution of values of a sum of additive arithmetic functions with shifted argument. The case of the Poisson limit distribution is studied. The functions considered take at most two values on the set of primes, 0 and 1, and satisfy some additional conditions. Some examples are given.      

平稳遍历马氏链部分和序列最小值分布的渐近估计及其应用

本文给出了有限状态平稳遍历Markov链部分和序列最小值分布的一个渐近估计式并利用它对一类Athreya-KarlinBPRE．灭种概率的渐近行为作出估计．