含小参数微分方程组的近似不变流形

本文首先用正则摄动方法给出一类含小参数常微分方程组的近似不变流形和近似分析解。然后,给出近似分析解的误差估计式;并且证明了在一定的条件下,这类近似不变流形是中心流形的近似表示式。     

鲁棒凸优化问题拟近似解的刻划

本文研究鲁棒凸优化问题拟近似解的最优性条件和对偶理论.首先利用鲁棒优化方法,在由约束函数的共轭函数的上图给出的闭凸锥约束规格条件下,建立了拟近似解的最优性充要条件.其次给出了鲁棒凸优化问题拟近似解在Wolf型和Mond-weir型对偶模型下的强(弱)对偶定理.最后给出具体实例验证了本文获得的结果.     

自治差分方程的稳定性

本文首先给出了文献[2]中关于一次近似系统不稳定性定理的一个反例,然后给出了关于自治差分方程利用其一次近似系统的不稳定性来判别原系统不稳定性的判别定理。     

平面上的近似连续积分

近似连续Ｐｅｒｒｏｎ 积分首先由Ｂｕｒｋｉｌｌ提出,称之为ＡＰ积分,由于近似复盖具有划分性质,因而可以给出ＡＰ积分的Ｒｉｅｍ ａｎｎ型定义． 但是,在平面上近似复盖是否具有划分性质还不知道． 本文给出了一种方法,用这种方法可以在平面上给出ＡＰ积分的Ｒｉｅｍ ａｎｎ型定义而不需要证明划分的存在性．     

连续年龄分布下两种群系统的近似可控性

本文研究一类具有年龄结构的非线性两种群系统的适定性和近似可控制性,运用不动点原理给出了近似可控性条件.     

基于二项分布的短期聚合风险模型

重点讨论了索赔次数服从于二项分布的情况下单个险种和多个险种的聚合风险模型,得出了在此情况下求其分布函数的若干方法,并给出聚合理赔量的两种近似模型,正态近似和平移伽马近似.最后给出了一个数值例子,验证了本文的分布函数的若干求法.     

一类带平行机的两阶段柔性流水调度近似算法

本文研究一类柔性流水调度与平行机调度相结合的两阶段流水调度模型,模型中第1阶段有1台机器,第2阶段有m台同构并行机,每个任务在第2阶段需要size_i台机器同时并行执行.目标是所有任务都完成的完工时间最小化.该模型已被证明出是强NP难的,并给出了在某种特定情况下近似比为3的近似算法.本文首先详细分析了前人近似算法基本过程,给出该算法近似比分析的局限性;接着给出了一个近似比为3的算法,摒弃了前人给出的近似比为3时的约束条件;最后研究了当第2阶段机器数为2和3时的两种特定情况,采用列表调度思想,给出了近似比为2.5和2.67的近似算法.     

混合效应模型的最优区组设计

本文对将驻点个数等于参数个数的混合效应模型做D-最优设计.以两种误差分布的组合为例做近似区组设计,然后给出精确区组设计的设计点位置及区组的权数的解析方程组,给出精确设计比近似设计好的条件.最后,给出数值结果及效率比较.     

非线性不适定问题的正则化方法

本文考虑非线性不适定问题Tx=y的近似求解,利用Тихоноь正则化方法来逼近问题的x-极小模解,当算子和右端都近似已知时,给出一种决定正则化参数的方法,并给出正则解的收效性和渐近收敛阶估计。     

无解的模糊关系方程的最优近似解

对无解的模糊关系方程给出了最优近似解的定义,证明了最优近似解的存在性,给出了求最优近似解的算法     

积分波动率二阶变差估计量分析:鞍点算法

本文利用鞍点逼近方法对Black-Scholes模型的积分波动率的二阶变差估计量的估计误差进行分析,得到了相对于中心极限定理更为精细的结果,并且给出了逼近的鞍点算法。结果表明鞍点逼近是中心极限定理的纠正。模拟结果表明鞍点算法给出的估计误差分布相对于正态逼近更合理。该结果在对积分波动率进行统计假设检验时是有意义的。     

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.     

A Saddlepoint Approximation to the Distribution of Inhomogeneous Discounted Compound Poisson Processes

In this article we propose an accurate approximation to the distribution of the discounted total claim amount, where the individual claim amounts are independent and identically distributed and the number of claims over a specified period is governed by an inhomogeneous Poisson process. More precisely, we compute cumulant generating functions of such discounted total claim amounts under various intensity functions and individual claim amount distributions, and invert them by the saddlepoint approximation. We provide precise conditions under which the saddlepoint approximation holds. The resulting approximation is numerically accurate, computationally fast and hence more efficient than Monte Carlo simulation.     

An efficient third-moment saddlepoint approximation for probabilistic uncertainty analysis and reliability evaluation of structures

This paper presents an efficient third-moment saddlepoint approximation approach for probabilistic uncertainty analysis and reliability evaluation of random structures. By constructing a concise cumulant generating function (CGF) for the state variable according to its first three statistical moments, approximate probability density function and cumulative distribution function of the state variable, which may possess any types of distribution, are obtained analytically by using saddlepoint approximation technique. A convenient generalized procedure for structural reliability analysis is then presented. In the procedure, the simplicity of general moment matching method and the accuracy of saddlepoint approximation technique are integrated effectively. The main difference of the presented method from existing moment methods is that the presented method may provide more detailed information about the distribution of the state variable. The main difference of the presented method from existing saddlepoint approximation techniques is that it does not strictly require the existence of the CGFs of input random variables. With the advantages, the presented method is more convenient and can be used for reliability evaluation of uncertain structures where the concrete probability distributions of input random variables are known or unknown. It is illustrated and examined by five representative examples that the presented method is effective and feasible.     

Saddlepoint approximation at the edges of a conditional sample space

Saddlepoint methods present a convenient way to approximate probabilities associated with canonical sufficient statistic vectors in generalized linear models. Implementing saddlepoint approximations requires calculating maximum likelihood estimators for the associated parameters. When the sufficient statistic vector lies at the edge of the sample space, maximum likelihood estimators may not exist. This paper describes how to modify saddlepoint approximation to work in these cases.     

Saddlepoint approximations for tests of dispersion

The nonparametric class of tests for dispersion is widely used for testing the equality of the scale parameters of two populations. This class includes Mood, Siegel–Tukey, Klotz and Moses tests. This paper uses the double saddlepoint approximation to calculate analytical mid-p-values for these tests that are almost exact as the permutation simulation method. The performance of the saddlepoint method is assessed using an extensive simulation study. The speed and accuracy of the saddlepoint method enable us to invert the dispersion tests to calculate (1 ? α)100% confidence intervals for the dispersion parameter.     

A numerical comparison of the normal and some saddlepoint approximations to a distribution-free test for stochastic ordering in the competing risks model

Calculating the exact critical value of the test statistic is important in nonparametric statistics. However, to evaluate the exact critical value is difficult when the sample sizes are moderate to large. Under these circumstances, to consider more accurate approximation for the distribution function of a test statistic is extremely important. A distribution-free test for stochastic ordering in the competing risks model has been proposed by Bagai et al. (1989). Herein, we performed a saddlepoint approximation in the upper tails for the Bagai statistic under finite sample sizes. We then compared the saddlepoint approximations with the Bagai approximation and investigate the accuracy of the approximations. Additionally, the orders of errors of the saddlepoint approximations were derived.     

Some Properties and Improvements of the Saddlepoint Approximation in Nonlinear Regression

We summarize properties of the saddlepoint approximation of the density of the maximum likelihood estimator in nonlinear regression with normal errors: accuracy, range of validity, equivariance. We give a geometric insight into the accuracy of the saddlepoint density for finite samples. The role of the Riemannian curvature tensor in the whole investigation of the properties is demonstrated. By adding terms containing this tensor we improve the saddlepoint approximation. When this tensor is zero, or when the number of observations is large, we have pivotal, independent, and ² distributed variables, like in a linear model. Consequences for experimental design or for constructions of confidence regions are discussed.     

Saddlepoint Approximation for the Distribution Function Near the Mean

We repair numerical difficulties in applying saddlepoint tail probability approximations when the ordinate at which the approximation is evaluated is near the mean of the distribution approximated. These modifications apply to double saddlepoint approximations to conditional distributions as well.     

Saddlepoint approximations for curved exponential families

An approximation of the density of the maximum likelihood estimator in curved exponential families is derived using a saddlepoint expansion. The approximation is particularly simple in nonlinear regression. An example is considered.