Stationary Distribution Function Estimation for Ergodic Diffusion Process

The problem of nonparametric stationary distribution function estimation by the observations of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.     

On characterizations of distributions by mean absolute deviation and variance bounds

In this paper we present a bound for the mean absolute deviation of an arbitrary real-valued function of a discrete random variable. Using this bound we characterize a mixture of two Waring (hence geometric) distributions by linearity of a function involved in the bound. A double Lomax distribution is characterized by linearity of the same function involved in the analogous bound for a continuous distribution. Finally, we characterize the Pearson system of distributions and the generalized hypergeometric distributions by a quadratic function involved in a similar bound for the variance of a function of a random variable.     

单调递增分形函数上、下界的估计

Mass distribution principle is one of important tools in studying Hausdorff dimension and Hausdorff measure.In this paper we will give a numerical approximate method of upper bound and lower bound of mass distribution function f(x)(it is a monotone increasing fractal function)and its some applications.     

Lower bounds for the distribution function of a k-dimensionaln-extensible exchangeable process

A lower bound for the distribution function of a k-dimensional, n-extensible exchangeable process is provided when the marginals are uniform on the unit segment. The result is obtained by means of standard linear programming techniques. The lower bound for infinitely extendible exchangeable processes is the distribution of independent random variables.     

Binomial Moments of the Distance Distribution and the Probability of Undetected Error

In [1] K. A. S. Abdel-Ghaffar derives a lower bound on the probability of undetected error for unrestricted codes. The proof relies implicitly on the binomial moments of the distance distribution of the code. We use the fact that these moments count the size of subcodes of the code to give a very simple proof of the bound in Abdel by showing that it is essentially equivalent to the Singleton bound. This proof reveals connections of the probability of undetected error to the rank generating function of the code and to related polynomials (Whitney function, Tutte polynomial, and higher weight enumerators). We also discuss some improvements of this bound.Finally, we analyze asymptotics. We show that an upper bound on the undetected error exponent that corresponds to the bound of Abdel improves known bounds on this function.     

Two approximations of the present value distribution of a disability annuity

The distribution function of the present value of a cash flow can be approximated by means of a distribution function of a random variable, which is also the present value of a sequence of payments, but with a simpler structure. The corresponding random variable has the same expectation as the random variable corresponding to the original distribution function and is a stochastic upper bound of convex order. A sharper upper bound can be obtained if more information about the risk is available. In this paper, it will be shown that such an approach can be adopted for disability annuities (also known as income protection policies) in a three state model under Markov assumptions. Benefits are payable during any spell of disability whilst premiums are only due whenever the insured is healthy. The quality of the two approximations is investigated by comparing the distributions obtained with the one derived from the algorithm presented in the paper by Hesselager and Norberg [Insurance Math. Econom. 18 (1996) 35–42].     

Numerical algorithms for Panjer recursion by applying Bernstein approximation

In actuarial science, Panjer recursion (1981) is used in insurance to compute the loss distribution of the compound risk models. When the severity distribution is continuous with density function, numerical calculation for the compound distribution by applying Panjer recursion will involve an approximation of the integration. In order to simplify the numerical algorithms, we apply Bernstein approximation for the continuous severity distribution function and obtain approximated recursive equations, which are used for computing the approximated values of the compound distribution. The theoretical error bound for the approximation is also obtained. Numerical results show that our algorithm provides reliable results.     

On the remained term for the central limit theorem

Summary An upper bound for the remainder term of the Edgeworth expansion for the distribution of the normalized sum of independent and identically distributed random variables is given in terms of 3rd and 4th order moments, together with the total variation of the probability density function of the underlying distribution. The Institute of Statistical mathematics     

基于污染Gamma分布的聚合风险模型及其在风险分类中的应用

分析了污染Gamma分布及其性质,讨论了基于污染Gamma分布的聚合风险模型.对模型的概率特性和参数估计进行了分析,并对该模型在风险分类中的应用进行了讨论.为克服索赔总量的分布函数在计算上的困难,利用同单调性理论得到了随机凸序意义下索赔总量随机变量S的随机上界Sc,对Sc的分布函数及限额损失保费进行了讨论.通过一个例子对所述结论的有效性进行验证.     

Stability analysis and optimization of an inventory system with bounded orders

This paper addresses the control of a one-item inventory system subject to random order lead time and random demand. The key parameter of the control policy is the objective inventory. In each period, the order to be placed brings the inventory position as close as possible to the objective inventory. The order of each period is kept between a lower bound and an upper bound. We show that the distribution of the inventory level converges to its stationary distribution provided that the lower bound is smaller than the average demand, the upper bound is greater than the average demand and some regularity conditions hold. The average inventory cost is shown to be a convex function of the objective inventory level. A simulation-based approach is proposed for the determination of the optimal objective inventory. A method of bisection with derivative is then used to determine the optimal objective inventory. The derivatives needed in various iterations of this method are estimated using a single sample path with respect to a given objective inventory. Numerical results are provided.     

A Berry-Esséen Bound for the Product-Limit Estimator under Left-Truncation and Right-Censoring

For left-truncated and right-censored data, the product-limit estimator F?_xis a well-known nonparametric estimator for the distribution function F_x of the target variable X such as the survival time. Since F?_xas a very complicated product form we establish first the Berry-Esseen bound for the cumulative hazard estimator of F_x The cumulative hazard estimator can be represented as a U-statistic. By using the result in Helmers and van Zwet [6], we derive the Berry-esséen bound for this U-statistic. Then Berry-Esseen bounds for the distribution of the cumulative hazard estimator and the normal distribution and the distribution of the product-limit estimator and the normal distribution are obtained.     

Sequential change-point detection for mixing random sequences under composite hypotheses

The problem of sequential detection of a change-point in the density function of one-dimensional distribution of observations from a mixing random sequence is considered when both before and after a change-point this density function belongs to a certain family of distributions, i.e. in the situation of composite hypotheses. A new quality criterion for change-point detection is proposed. The asymptotic a priori lower bound for this criterion is proved for wide class of methods of change-point detection. An asymptotically optimal method of change-point detection is proposed for which this lower bound is attained asymptotically. In particular, for the case of a simple hypothesis before a change-point, this method coincides with the generalized cumulative sums (CUSUM) method.      

A storage model with self-similar input

A storage model with self-similar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given.     

Note on the Essential Norm of Composition Operators from HP（BN） to Hq（BN）

The authors give an upper bound of the essential norms of composition operators between Hardy spaces of the unit ball in terms of the counting function in the higher dimensional value distribution theory defined by Professor S.S.Chern.The sufcient condition for such operators to be bounded or compact is also given.     

基于多重工作休假的成批到达离散时间排队的性能分析

研究了一个成批到达的离散时间 Geom$^{[X]}$/Geom/1 多重工作休假排队. 首先,建立了模型的二维马尔可夫链,利用矩阵分析的方法, 导出了稳态队长复杂的概率母函数. 其次, 为了展示此模型与经典无休假Geom$^{[X]}$/Geom/1排队的联系, 给出稳态队长的随机分解结果. 尤其重要的是,发现了条件负二项分布的双参数加法定理, 利用这些结论,得到了矩母函数序下的稳态等待时间的上下界. 进一步,求出了平均队长和平均等待时间的上下界. 最后,提出一些数值例子以验证结论.     

Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities

Introducing the discrete probability distribution by means of the Prabhakar (or the three-parameter Mittag–Leffler) function, we establish explicit expressions for raw and factorial moments and also general fractional-order moments. Applying an elementary moment inequality we obtain functional upper bounds for the Turánian difference for Prabhakar function. Finally, a Laguerre inequality is proved and functional upper bound has been given for Laguerreian difference for Prabhakar function.     

Bounds to optimal burn‐in and optimal work size

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we consider the problem of determining the optimal burn‐in time and optimal work size maximizing the long‐run average amount of work saved per time unit in the computer applications. Assuming that the underlying lifetime distribution of the computer has an initially decreasing or/and eventually increasing failure rate function, an upper bound for the optimal burn‐in time is derived for each fixed work size and a uniform (with respect to the burn‐in time) upper bound for the optimal work size is also obtained. Furthermore, it is shown that a non‐trivial lower bound for the optimal burn‐in time can be derived if the underlying lifetime distribution has a large initial failure rate. Copyright © 2005 John Wiley & Sons, Ltd.     

Distribution of Artin–Schreier extensions

The article at hand contains exact asymptotic formulas for the distribution of conductors of elementary abelian p-extensions of global function fields of characteristic p. As a consequence for the distribution of discriminants, this leads to an exact asymptotic formula for simple cyclic extensions and an interesting lower bound for noncyclic elementary abelian extensions.     

某些一致分布的点集序列

本文借助于“倒根函数”和矩阵构造定义了［０,１）Ｓ（Ｓ≥１）中的一些有限点集,给出了它们的偏差的上界估计,从而证明了由它们组成的点集序列是一致分布的．     

Multi-stage portfolio selection problem with dynamic stochastic dominance constraints

We study the multi-stage portfolio selection problem where the utility function of an investor is ambiguous. The ambiguity is characterized by dynamic stochastic dominance constraints, which are able to capture the dynamics of the random return sequence during the investment process. We propose a multi-stage dynamic stochastic dominance constrained portfolio selection model, and use a mixed normal distribution with time-varying weights and the K-means clustering technique to generate a scenario tree for the transformation of the proposed model. Based on the scenario tree representation, we derive two linear programming approximation problems, using the sampling approach or the duality theory, which provide an upper bound approximation and a lower bound approximation for the original nonconvex problem. The upper bound is asymptotically tight with infinitely many samples. Numerical results illustrate the practicality and efficiency of the proposed new model and solution techniques.