On Edgeworth Expansions in Generalized Urn Models

The random vector of frequencies in a generalized urn model can be viewed as conditionally independent random variables, given their sum. Such a representation is exploited here to derive Edgeworth expansions for a “sum of functions of such frequencies,” which are also called “decomposable statistics.” Applying these results to urn models such as with- and without-replacement sampling schemes as well as the multicolor Pólya–Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without-replacement scheme, and for the so-called Dixon statistic that is useful in comparing two samples.     

一类相合的分位点型检验

广义非参数似然比检验统计量是一类很广的统计量,包含了众多重要的检验统计量,如Anderson-Darling(AD)等.利用Rubin的随机经验分布函数替代经验分布函数的方法,得到了广义非参数似然比检验统计量的新版本,构造了新的检验统计量.由于新的检验统计量在给定样本下仍然是随机变量,选择了它的分位点和期望作为检验统计量,分别称之为分位点型检验统计量和期望型检验统计量.在简单假设情况下,证明了分位点型检验统计量和期望型检验统计量在固定备择下的相合性.模拟结果显示,在某些备择下,新的检验的功效明显高于原有的基于经验分布函数的检验的功效.     

Lorenz ordering of power-function order statistics

Based on Lorenz comparisons of two random variables from the four-parameter generalized beta distribution of the first kind, Lorenz-order relationships among order statistics from different power-function samples with varying sizes are obtained.     

Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data

In this paper, we establish some results for the increasing convex comparisons of generalized order statistics. First, we prove that if the minimum of two sets of generalized order statistics are ordered in the increasing convex order, then the remaining generalized order statistics are also ordered in the increasing convex order. This result is extended to the increasing directionally convex comparisons of random vectors of generalized order statistics. For establishing this general result, we first prove a new result in that two random vectors with a common conditionally increasing copula are ordered in the increasing directionally convex order if the marginals are ordered in the increasing convex order. This latter result is, of course, of interest in its own right.     

Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order

In this paper, we characterize counter-monotonic and upper comonotonic random vectors by the optimality of the sum of their components in the senses of the convex order and tail convex order respectively. In the first part, we extend the characterization of comonotonicity by  Cheung (2010) and show that the sum of two random variables is minimal with respect to the convex order if and only if they are counter-monotonic. Three simple and illuminating proofs are provided. In the second part, we investigate upper comonotonicity by means of the tail convex order. By establishing some useful properties of this relatively new stochastic order, we prove that an upper comonotonic random vector must give rise to the maximal tail convex sum, thereby completing the gap in  Nam et al. (2011)’s characterization. The relationship between the tail convex order and risk measures along with conditions under which the additivity of risk measures is sufficient for upper comonotonicity is also explored.     

关于次序统计量数学期望的两个恒等式的简证及其应用

给出了最大最小次序统计量分布函数和密度函数的恒等式,从而在随机变量独立同分布情形下,给出了关于次序统计量数学期望的两个恒等式的简捷证法.最后将几个恒等式应用于威布尔分布的最大次序统计量的分布及其数字特征的计算.     

Post buckling response of laminated composite plate on elastic foundation with random system properties

This paper deals with second order statistics of post buckling load of shear deformable laminated composite plates resting on linear elastic foundation with random system properties. The formulation is based on higher order shear deformation plate theory in general von Karman sense, which includes foundation effect using two-parameter Pasternak model. The random system equations are derived using the principal of virtual work. A finite element method is used for spatial descretization of the laminate with a reasonable accuracy. A perturbation technique has been the first time successfully combined with direct iterative technique by neglecting the changes in nonlinear stiffness matrix due to random variation of transverse displacements during iteration. The numerical results for the second order statistics of post buckling loads are obtained. A detailed study is carried out to highlight the characteristics of the random response and its sensitivity to different foundation parameters, the plate thickness ratio, the plate aspect ratio, the support condition, the stacking sequence and the lamination angle on the post buckling response of the laminate. The results have been compared with existing results and an independent Monte Carlo simulation.     

Recurrence relations among moments of order statistics from two related sets of independent and non-identically distributed random variables

Some recurrence relations among moments of order statistics from two related sets of variables are quite well-known in the i.i.d. case and are due to Govindarajulu (1963a, Technometrics, 5, 514–518 and 1966, J. Amer. Statist. Assoc., 61, 248–258). In this paper, we generalize these results to the case when the order statistics arise from two related sets of independent and non-identically distributed random variables. These relations can be employed to simplify the evaluation of the moments of order statistics in an outlier model for symmetrically distributed random variables.     

非负NOD随机变量序列的逆矩

仅在一阶矩有限的条件下获得了非负NOD随机变量序列正则和的逆矩的渐近逼近,特别得到了部分和的逆矩的渐近逼近.本文的结论推广了已有的结果.     

Asymptotics for Ratios with Applications to Reinsurance

This paper provides an overview of results pertaining to moment convergence for certain ratios of random variables involving sums, order statistics and extreme terms in the sense of modulus. Most of the literature on this matter originates from Darling (1952) who gave a criterion for the convergence in probability to 1 of the ratio of the maximum to the sum in case of nonnegative random variables. Sophie A. Ladoucette is supported by the grant BDB-B/04/03 of the Catholic University of Leuven.     

Order statistics from mixed exchangeable random variables

Two different exchangeable samples are considered and these two samples are assumed to be independent of each other. From these two samples a new sample is combined and treated as a single set of observations. The distribution of a single order statistic and the joint distribution of two order statistics for a new mixed sample are derived and expressed in terms of joint distribution functions. As a special case the distribution of a single order statistic and the joint distribution of two nonadjacent order statistics from exchangeable random variables are obtained. The results presented in this paper allows widespread applications in modelling of various lifetime data, biomedical sciences, reliability and survival analysis, actuarial sciences etc., where the assumption of independence of data cannot be accepted and the exchangeability is a more realistic assumption.     

Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C⁰ finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.     

Representations of ordered random variables in the terms of the sums or products of independent variables

This is a survey of the available representations capable of expressing dependent ordered variables like order statistics and records as sums of independent summands or as products of independent random multipliers. Some new similar representations are obtained that in a similar manner express the common distributions of representatives of these two types of ordered random variables.     

ʲôʱ�����һ������ֵ��ɢ�ֲ��IJ�������С���ͳ����?

??We prove three theorems for iid discrete randomvariables taking two values, three values, and k (3\leq k<\infty) valuesby using the technique of indicator function. Under some specifications of the probabilities, we prove that the sum is a minimal sufficient statistics for the unknown parameter of interest of the discrete random variable taking two values, three values, and k (3\leq k<\infty) values. For the dice example, a figure shows that the specifications of the six probabilities are between 0 and 1 and sum to 1, and a fair dice is possible.     

Sign or root of unity ambiguities of certain Gauss sums

Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign (or a root of unity) ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign (root of unity) ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

When the Sum is a Minimal Sufficient Statistics for the Parameter of a Discrete Distribution with Finite Values?

We prove three theorems for iid discrete randomvariables taking two values, three values, and k (3\leq k<\infty) valuesby using the technique of indicator function. Under some specifications of the probabilities, we prove that the sum is a minimal sufficient statistics for the unknown parameter of interest of the discrete random variable taking two values, three values, and k (3\leq k<\infty) values. For the dice example, a figure shows that the specifications of the six probabilities are between 0 and 1 and sum to 1, and a fair dice is possible.     

Controllability for Semilinear Functional and Neutral Functional Evolution Equations with Infinite Delay in Fréchet Spaces

The controllability of mild solutions defined on the semi-infinite positive real interval for two classes of first order semilinear functional and neutral functional differential evolution equations with infinite delay is studied in this paper. Our results are obtained using a recent nonlinear alternative due to Avramescu for sum of compact and contraction operators in Fréchet spaces, combined with the semigroup theory.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

Minimal cyclic random motion in and hyper-Bessel functions

We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity c in . The n+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time t), the geometrical features of which are here investigated.The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order n+1. These distributions are proved to be connected with (n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion in R³ with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.