Ordering Gini indexes of multivariate elliptical risks

Gini index is a well-known tool in economics that is often used for measuring income inequality. In insurance, the index and its modifications have been used to compare the riskiness of portfolios, to order reinsurance contracts, and to summarize insurance scores (relativities). In this paper, we establish several stochastic orders between the Gini indexes of multivariate elliptical risks with the same marginals but different dependence structures. This work is motivated by the applied studies of Brazauskas et al. (2007) and Samanthi et al. (2015), who employed the Gini index to compare the riskiness of insurance portfolios. Based on extensive Monte Carlo simulations, these authors have found that the power function of the associated hypothesis test increases as portfolios become more positively correlated. The comparison of the Gini indexes (of empirically estimated risk measures) presented in this paper provides a theoretical explanation to this statistical phenomenon. Moreover, it enriches the studies of the problem of central concentration of elliptical distributions and generalizes the pd-1 order proposed by Shaked and Tong (1985).     

Exit times for multivariate autoregressive processes

We study exit times from a set for a family of multivariate autoregressive processes with normally distributed noise. By using the large deviation principle, and other methods, we show that the asymptotic behavior of the exit time depends only on the set itself and on the covariance matrix of the stationary distribution of the process. The results are extended to exit times from intervals for the univariate autoregressive process of order n$n$, where the exit time is of the same order of magnitude as the exponential of the inverse of the variance of the stationary distribution.     

Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications

The aggregate claim amount in a particular time period is a quantity of fundamental importance for proper management of an insurance company and also for pricing of insurance coverages. In this paper, we show that the proportional hazard rates (PHR) model, which includes some well-known distributions such as exponential, Weibull and Pareto distributions, can be used as the aggregate claim amount distribution. We also present some conditions for the use of exponentiated Weibull distribution as the claim amount distribution. The results established here complete and extend the well-known result of Khaledi and Ahmadi (2008).     

Gompertz分布顺序统计量的一些随机比较

考虑两组相互独立的来自非齐次总体Gompertz分布的样本,给出了最小顺序统计量的反向失效率序、散度序以及凸变换序之间的比较和最大顺序统计量的普通随机序的比较.     

Stochastic Ordering of Multivariate Normal Distributions

We show an interesting identity for Ef(Y) – Ef(X), where X, Yare normally distributed random vectors and f is a function fulfilling some weak regularity condition. This identity will be used for a unified derivation of sufficient conditions for stochastic ordering results of multivariate normal distributions, some well known ones as well as some new ones. Moreover, we will show that many of these conditions are also necessary. As examples we will consider the usual stochastic order, convex order, upper orthant order, supermodular order and directionally convex order.     

Tail asymptotic results for elliptical distributions

Let (X,Y) be a bivariate elliptical random vector with associated random radius in the Gumbel max-domain of attraction. In this paper we obtain a second order asymptotic expansion of the joint survival probability and the conditional probability , for x,y large.     

关于正态分布的次序统计量的随机序

设$X_1,X_2,\cdots,X_n$和$X^*_1,X^*_2,\cdots,X^*_n$分别服从正态分布$N(\mu_i,\sigma^2)$和$N(\mu^*_i,\sigma^2)$,以$X_{(1)}$,$X^*_{(1)}$分别表示$X_1,\cdots,X_n$和$X^*_1,\cdots,X^*_n$的极小次序统计量,以$X_{(n)}$, $X^*_{(n)}$分别表示$X_1,\cdots,X_n$和$X^*_1,\cdots$,$X^*_n$的极大次序统计量. 我们得到了如下结果:(i)\,如果存在严格单调函数$f$使得$(f(\mu_{1}),\cdots,f(\mu_{n}))\succeq_{\text{m}}$ $(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,且$f'(x)f'(x)\!\geq\!0$, 则$X_{(1)}\!\leq_{\text{st}}\!X^*_{(1)}$;(ii)\,如果存在严格单调函数$f$使得$(f(\mu_{1})$,$\cdots,f(\mu_{n}))\succeq_{\text{m}}(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,且$f'(x)f'(x)\leq 0$, 则$X_{(n)}\geq_{\text{st}}X^*_{(n)}$.(iii)\,设$X_{1},X_{2},\cdots,X_{n}$和\, $X^*_{1},X^*_{2},\cdots,X^*_{n}$分别服从正态分布$N(\mu,\sigma_i^2)$和$N(\mu,\sigma_i^{*2})$,若$({1}/{\sigma_{1}},\cdots,{1}/{\sigma_{n}})\succeq_{\text{m}}({1}/{\sigma^{*}_{1}},\cdots,{1}/{\sigma^{*}_{n}})$,则有$X_{(1)}\leq_{\text{st}}X^*_{(1)}$和$X_{(n)}\geq_{\text{st}}X^*_{(n)}$同时成立.     

逆Weibull分布随机变量的随机比较

研究了两个相互独立的逆Weibull分布随机变量间的随机序,似然比序,危险率序以及凸序之间的相互关系,给出了两个相互独立但不同分布的随机变量满足各种随机序时其分布所含参数间的相应关系．也给出了两组相互独立但不同分布的随机变量极值间在一般随机序下的大小关系．     

关于Gamma分布的秩序统计量的随机比较

对于独立不同分布的两个Gamma样本，当它们相同的形状参数大于或等于1时，最近Korwar(2002)证明了，当它们的尺度参数满足优化序时，样本的卷积就满足似然比序．本文我们证明了，当Gamma分布的形状参数小于1时，样本的对应秩序统计量之间存在一致的一般随机序；然而当形状参数大于1时，样本对应的极大值和极小值统计量有着相反的一般随机序。     

Asymptotic behaviors of the Lorenz curve and Gini index in sampling from a length-biased distribution

In this work, we consider the nonparametric estimators of the Lorenz curve and Gini index based on a sample from the corresponding length-biased distribution. We show that this estimators are strongly consistent for the associated Lorenz curve and Gini index. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. We apply the strong Gaussian approximation technique to obtain a functional law for the iterated logarithm for the Lorenz curve. Also, we obtain an asymptotic normality for the corresponding Gini index.     

逆Weibull分布随机变量的随机比较(英文)

研究了两个相互独立的逆Weibull分布随机变量间的随机序,似然比序,危险率序以及凸序之间的相互关系,给出了两个相互独立但不同分布的随机变量满足各种随机序时其分布所含参数间的相应关系.也给出了两组相互独立但不同分布的随机变量极值间在一般随机序下的大小关系.     

Characterizations of the exponential distribution by stochastic ordering properties of the geometric compound

Under the reliability NBU/NWU conditions, the exponential distribution is characterized by stochastic ordering properties which link the geometric compound with minimum order statistics or spacings of order statistics. This somewhat answers a question posed by Kakosyan, Klebanov and Melamed (1984,Characterization of Distributions by the Method of intensively Monotone Operators, Springer, New York). We also show the related results based on the residual life in a renewal process and on record values. Finally, some fundamental properties of the NBUC/NWUC classes of life distributions are investigated.     

随机Hamilton系统同胚流的大偏差原理

在非Lipschitz条件下得到随机微分方程同胚流的大偏差原理.作为应用,本文同时给出了随机Hamilton系统同胚流的大偏差原理.特别地,以下二阶非线性随机振荡方程同胚流的大偏差原理也同样成立:Z_t=C_0Z_t-Z_t~3+Θ(Z_t)W_t,(Z_0,Z_0)=(z,u)∈R~2,其中C_0为任意常数,Θ为一阶导数有界的二阶连续可微函数,W_t是一维Brown白噪声.     

On Stochastic Orders for Sums of Independent Random Variables

In this paper, it is shown that a convolution of uniform distributions (a) is more dispersed and (b) has a smaller hazard rate when the scale parameters of the uniform distributions are more dispersed in the sense of majorization. It is also shown that a convolution of gamma distributions with a common shape parameter greater than 1 is larger in (a) likelihood ratio order and (b) dispersive order when the scale parameters are more dispersed in the sense of majorization.     

板梁组合结构可靠性分析的随机边界元法

本文用随机边界元法分析了随机荷载作用下具有随机边界条件的正交各向异性板、梁组合结构的可靠性.文中首先给出正交各向异性板、梁组合结构的边界积分方程,进而基于随机边界元法建立了随机结构可靠性分析方法和得到用于计算正交各向异性板、梁组合结构可靠性指标的公式.算例表明了本文方法的有效性.     

Distributions of maximum likelihood estimators of Lorenz curve and Gini index of exponential distribution

Summary This paper presents the maximum likelihood estimators (MLEs) of the Lorenz curve and Gini index of the exponential distribution, their exact distributions and moments. All these MLEs are shown to converge almost surely and in therth mean. Further their asymptotic distributions are obtained. Here we use only very simple arguments to derive certain results that are very useful in statistical study of ‘inequality’.     

相依随机序列加权和滑动平均的若干小偏差定理

引入随机序列滑动似然比作为任意二值随机序列相对于Bernoulli分布的独立随机变量序列偏差的一种随机性度量,通过滑动相对熵给出了样本空间的一个子集.在此子集上得到了一类关于随机序列滑动平均的用不等式表示的强极限定理,即小偏差定理,推广了文献[5],[6]等关于随机序列算术平均的结果,这些结果蕴含近期诸多文献的主要结果.     

随机需求的车辆路线问题的新模型

本主要研究随机需求的VRP问题,其中服务需求量满足二项式分布,根据期望值的大小我们提出了在一条路线上理想最大服务点数的新概念,并在此基础上建立了三种VRP问题的新模型,由于允许服务失败两次和部分服务使得模糊能适应多种实际问题,以模拟退火思想为基础的两阶段方法经修正后用于解新模型并取得较好的数值结果,理论分析和数值结果表明,新模型较好地描述随机需求的VRP问题,并且容易求解。     

Comparisons and bounds for expected lifetimes of reliability systems

Sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non-identically distributed lifetimes are expressed in terms of expected lifetimes of components. Similar evaluations are concluded for the respective mean residual lifetimes. In the IID case, improved inequalities dependent on a concentration parameter connected to the Gini dispersion index are obtained. The results can be used to compare systems with component lifetimes ordered in the convex ordering. In the INID case, some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes, and the expected lifetime of single unit for the equivalent systems with IID components. The latter can be further simplified in the case of weak Schur-concavity and Schur-convexity of the system generalized domination polynomial.     

任意随机序列关于非齐次马氏链的随机和的一类随机偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic sequence, as a measure of dissimilarity between the joint distribution on measure P and the Markov distribution on measure Q, is introduced. A class of random approximation theorems for arbitrary stochastic dominated sequence are obtained by using the tools of generating functions and the tailed-probability generating functions.