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。     

Mean residual life order of convolutions of heterogeneous exponential random variables

In this paper, we study convolutions of heterogeneous exponential random variables with respect to the mean residual life order. By introducing a new partial order (reciprocal majorization order), we prove that this order between two parameter vectors implies the mean residual life order between convolutions of two heterogeneous exponential samples. For the 2-dimensional case, it is shown that there exists a stronger equivalence. We discuss, in particular, the case when one convolution involves identically distributed variables, and show in this case that the mean residual life order is actually associated with the harmonic mean of parameters. Finally, we derive the “best gamma bounds” for the mean residual life function of any convolution of exponential distributions under this framework.     

On the right spread order of convolutions of heterogeneous exponential random variables

A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. This gives a new insight into the distribution theory of convolutions of independent random variables. A sufficient condition is also derived for comparing such convolutions in terms of Lorenz order.     

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In this paper, we study the stochastic comparisons of order statistics from generalized normal distributions. We obtain some sufficient conditions for ordering results based on parameter matrix and vector majorization comparisons. These conditions are necessary in some cases.     

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??In this paper, we compare the smallest order statistics arising from multiple-outlier models when the numbers of independent and identically distributed random variables are different. Let and denote the smallest order statistics among, and, respectively, whereand. We then prove that $ and are ordered in terms of the usual stochastic order, hazard rate order and likelihood ratio order under the majorization relationship between and.     

On stochastic comparisons of series systems with heterogeneous dependent and independent location-scale family distributed components

This paper carries out comparisons of heterogeneous series systems with location-scale family distributed components It is shown that the systems with dependent components in series sharing Archimedean copula with more dispersion in the location or scale parameters result in better performance in the sense of the usual stochastic order. Moreover, if the components are independently distributed, it is possible to obtain more generalized results as compared to the dependent set-up.     

Comparisons on aggregate risks from two sets of heterogeneous portfolios

In this paper, we stochastically compare the aggregate risks from two heterogeneous portfolios. It is shown that under suitable conditions the more heterogeneities among aggregate risks would result in larger aggregate risks in the sense of the stochastic order. The stochastic properties of aggregate risks when the claims follow proportional hazard rates models or scale models are studied. We also provide sufficient conditions for comparing the aggregate risks arising from two sets of heterogeneous portfolios with claims having gamma distributions. In particular, the aggregate risks of portfolios from dependent samples with comonotonic dependence structures or arrangement increasing density functions are discussed. The new results established strengthen and generalize several results known in the literature including Ma (2000), Khaledi and Ahmadi (2008), Xu and Hu (2011), Xu and Balakrishnan (2011), Pan et al. (2013) and Barmalzan et al. (2015).     

On the Convergence Rates of Extreme Generalized Order Statistics

A classical result of extreme value theory yields that in case of a linear normalization three possible types of limit distributions are possible. As proved recently a similar classification of the limit distributions holds for extreme generalized order statistics which provide a general concept of ordered random variables. In this paper, we derive results for the convergence rates of the nth and (n-r+1)st generalized order statistic, respectively. It turns out that the rate is highly influenced by the choice of the normalizing sequence. Moreover, we show that a uniform bound of order 1/n holds for underlying generalized Pareto distributions, whereas for the standard normal distribution the convergence might be very slow. Similar results for ordinary order statistics are included.     

Convolution on distribution spaces characterized by regularization

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.     

Unimodality of uniform generalized order statistics,with applications to mean bounds

We prove that uniform generalized order statistics are unimodal for an arbitrary choice of model parameters. The result is applied to establish optimal lower and upper bounds on the expectations of generalized order statistics based on nonnegative samples in the population mean unit of measurement. The bounds are attained by two-point distributions.     

Comonotonic convex upper bound and majorization

When the dependence structure among several risks is unknown, it is common in the actuarial literature to study the worst dependence structure that gives rise to the riskiest aggregate loss. A central result is that the aggregate loss is the riskiest with respect to convex order when the underlying risks are comonotonic. Many proofs were given before. The objective of this article is to present a new proof using the notions of decreasing rearrangement and the majorization theorem, and give clear explanation of the relation between convex order, the theory of majorization and comonotonicity.     

The use of order statistic numerical simulation algorithms

A modification of the standard algorithm for the simulation of order statistics for a uniform distribution is proposed that uses confidence intervals. It is found that one of the applications of the algorithms for the simulation of order statistics (namely, simulation of the beta distribution with integer parameters) gives more efficient methods for the simulation of order statistics than the algorithm based on confidence intervals. It is shown that the resulting algorithm can be used for the efficient simulation of random variables with polynomial density and of beta distributed random variables with large noninteger parameters.     

多项分布与Dirichlet分布概率的相互表示

讨论了四种多项分布尾概率与四种Dirichlet分布尾概率的相互表示,并将结果应用于Majorization理论,得到了多项分布和Dirichlet分布对应的一些性质.同时,结果可应用于多项分布最大、最小参数的贝叶斯推断,在佛罗里达州沃尔顿县白人和黑人的职业状态调查结果中,求出最大、最小参数的后验分布函数以及95%贝叶斯区间估计,模拟结果表明提供的方法具有较好的表现.     

The Maximum Surplus Distribution before Ruin in an Erlang（n） Risk Process Perturbed by Diffusion

In this paper, we consider the distribution of the maximum surplus before ruin in a generalized Erlang(n) risk process (i.e., convolution of n exponential distributions with possibly different parameters) perturbed by diffusion. It is shown that the maximum surplus distribution before ruin satisfies the integro-differential equation with certain boundary conditions. Explicit expressions are obtained when claims amounts are rationally distributed. Finally, the surplus distribution at the time of ruin and the surplus distribution immediately before ruin are presented.     

Independent scale-free spacings for the exponential and uniform distributions

It is shown that, by suitable random normalisation, the spacings (differences between order statistics) of exponential and uniform samples are independent. These normalised spacings can be used for consecutive discordancy testing when scale (and possibly also location) parameters are unknown.     

Improved estimation under Pitman's measure of closeness

Stein-type and Brown-type estimators are constructed for general families of distributions which improve in the sense of Pitman closeness on the closest (in a class) estimator of a parameter. The results concern mainly scale parameters but a brief discussion on improved estimation of location parameters is also included. The loss is a general continuous and strictly bowl shaped function, and the improved estimators presented do not depend on it, i.e., uniform domination is established with respect to the loss. The normal and inverse Gaussian distributions are used as illustrative examples. This work unifies and extends previous relevant results available in the literature.     

The convolution and multiplication of one‐dimensional associated homogeneous distributions

The set of associated homogeneous distributions (AHDs) with support in R is an important subset of the tempered distributions because it contains the majority of the (one‐dimensional) distributions typically encountered in physics applications (including the δ distribution). In a previous work of the author, a convolution and multiplication product for AHDs on R was defined and fully investigated. The aim of this paper is to give an easy introduction to these new distributional products. The constructed algebras are internal to Schwartz’ theory of distributions and, when one restricts to AHDs, provide a simple alternative for any of the larger generalized function algebras, currently used in non‐linear models. Our approach belongs to the same class as certain methods of renormalization, used in quantum field theory, and are known in the distributional literature as multi‐valued methods. Products of AHDs on R, based on this definition, are generally multi‐valued only at critical degrees of homogeneity. Unlike other definitions proposed in this class, the multi‐valuedness of our products is canonical in the sense that it involves at most one arbitrary constant. A selection of results of (one‐dimensional) distributional convolution and multiplication products are given, with some of them justifying certain distributional products used in quantum field theory. Copyright © 2012 John Wiley & Sons, Ltd.     

同单调相依结构下两重生命模型的概率分布

在寿险实务中,在处理涉及到多个生命的问题时往往假设各个生命之间是独立的,但事实上,因为受某些相同因素影响的生命之间总是存在一定的正相依性．本文证明了在给定边际分布的二维随机向量中,同单调相依结构是在相关序意义下最强的正相依结构,研究了在此相依结构下的两重生命模型的概率分布,并给出了随机序意义下两个状态消亡时间的随机上界和随机下界．     

线性核Toader平均的Schur凸性和Schur几何凸性

为了研究线性核Toader平均Mr(a,b)在R_(++)²上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论:当r≥1时,M_r(a,b)在R_(++)2上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论:当r≥1时,M_r(a,b)在R_(++)²上是Schur凸函数;当r≤1时,Mr(a,b)在R_(++)2上是Schur凸函数;当r≤1时,Mr(a,b)在R_(++)²上是Schur凹函数;当r≥1/2时,M_r(a,b)在R_(++)2上是Schur凹函数;当r≥1/2时,M_r(a,b)在R_(++)²上是Schur几何凸函数.最后,依据M_r(a,b)的Schur凸性和Schur几何凸性建立了新的不等式.     

Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small

We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(−n ^d ), 0<d<1, wheren is the sample size.