SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

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Some Classes of Life Distributions Characterized by Concave Ordering

1.IntroductionandDefintionsInreliabilitytheory,maintenancetheory,biometricsandeconometrics,variousclassesoflifedistributionsbasedondifferentconceptsofagingareveryuseful.Recently,anumberofclassesoflifedistributionshavebeenintroducedbymeansofdifferentapproachestocharacterizingagingproperty.Oneoftheseapproachesisbasedonthefactthatmanyclassesoflifedistributionmaybecharacterizedbyvariousstochasticorderings,see,e.g.[if.[2],[3]and[4],etc.LetFbethedistributionfunctionofanon-negativerandomvariableXre…     

Risk aggregation with dependence uncertainty

Risk aggregation with dependence uncertainty refers to the sum of individual risks with known marginal distributions and unspecified dependence structure. We introduce the admissible risk class to study risk aggregation with dependence uncertainty. The admissible risk class has some nice properties such as robustness, convexity, permutation invariance and affine invariance. We then derive a new convex ordering lower bound over this class and give a sufficient condition for this lower bound to be sharp in the case of identical marginal distributions. The results are used to identify extreme scenarios and calculate bounds on Value-at-Risk as well as on convex and coherent risk measures and other quantities of interest in finance and insurance. Numerical illustrations are provided for different settings and commonly-used distributions of risks.     

Fuzzy Optimization Problems Based on Ordering Cones

The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients.     

Existence theory for generalized nonlinear complementarity problems

The nonlinear complementarity problem is generalized by replacing the usual nonnegative ordering ofR ⁿ by an ordering generated by a convex cone. Two new classes of operators are introduced, each of which is used to guarantee the existence of a solution to the generalized problem. The new classes can be seen to be broader than previously studied classes. In addition, conditions are presented under which the solution set of the generalized linear complementarity problem is shown to have at most a finite number of solutions.This research was partially supported by National Science Foundation, Grant No. GP-16293, and constitutes part of the junior author's doctoral thesis. The authors are indebted to Dr. Carlton E. Lemke for many helpful discussions.     

Directionally Convex Comparison of Correlated First Passage Times

Many important classes of multivariate distributions arising from reliability modeling are the distributions of correlated first passage times of certain multivariate point processes. In this paper, we obtain results that compare variability and dependence structure of these correlated first passage times, in the sense of directionally convex ordering. Under certain conditions, we also obtain some easily computable distributional bounds for the first passage times whose joint distributions can not be expressed explicitly.     

Bondesson's functions in reliability theory

Bondesson's functions in reliability theory are shown to be related to a recursive sequence of probability distributions. These are the ‘higher-order’ versions of the mean remaining lifetime in an equilibrium renewal process. Based on these functions, classes of distribution functions can be defined. This paper will investigate these classes and place Bondesson's work in the content of the other work done in reliability theory. Connections are made with the decreasing variance residual lifetime class and stochastic ordering. Copyright © 1999 John Wiley & Sons, Ltd.     

The extreme points of the set of decreasing failure rate distributions

Summary Since the class of extended decreasing failure rate (EDFR) life distributions (i.e., distributions with support in [0, ]) is compact and convex, it follows from Choquet's Theorem that every EDFR life distribution can be represented as a mixture of extreme points of the EDFR class. We identify the extreme points of this class and of the standard class of decresing failure rate (DFR) life distributions. Further, we show that even though the convex class of DFR life distributions is not compact, every DFR life distribution can be represented as a mixture of extreme points of the DFR class.Research sponsored by the Air Force Office of Scientific Research, AFSC, USAF, under Grant AFOSR 78-3678.Research sponsored by the National Science Foundation MCS-7904698.     

多项分布总体的相等对增凸序问题的多样本检验

概率分布间随机序在实践中已经得到了广泛的应用,而且似然比检验是用以检验涉及随机序问题的最普遍的检验方法．但是,关于多个多项式总体间的增凸序约束的统计推断问题并没有得到充分发展．多样本的增凸序对无约束的检验问题已被研究．然而,多总体的相等性对增凸序的假设检验问题似乎更有研究意义．并且分布的相等对随机序的假设检验问题往往是统计学家最为普遍地考虑．对多样本的情况,本文考虑了分布的相等对增凸序的假设检验问题,并且获得似然比检验统计量的零渐近分布,它是一组x~2分布随机变量的加权和,即■~2分布．     

COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.     

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.     

How Do Cultural Classes Emerge from Assimilation and Distinction? An Extension of the Cucker-Smale Flocking Model

When cultural tastes are not neutral but hierarchically matched to social status, people assimilate themselves to higher status by consuming cultural goods while distinguishing themselves from lower status by developing new tastes. Extending the Cucker-Smale model for mutual influence among agents, we examine when and how many cultural classes emerge from continuous distributions of tastes and what conditions those classes satisfy, through the assimilation-distinction mechanism. We simulate the models with different initial distributions of tastes (uniform, normal, and chi-square), given various ranges of 2 parameters: (a) the strength and (b) the range of distinction relative to assimilation. Tastes are flocking and cultural classes emerge when the range of assimilation is much larger than that of distinction. The number of classes increases with the strength of distinction, whereas the distance between classes equals the range of distinction. Some properties of emergent classes are mathematically proved. First, in a two-class system, the stronger distinction, the larger the upper class. Second, in a three-class system, the middle class is necessarily larger than the lower class and likely larger than the upper class. Third, a 3-class system cannot emerge if distinction is weaker than assimilation. These properties are universal and do not depend on the initial distribution of cultural tastes. This independence predicts homogeneous cultural classes emerging across different social conditions. Also, the cultural middle class as the largest group may explain why subjective class consciousness is often higher than objective position. Unless assimilating efforts can reach an infinite range, there emerges a cultural outcast at the lowest end of the cultural hierarchy.     

Application of the phase‐type mortality law to life contingencies and risk management

The class of phase‐type distributions has recently gained much popularity in insurance applications due to its mathematical tractability and denseness in the class of distributions defined on positive real line. In this paper, we show how to use the phase‐type mortality law as an efficient risk management tool for various life insurance applications. In particular, pure premiums, benefit reserves, and risk‐loaded premiums using CTE for standard life insurance products are shown to be available in analytic forms, leading to efficient computation and straightforward implementation. A way to explicitly determine provisions for adverse deviation for interest rate and mortality is also proposed. Furthermore, we show how the interest rate risk embedded in life insurance portfolios can be analyzed via interest rate sensitivity index and diversification index which are constructed based on the decomposition of portfolio variance. We also consider the applicability of phase‐type mortality law under a few non‐flat term structures of interest rate. Lastly, we explore how other properties of phase‐type distributions may be applied to joint‐life products as well as subgroup risk ordering and pricing within a given pool of insureds. Copyright © 2017 John Wiley & Sons, Ltd.     

Positive dependence orderings

Summary This paper presents a systematic basis for studying orderings of bivariate distributions according to their degree of positive dependence. The general concept of a positive dependence ordering (PDO) is introduced and its properties discussed. Based on this concept, a new ordering of bivariate distributions according to their degree of total positivity of order two (TP₂) is presented, and is shown to be a PDO. Properties of this TP₂ ordering are derived and numerous applications are presented. The work of this author is supported by the National Science Foundation under Grant MCS-8301361. The work of this author is sponsored by the Air Force Office of Scientific Research, Air Force Systems Command under Contract F49629-82-K-001. Reproduction in whole of in part is permitted for any purpose of the United States Government.     

Free Triangle Orders

We define a new class of ordered sets, called free triangle orders. These are ordered sets represented by a left-to-right ordering on geometric objects contained in a horizontal strip in the plane. The objects are called 'free triangles', and have one vertex on each of the two boundaries of the strip and one vertex in its interior. These ordered sets generalize the classes of trapezoid and triangle orders studied by Bogart, Möhring, and Ryan, represented by trapezoids and triangles respectively, contained within a strip in the plane, and are a special case of the orders of tube dimension 2 introduced by Habib and co-workers, which are represented by any set of convex bodies contained within a strip in the plane. Our main result is that the class of free triangle orders strictly contains the class of trapezoid orders.     

Generalized Vector Equilibrium Problems in Generalized Convex Spaces

In this paper, we introduce and study a class of abstract generalized vector equilibrium problems (AGVEP) in generalized convex spaces which includes most vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems, and generalized vector variational inequality problems as special cases. By using the generalized GKKM and generalized SKKM type theorems due to the first author, some new existence results of equilibrium points for the AGVEP are established in noncompact generalized convex spaces. As consequences, some recent results in the literature are obtained under much weaker assumptions.     

SOME NEW PARTIAL ORDERINGSDESCRIBING AGING PROPERTY

The concept of aging, which is very users in relability describes the adverse orbeneficial effects of age on the lifetime of systems. Several partial orderings of life distributionshave been introduced. In this article,some more partial orderings of life distributions are imtroduced mainly by means of the equilibrium distribution. And conversely, a group of new aging clasps oflife distrbutions can be derived from these orderings.     

Estimation of two ordered bivariate mean residual life functions

Situations occur frequently in which the mean residual life (mrl) functions of two populations must be ordered. For example, if a mechanical device is improved, the mrl function for the improved device should not be less than that of the original device. Also, mrl functions for medical patients should often be ordered depending on the status of concomitant variables. This paper proposes nonparametric estimators of the bivariate mrl function under a mrl ordering. The estimators are shown to be asymptotically unbiased, strongly uniformly consistent and weakly convergent to a bivariate Gaussian process. The estimators are shown to be the projections, in a sense to be made precise, of the empirical mrl function onto an appropriate convex set of mrl functions. In the one-sample problem, the new estimators dominate the empirical mrl function in terms of risk with respect to a wide class of loss functions.     

On HNBUE class after specific age

In this article we introduce new classes of life distributions namely harmonic new better (worse) than used in expectation after specific age t₀ HNBUE¹t₀ (HNWUE¹t₀). The closure properties under various reliability operations such as convolution, mixture, mixing and the homogeneous Poisson shock model of these classes are studied. Furthermore, nonparametric test is proposed to test exponentiality vs. the HNBUE¹t₀ class. The critical values and the powers of this test are calculated to assess the performance of the test. It is shown that the proposed test have high efficiencies for LFR and Weibull distributions. Sets of real data are used as examples to elucidate the use of the proposed test for practical problems.     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets.