Ordering of risks: a review

In this review we will consider and discuss the most important partial orderings of riks, namely consistent partial ordering and net-stop-loss ordering. More especially we will study the consequencies of ordering of risks for the compound risk: S = X₁+X₂+···+X_N.The impact of orderings of claim size distributions (F_X) and claim intensities (F_N) on orderings of claim amounts (F_S) is examined. The consequencies of these kind of orderings on orderings of risks by means of premium calculation principles is also discussed.In this framework the influence of the dangerousness of distributions on orderings of risks is given.In analogy with the notion of stochastic dominance appearing in the theory of finance, the notion of stop-loss dominance is introduced.     

Partial orderings of matrices referring to singular values or eigenvalues

A new partial ordering in the set of complex matrices is defined, which is weaker than the star ordering introduced by Drazin in 1978 and stronger than the minus ordering introduced by Hartwig in 1980. This ordering refers to singular values of matrices, and the interest in it was generated by canonical interpretations of the minus and star orderings, given by Hartwig and Styan in 1986. For Hermitian matrices, a similar ordering referring to eigenvalues is also considered, and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out.     

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.     

Characterization of lifetime distributions in the Ls-sense by a generalized spread

We characterize partial orderings between lifetime distributions by L_s-tailweight. We obtain some ageing properties by comparing distributions with the negative exponential distribution with unit mean.     

Weighted-mean regions of a probability distribution

In this paper we investigate a new class of central regions for probability distributions on R^d, called weighted-mean regions. Their restrictions to an empirical distribution are the weighted-mean trimmed regions investigated by Dyckerhoff and Mosler (2011) for d-variate data. Furthermore a new class of stochastic orderings of variability, the weighted-mean orderings, is introduced.     

Proximity between life distributions and exponential distributions (I)

Many classes of life distributions have been introduced into reliability theory. Because of the importance of exponential distributions in reliability theory, it is interesting to study the difference between life distributions and exponential distributions. In this paper, we study the proximity between the life distribution in various classes and the exponential distribution. We shall give some simple upper bounds.This research was partially supported by the National Natural Science Foundation of China.     

Stochastic orderings, folded beta distributions and fairness in coin flips

This paper studies stochastic orderings for folded beta distributions. A necessary and sufficient condition for one folded beta random variable to be larger than another in likelihood ratio order is obtained. The work is motivated by the recent results of Porzio and Ragozini on stochastic orderings for folded binomial distributions. Beta distributions are commonly used to model proportions or probabilities, particularly as conjugate distributions for binomial random variables. Stochastic orderings may arise, for instance, in considerations of fairness in coin flips.     

An application of Kendall distributions and alternative dependence measures: SPX vs. VIX

Most of the recently-defined notions of positive or negative dependence rely upon a variety of orderings of bivariate random vectors. These orderings are generally partial orders, and thus there are many pairs of random vectors which are not comparable. By using a weakened version of stochastic domination and the concepts of Kendall distributions and metacopulas, an entirely new class of orderings, in which the comparability issue is resolved, has been recently created. Each ordering in this class can be used to construct a measure of dependence. A detailed example will be given, using data from the Standard & Poor’s 500 index and Chicago Board of Trades index for implied volatility.     

Some partial orderings of exchangeable random variables by positive dependence

Some partial orderings of positively dependent exchangeable random variables are introduced. The interrelations among them, the inequalities which follow from them and two models which yield such partial orderings are then discussed. Particular examples include ordering multivariate normal, t, χ², Cauchy, exponential, binomial, Poisson, gamma and Farlie-Gumbel-Morgenstern random vectors. Applications to genetic selection and choice of sampling procedures are given.     

Stochastic comparisons of multivariate mixtures

In this paper we establish multivariate hazard rate, multivariate reverse hazard rate, and multivariate likelihood ratio stochastic orderings among multivariate random mapping (mixture) distributions. The new results streamline and simplify the proofs of some partial results that have recently appeared in the literature. Some applications in reliability theory and risk management are described.     

Multivariate new better than used in expectation distributions

Various definitions of multivariate new better than used in expectation (MNBUE) life distributions are introduced and their interrelationships are studied. The definitions are multivariate generalizations of the very useful univariate aging property new better than used in expectations (NBUE), which is weaker than new better than used (NBU) and decreasing mean residual life (DMRL) properties. Various closure properties of the different MNBUE classes are proved. Some examples are given to illustrate the relationship between the proposed MNBUE properties and MNBUE properties presented by Buchanan and Singpurwalla (1977). Examples of well known life distributions, that give rise to the MNBUE distributions are discussed. We also study the dual multivariate new worse than used in expectation (MNWUE) distributions.     

A family of kurtosis orderings for multivariate distributions

In this paper, a family of kurtosis orderings for multivariate distributions is proposed and studied. Each ordering characterizes in an affine invariant sense the movement of probability mass from the “shoulders” of a distribution to either the center or the tails or both. All even moments of the Mahalanobis distance of a random vector from its mean (if exists) preserve a subfamily of the orderings. For elliptically symmetric distributions, each ordering determines the distributions up to affine equivalence. As applications, the orderings are used to study elliptically symmetric distributions. Ordering results are established for three important families of elliptically symmetric distributions: Kotz type distributions, Pearson Type VII distributions, and Pearson Type II distributions.     

SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

ＳＯＭＥＭＵＬＴＩＶＡＲＩＡＴＥＤＭＲＬＡＮＤＮＢＵＥＤＥＦＩＮＩＴＩＯＮＳＢＡＳＥＤＯＮＣＯＮＤＩＴＩＯＮＡＬＳＴＯＣＨＡＳＴＩＣＯＲＤＥＲＷＡＮＧＹＵＥＤＯＮＧ（王跃东）ＣＡＯＪＩＮＨＵＡ（曹晋华）（ＤｅｐａｒｔｍｅｎｔｏｆＳｔａｔｉｓｔｉｃｓ，...     

Comparison of Inhomogeneous Poisson Processes

This note illustrates the use of various partial orderings for point processes by comparing inhomogeneous Poisson processes, and studies the relationships among these orderings mainly in the ease of inhomogeneous Poisson processes. Whitt has detailed several definitions of partial orderings which have been applied mainly to renewal processes (see, e. g., Whitt [14] and Miller[4]) and semi-Markov proeesse (see, e. g., Sonderman [10]).     

Relationships between pure tail orderings of lifetime distributions and some concepts of residual life

The concepts of pure-tail orderings as defined by sq- and D-orderings are shown to order the family of reliability life distributions which age smoothly in a natural way. This ordering extends to comparisons regarding the limiting behavior of the residual life, mean residual life, sojourn time between perfect repairs in repairable systems, failure rate and, through the preservation of sq- and D-orderings by various reliability operations, to certain coherent systems of components that age smoothly. Possible applications of the results to the industrial practice of cannibalization are also noted.     

THE NBELC AND NWELC CLASSES OF LIFE DISTRIBUTIONS

ＴＨＥＮＢＥＬＣＡＮＤＮＷＥＬＣＣＬＡＳＳＥＳＯＦＬＩＦＥＤＩＳＴＲＩＢＵＴＩＯＮＳ￥ＣＡＯＪＩＮＨＵＡ；ＷＡＮＧＹＵＥＤＯＮＧ（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅ，Ｂｅｉｊｉｎ...     

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．     

Symmetrische Zerlegung von Kettenprodukten

A general theorem on the existence of partial matchings induced by linear orderings is proved which in the case of the natural orderings on the levels of a chainproduct yields full matchings and the symmetric decompositions of de Bruijn. In addition an explicit formula for these matchings is provided.     

Partial Orderings in the Category of Matrices over the Quaternion Field

ＰａｒｔｉａｌＯｒｄｅｒｉｎｇｓｉｎｔｈｅＣａｔｅｇｏｒｙｏｆＭａｔｒｉｃｅｓｏｖｅｒｔｈｅＱｕａｔｅｒｎｉｏｎＦｉｅｌｄＺｈｕａｎｇＷａｊｉｎ（庄瓦金）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈａｎｇＮｏｒｍａｌＣｏｌｌｅｇｅ，Ｆｕｊｉ...     

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.