Local dependence functions for some families of bivariate distributions and total positivity

The purpose of this paper is to investigate a very useful application of a certain local dependence function γ_f(x,y), which was considered recently by Holland and Wang [20]. An interesting property of γ_f(x,y) is that the underlying joint density f(x,y) is TP₂ (that is, totally positive of order 2) if and only if . This gives an elegant way to investigate the TP₂ property of any bivariate distribution. For the Saramanov family, the Ali-Mikhail-Haq family of bivariate distributions and the family of bivariate elliptical distributions, we derive the local dependence function and obtain conditions for f(x,y) to be TP₂. These families are quite rich and include many other large classes of bivariate distributions as their special cases. Similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with Pareto conditionals.     

A framework for positive dependence

This paper presents, for bivariate distributions, a unified framework for studying and relating three basic concepts of positive dependence. These three concepts are positive dependence orderings, positive dependence properties and measures of positive dependence. The latter two concepts are formally defined and their properties discussed. Interrelationships among these three concepts are given, and numerous examples are presented.Supported by the National Science Foundation under Grant DMS-8301361.Supported by the Air Force Office of Scientific Research under Contract 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A concept of negative dependence using stochastic ordering

A concept of negative dependence called negative dependence by stochastic ordering is introduced. This concept satisfies various closure properties. It is shown that three models for negetive dependence satisfy it and that it implies the basic negative orthant inequalities. This concept is also satisfied by the multinomial, multivariate hypergeometric. Dirichlet and Dirichlet compound multinomial distributions. Furthermore, the joint distribution of ranks of a sample and the multivariate normal with nonpositive pairwise correlations also satisfy this condition. The positive dependence analog of this condition is also studied.     

Some concepts of positive dependence for bivariate interchangeable distributions

In this work we consider some familiar and some new concepts of positive dependence for interchangeable bivariate distributions. By characterizing distributions which are positively dependent according to some of these concepts, we indicate real situations in which these concepts arise naturally. For the various families of positively dependent distributions we prove some closure properties and demonstrate all the possible logical relations. Some inequalities are shown and applied to determine whether under- (or over-) estimates, of various probabilistic quantities, occur when a positively dependent distribution is assumed (falsely) to be the product of its marginals (that is, when two positively dependent random variables are assumed, falsely, to be independent). Specific applications in reliability theory, statistical mechanics and reversible Markov processes are discussed. This work was partially supported by National Science Foundation GP-30707X1. It is part of the author's Ph.D. dissertation prepared at the University of Rochester and supervised by A. W. Marshall. Now at Indiana University.     

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.     

On banach spaces whose duals areL 1 spaces

A structure theorem for Banach spaces whose duals areL ₁ spaces, is proved. The research of the second named author has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Stochastic Comparisons and Dependence among Concomitants of Order Statistics

Let (X_i, Y_i) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X_(r) denotes the rth ordered X-variate then the Y-variate, Y_[r], paired with X_(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP₂ dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering.     

Locking effects in the finite element approximation of elasticity problems

Summary We consider the finite element approximation of the 2D elasticity problem when the Poisson ratiov is close to 0.5. It is well-known that the performance of certain commonly used finite elements deteriorates asv0, a phenomenon calledlocking. We analyze this phenomenon and characterize the strength of the locking androbustness of varioush-version schemes using triangular and rectangular elements. We prove that thep-andh-p versions are free of locking with respect to the error in the energy norm. A generalization of our theory to the 3D problem is also discussed.The work of this author was supported in part by the Office of Naval Research under Naval Research Grant N00014-90-J-1030The work of this author was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, U.S. Air Force, under grant AFOSR 89-0252     

Marshall and Olkin’s Distributions

A review is provided of the continuous and discrete distributions introduced by the eminent Professors Marshall and Olkin. The topics reviewed include: bivariate geometric distribution, extreme value behavior, bivariate negative binomial distribution, bivariate exponential distribution, concomitants, reliability, distributions of sums and ratios, Ryu’s bivariate exponential distribution, bivariate Pareto distribution and generalized exponential and Weibull distributions. Some hitherto unknown results about these distributions are also mentioned. This is a tribute to the work of Professors Marshall and Olkin.     

Tolerance intervals for quantiles of bivariate risks and risk measurement

This paper considers joint distributions of order statistics for risk variables and their concomitants for actuarial risk analysis under dependence. With this purpose, bivariate integral transformations are performed and some examples are presented using copulas, the FGM copulas in particular. Quantiles of the distributions concerned are discussed and their tolerance intervals are constructed. Risk measures such as VaR in the set up of the tolerance intervals are included in the discussions.     

A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Algorithms for a class of nondifferentiable problems

A nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities are due to terms of the form min(f ₁(x),...,f _n(x)), which may enter nonlinearly in the cost and the constraints. Necessary and sufficient conditions are developed. Two algorithms for solving this problem are described, and their convergence is studied. A duality framework for interpretation of the algorithms is also developed.This work was supported in part by the National Science Foundation under Grant No. ENG-74-19332 and Grant No. ECS-79-19396, in part by the U.S. Air Force under Grant AFOSR-78-3633, and in part by the Joint Services Electronics Program (U.S. Army, U.S. Navy, and U.S. Air Force) under Contract N00014-79-C-0424.     

Tail dependence for elliptically contoured distributions

The relationship between the theory of elliptically contoured distributions and the concept of tail dependence is investigated. We show that bivariate elliptical distributions possess the so-called tail dependence property if the tail of their generating random variable is regularly varying, and we give a necessary condition for tail dependence which is somewhat weaker than regular variation of the latter tail. In addition, we discuss the tail dependence property for some well-known examples of elliptical distributions, such as the multivariate normal, t, logistic, and Bessel distributions.     

Reliability Studies of Bivariate Distributions with Pareto Conditionals

In this paper we study Arnold's (1987, Statist. Probab. Lett.5, 263–266) class of bivariate distributions with Pareto conditionals from a reliability point of view. Failure rates and mean residual life function of the marginal distributions and their monotonic properties are studied. The hazard components and their properties are investigated and their relationships with some measures of dependence are established. Finally, the failure rate of the minimum of the two components is examined and its monotonicity is investigated. Some of the results presented here are general and would be useful in studying the dependence structure in other classes of bivariate distributions.     

Preservation of certain dependent structures under bivariate homogeneous Poisson shock models

A bivariate Poisson shock model resulting from two devices receiving shocks from two independent sources is shown to preserve certain bivariate dependent structures such as total positivity of order 2 (TP₂), stochastic increasing (SI), right tail increasing (RTI) etc. However, when two devices are subjected to the same source of shocks it is observed through a counter example that some of these preservation results do not hold any more. In such cases sufficient conditions are given under which the bivariate random vector denoting the life lengths of two devices is shown to have the above-mentioned bivariate dependent structures.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.     

Limiting properties of the occurrence/exposure rate and simple risk rate

In this paper, we study the asymptotic distributions of the functions of the occurrence/exposure rates of several groups of patients as well as Berry-Esseen bound on the distribution function of the occurrence/exposure rate. Asymptotic distributions of functions of the simple risk rates are also derived. The results are useful in not only medical research but also in the area of reliability.The work of the first two authors was supported by Contract F49620-C5-C-0008 of the Air Force Office of Scientific Research. The United States Government is authorized to reproduce and distribute reprints for governmental purposes.     

A unified view of interior point methods for linear programming

The paper shows how various interior point methods for linear programming may all be derived from logarithmic barrier methods. These methods include primal and dual projective methods, affine methods, and methods based on the method of centers. In particular, the paper demonstrates that Karmarkar's algorithm is equivalent to a classical logarithmic barrier method applied to a problem in standard form.Invited paper presented at the Workshop on Supercomputers in Optimization, Minneapolis, Minn., May 1988.The work of this author was supported by the Air Force Office of Scientific Research, Air Force Systems Command, USA, under Grants AFOSR-87-0215 and AFOSR-85-0271. The US Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation thereon.     

On extreme points inl 1

It is proved that every bounded closed and convex subset ofl ₁ is the closed convex hull of its extreme points. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Computational experience with a globally convergent primal—dual predictor—corrector algorithm for linear programming

Kojima, Megiddo, and Mizuno proved global convergence of a primal—dual algorithm that corresponds to methods used in practice. Here, the numerical efficiency of a predictor—corrector extension of that algorithm is tested. Numerical results are extremely positive, indicating that the safety of a globally convergent algorithm can be obtained at little computational cost. The algorithm is tested on infeasible problems with less success. Finally, the algorithm is applied to a warm started problem, with very encouraging preliminary results.Corresponding author. The research of this author is sponsored by the Air Force Office of Scientific Research, Air Force System Command under Grant AFOSR-92-J0046. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notations thereon.