Relationships Among Some Concepts of Bivariate Dependence
01 Apr 1972-Annals of Mathematical Statistics (ANNALS OF MATHEMATICAL STATISTICS)-Vol. 43, Iss: 2, pp 651-655
TL;DR: In this paper, the authors consider some unresolved relationships among various notions of bivariate dependence and show that for any non-decreasing $f$ and $g$ the associated notions of dependence are associated.
Abstract: We consider some unresolved relationships among various notions of bivariate dependence. In particular we show that $P\lbrack T > t \mid S > s\rbrack \uparrow$ in $s$ (or alternately, $P\lbrack T \leqq t \mid S \leqq s\rbrack \downarrow$ in $s$) implies $S, T$ are associated, i.e. $\operatorname{Cov} \lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all non-decreasing $f$ and $g$.
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TL;DR: In this article, a function f(x) defined on X = X 1 × X 2 × × × X n where each X i is totally ordered satisfying f (x ∨ y) f(xi ∧ y) ≥ f(y) f (y), where the lattice operations ∨ and ∧ refer to the usual ordering on X, is said to be multivariate totally positive of order 2 (MTP2).
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04 Nov 2005
TL;DR: In this article, the authors provide an essential guide to managing modern financial risk by combining coverage of stochastic order and risk measure theories with the basics of risk management, including dependence concepts and dependence orderings.
Abstract: The increasing complexity of insurance and reinsurance products has seen a growing interest amongst actuaries in the modelling of dependent risks. For efficient risk management, actuaries need to be able to answer fundamental questions such as: Is the correlation structure dangerous? And, if yes, to what extent? Therefore tools to quantify, compare, and model the strength of dependence between different risks are vital. Combining coverage of stochastic order and risk measure theories with the basics of risk management and stochastic dependence, this book provides an essential guide to managing modern financial risk. * Describes how to model risks in incomplete markets, emphasising insurance risks. * Explains how to measure and compare the danger of risks, model their interactions, and measure the strength of their association. * Examines the type of dependence induced by GLM-based credibility models, the bounds on functions of dependent risks, and probabilistic distances between actuarial models. * Detailed presentation of risk measures, stochastic orderings, copula models, dependence concepts and dependence orderings. * Includes numerous exercises allowing a cementing of the concepts by all levels of readers. * Solutions to tasks as well as further examples and exercises can be found on a supporting website.
590 citations
TL;DR: The authors show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case.
Abstract: The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.
427 citations
Cites background from "Relationships Among Some Concepts o..."
...Most common are the following notions, introduced by Lehmann (1966), as well as Esary and Proschan (1972)....
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TL;DR: A new and general numerical method for calculating the appropriate convolutions of a wide range of probability distributions using lower and upper discrete approximations to the quantile function (the quasi-inverse of the distribution function) and has advantages over other methods previously proposed.
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TL;DR: In this paper, various notions of multivariate negative dependence are introduced and their interrelationship is studied, and examples are given to illustrate these concepts in statistics and probability are given.
Abstract: Various notions of multivariate negative dependence are introduced and their interrelationship is studied. Examples are given to illustrate these concepts. Applications of the results in statistics and probability are given.
219 citations
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TL;DR: In this article, the authors give three successively stronger definitions of positive dependence, and investigate their consequences, explore the strength of each definition through a number of examples, and give some statistical applications.
Abstract: Problems involving dependent pairs of variables (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 × 2 tables. This is due primarily to the importance of these cases but perhaps partly also to the fact that they exhibit only a particularly simple form of dependence. (See Examples 9(i) and 10 in Section 7.) Studies involving the general case center mainly around two problems: (i) tests of independence; (ii) definition and estimation of measures of association. In most treatments of these problems, there occurs implicitly a concept which is of importance also in other contexts (for example, the evaluation of the performance of certain multiple decision procedures), the concept of positive (or negative) dependence or association. Tests of independence, for example those based on rank correlation, Kendall’s Z-statistic, or normal scores, are usually not omnibus tests (for a discussion of such tests see [4], [15] and [17], but designed to detect rather specific types of alternatives, namely those for which large values of Y tend to be associated with large values of X and small values of Y with small values of X (positive dependence) or the opposite case of negative dependence in which large values of one variable tend to be associated with small values of the other. Similarly, measures of association are typically designed to measure the degree of this kind of association. The purpose of the present paper is to give three successively stronger definitions of positive dependence, to investigate their consequences, explore the strength of each definition through a number of examples, and to give some statistical applications.
1,682 citations
TL;DR: In this paper, it was shown that a random variable can be associated with another random variable if the test functions are either (a) binary or (b) bounded and continuous.
Abstract: It is customary to consider that two random variables $S$ and $T$ are associated if $\operatorname{Cov}\lbrack S, T\rbrack = EST - ES\cdot ET$ is nonnegative. If $\operatorname{Cov}\lbrack f(S), g(T)\rbrack \geqq 0$ for all pairs of nondecreasing functions $f, g$, then $S$ and $T$ may be considered more strongly associated. Finally, if $\operatorname{Cov}\lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all pairs of functions $f, g$ which are nondecreasing in each argument, then $S$ and $T$ may be considered still more strongly associated. The strongest of these three criteria has a natural multivariate generalization which serves as a useful definition of association: DEFINITION 1.1. We say random variables $T_1,\cdots, T_n$ are associated if \begin{equation*}\tag{1.1}\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack \geqq 0\end{equation*} for all nondecreasing functions $f$ and $g$ for which $Ef(\mathbf{T}), Eg(\mathbf{T}), Ef(\mathbf{T})g(\mathbf{T})$ exist. (Throughout, we use $\mathbf{T}$ for $(T_1,\cdots, T_n)$; also, without further explicit mention we consider only test functions $f, g$ for which $\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack$ exists.) In Section 2 we develop the fundamental properties of association: Association of random variables is preserved under (a) taking subsets, (b) forming unions of independent sets, (c) forming sets of nondecreasing functions, (d) taking limits in distribution. In Section 3 we develop some simpler criteria for association. We show that to establish association it suffices to take in (1.1) nondecreasing test functions $f$ and $g$ which are either (a) binary or (b) bounded and continuous. In Section 4 we develop the special properties of association that hold in the case of binary random variables, i.e., random variables that take only the values 0 or 1. These properties turn out to be quite useful in applications. We also discuss association in the bivariate case. We relate our concept of association in this case to several discussed by Lehmann (1966). Finally, in Section 5 applications in probability and statistics are presented yielding results by Robbins (1954), Marshall-Olkin (1966), and Kimball (1951). An application in reliability which motivated our original interest in association will be presented in a forthcoming paper.
1,246 citations
TL;DR: In this article, the minimal cut lower bound on the reliability of a coherent system, derived in Esary-Proschan [6] for the case of independent components not subject to maintenance, is shown to hold under a variety of component maintenance policies and in several typical cases of component dependence.
Abstract: In this article the minimal cut lower bound on the reliability of a coherent system, derived in Esary-Proschan [6] for the case of independent components not subject to maintenance, is shown to hold under a variety of component maintenance policies and in several typical cases of component dependence. As an example, the lower bound is obtained for the reliability of a “two out of three” system in which each component has an exponential life length and an exponential repair time. The lower bound is compared numerically with the exact system reliability; for realistic combinations of failure rate, repair rate, and mission time, the discrepancy is quite small.
97 citations
TL;DR: In this paper, a multivariate IHR distribution with a distribution function F(xI, x2,, xn) = P[X1 xl,..* X, > xn j X1 > xI', * * X,, > Xn'] is shown to be nondecreasing in x1', **, Xn' for every choice of xI, * * x n.
Abstract: 2. Multivariate IHR Distributions. Consider the randoih vector (X,, X2, , Xj) with distribution function F(xI, x2, , xn) = P[X1 xl, ..* X, > xn j X1 > xI', * * X,, > xn'] is nondecreasing in x1', * * , xn' for every choice of xI, * * x n. This generalises some notions of dependence that were studied by Lehmann [4] and Esary and Proschan [3]. Setting F(x,, X2, . * * Xn) = P[XI > XI, X2 > X2, , Xn, > Xn] we have:
85 citations
TL;DR: For a class of distributions with suitable monotony properties (in particular all distributions for which $f'(y)/f(y)$ is monotone decreasing, and all normal, exponential, gamma and beta distributions), this paper showed that the covariances of the order statistics in a sample of any chosen size are monotonically decreasing in either index separately.
Abstract: The distinction between efficiency in the asymptotic sense originally introduced by Fisher ([2], 1925, p. 703), and the finite sample sense sometimes used by others has been recently stressed by various writers (e.g., Berkson [1]). The technique of proof used below was originally developed to provide a simple example where the maximum likelihood estimate of location, though asymptotically efficient, was not of minimum variance for any finite sample size whatever. The (symmetrical) double exponental distribution with known scale, where the sample median is the maximum likelihood estimator of location, could easily be shown to be such an example. (While this result is useful in deflating unwarranted views about minimum variance properties of maximum likelihood estimates, Fisher's ([2], p. 716) results about intrinsic accuracy in the same situation are of more basic interest.) On examination, however, the technique used to provide this rather isolated and special result was found capable of showing, for a class of distributions with suitable monotony properties (in particular all distributions for which $f'(y)/f(y)$ is monotone decreasing, and all normal, exponential, gamma and beta distributions), that the covariances of the order statistics in a sample of any chosen size are monotone in either index separately.
70 citations