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Max-infinite divisibility and multivariate total positivity
Abstract
The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP 2 ). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.
... Such distributions and prominent subfamilies, like max-(or min-) stable laws, are well-established in the applied probability and statistics literature, see e.g. [3,46,2,28,49,16], and have recently gained interest in the modeling of spatial extremes, see [22,8,50,23]. In analytical terms, such probability distributions are canonically described by a so-called exponent measure and the work of [59,17] generalizes this framework to infinite sequences of random variables. ...
... We emphasize that H cannot be decomposed into H = H (1) + H (2) , where H (1) is always finite and independent of H (2) ∈ {0, ∞} R . Therefore, jumps to ∞ do not occur independently of the path behavior of the process in general. ...
... (2) t may take all finite values. Thus, in general, H cannot be decomposed into a finite process H (1) and a "killing" process H (2) ...
We establish a correspondence between exchangeable sequences of random variables whose finite-dimensional distributions are min- (or max-) infinitely divisible and non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of a min-id sequence is shown to be the sum of a very simple "drift measure" and a mixture of product probability measures, which corresponds uniquely to the L\'evy measure of a non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.
... It follows that C F is a copula in a fixed dimension d if and only if = − ln(F ) is d-monotone on (0, ∞). A geometric insight into the structure of 1 -norm symmetric exponent measures is provided in Section 4, which leads, in Section 5, to a stochastic representation for reciprocal Archimedean copulas in terms of Poisson point processes, and to an effective simulation algorithm. Section 6 presents a number of new examples and illustrations. ...
... , d and any u 1 , . . . , u d ∈ [0, 1], (1) , . . . , u π(d) ). ...
... Thus, for any F ∈ F d , the reciprocal Archimedean copula C F is precisely the dependence structure of a continuous max-id distribution whose exponent measure is 1 -norm symmetric and generated by = − ln(F ). In particular, therefore, reciprocal Archimedean copulas are multivariate totally positive in the sense of [1]. As they are also associated, they are both positive lower and upper orthant dependent; see, for example, Theorem 8.6 in [13]. ...
... where [1] d denotes the set {0, 1} d and J k := {i ∈ {1, . . . , d} : k i > 0}. ...
... , d} : k i > 0}. The survival function of τ can then be expressed in terms of the parameter multi-sequence {µ(k)} k∈ [1] d : the monotonicity of {µ(k)} k∈ [1] d is determined by the monotonicity of the survival function, and vice versa. While the family of G W d distributions is characterized by monotone {µ(k)} k∈ [1] d , MO d requires a stronger concept of monotonicity, namely logarithmic monotonicity. ...
... , d} : k i > 0}. The survival function of τ can then be expressed in terms of the parameter multi-sequence {µ(k)} k∈ [1] d : the monotonicity of {µ(k)} k∈ [1] d is determined by the monotonicity of the survival function, and vice versa. While the family of G W d distributions is characterized by monotone {µ(k)} k∈ [1] d , MO d requires a stronger concept of monotonicity, namely logarithmic monotonicity. ...
Working Paper: A unified formulation of the theory of d-variate wide-sense geometric and Marshall–Olkin exponential distributions is presented. Monotone multi-sequences of degree d with binary subscripts are shown to occupy a central role. ...
... for ∅ = S ⊆ {1, . . . , n}, where |S| indicates the cardinal of the set S. For a bivariate cdf F, we have that F is mini-infinitely divisible if and only if F is MT P 2 , where a function f : R n + → R is MT P 2 if f (x)(y) ≤ f (x∧y) f (x∨y) for all x, y ∈ R n + where ∧ and ∨ are the minimum and maximum componentwise of two vectors, respectively (see Alzaid and Proschan 1994;Joe 1997). Also, it is mentionable that if Z is a non negative integer valued random variable or the components of the random vector with sf F 0 are independent, then the MPH mixture model is well-defined. ...
In this paper, we study mixtures of the multivariate proportional hazard model, where the frailty (an unobservable non negative random variable) acts on the hazard rates of lifetimes at the same time to model common risk factors. We investigate ageing properties and dependence between components for the mixture model under study. In addition, we carry out stochastic comparisons assuming dependence between components of the baseline random vector. More specifically, these stochastic comparisons refer to mixture models that share the same frailty random variable but different baseline random vectors. The results are applied for inter epoch times of a non homogeneous mixed Poisson process.
... Max-id random vectors Z are associated (Resnick, 1987, Proposition 5.29), meaning that for any choice of nondecreasing functions h i : R D → R such that h i (Z) has finite second moments, i = 1, 2, one has cov{h 1 (Z), h 2 (Z)} ≥ 0 (Esary et al., 1967). In other words, a certain form of positive dependence prevails; see also Alzaid and Proschan (1994). This shows that any random vector with negative correlation between some of its components cannot be max-id. ...
Extreme-value theory for stochastic processes has motivated the statistical use of max-stable models for spatial extremes. However, fitting such asymptotic models to maxima observed over finite blocks is problematic when the asymptotic stability of the dependence does not prevail in finite samples. This issue is particularly serious when data are asymptotically independent, such that the dependence strength weakens and eventually vanishes as events become more extreme. We here aim to provide flexible sub-asymptotic models for spatially indexed block maxima, which more realistically account for discrepancies between data and asymptotic theory. We develop models pertaining to the wider class of max-infinitely divisible processes, extending the class of max-stable processes while retaining dependence properties that are natural for maxima: max-id models are positively associated, and they yield a self-consistent family of models for block maxima defined over any time unit. We propose two parametric construction principles for max-id models, emphasizing a point process-based generalized spectral representation, that allows for asymptotic independence while keeping the max-stable extremal-t model as a special case. Parameter estimation is efficiently performed by pairwise likelihood, and we illustrate our new modeling framework with an application to Dutch wind gust maxima calculated over different time units.
A unified formulation of the theory of d-variate wide-sense geometric (G_d^W) and Marshall–Olkin exponential (MO_d) distributions is presented in which d-monotone set functions occupy a central role. A semi-analytical derivation of G_d^W and MO_d distributions is deduced directly from the lack-of-memory property. In this context, the distributions are parametrized with d-monotone and d-log-monotone set functions arising from the univariate marginal distributions of minima and the d-decreasingness of the survival functions. In addition, a one-to-one correspondence is established between d-monotone (resp. d-log-monotone) set functions and d-variate (resp. d-variate min-infinitely divisible) Bernoulli distributions. The advantage of such a parametrization is that it makes the distributions highly tractable. As a showcase, we derive new results on the minimum stability and divisibility of the G_d^W family, and on the marginal equivalence in minima of G_d^W and distributions with geometric minima. Similarly, a surprisingly simple proof is given of the prominent result of Esary and Marshall (Ann. Statist., 1974) on the marginal equivalence in minima of multivariate exponential distributions.
In this paper we establish multivariate likelihood ratio, hazard rate, and reversed hazard rate stochastic orderings between two mixtures of multivariate distributions. The new results in this paper extend some recent results in the literature by assuming that the components of the mixture can be conditionally dependent.
Suppose Y is the strength of a system subjected to a sequence of stresses X1, X2, X3, ⋯. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of T, H̄(t), is given H̄(t) = ∑k=0 ∞ P(N(t) = k)P̄(k), where P̄(0) = 1 and $\bar{P}(k) = P(\max(X_{1}, \cdots,\, X_{k})
Necessary and sufficient conditions are given for a distribution function in R2 to be max-infinitely divisible. The d.f. F is max i.d. if Ft is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.
A family of concepts of stochastic dependence for bivariate distribution functions is introduced. Each concept gives rise to a family of bivariate distribution functions. We show the equivalence of some of these families with families of positively dependent distribution functions, which are known in the literature, and characterize some of them by notions from reliability theory. Interrelations among the various families are studied, and some moment inequalities are derived. Some examples and applications are discussed.
Let X1, …, Xn be a sequence of random variables with a given positive or negative dependence structure. In this article we exploit the assumed dependence structure to construct a sequence of bounds for the P(Xi ε Ci; i = 1, …, n), where Ci are infinite intervals of the same type. These bounds are superior to the well-known product bounds that are based solely on the marginal probabilities. Moreover, the new bounds can serve as respectable approximations for the P(Xi ε Ci; i = 1, …, n).
A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. This paper presents some meaningful derivations of a multivariate exponential distribution that serves to indicate conditions under which the distribution is appropriate. Two of these derivations are based on “shock models,” and one is based on the requirement that residual life is independent of age. It is significant that the derivations all lead to the same distribution.For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. A multivariate Weibull distribution is obtained through a change of variables.
A function f(x) defined on = 1 × 2 × … × n where each i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP2). A random vector Z = (Z1, Z2,…, Zn) of n-real components is MTP2 if its density is MTP2. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ−1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP2 properties. The MTP2 property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP2 kernels are presented and amplified by a wide variety of applications.
We consider some unresolved relationships among various notions of bivariate dependence. In particular we show that in s (or alternately, in s) implies are associated, i.e. for all non-decreasing f and g.
The theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial, Dirichlet and the multivariate hypergeometric. Some conditions of negative dependence that are satisfied by these distributions and which have practical meaning are introduced. Useful inequalities for some widely used distributions are obtained.
The univariate conditions of Gnedenko characterizing domains of attraction for univariate extreme value distributions are generalized to higher dimensions. In addition, it is shown that random variables with a multivariate extreme value distribution are associated. Applications are given to a number of parametric families of joint distributions with given marginal distributions.
A new concept of positive association is introduced. Some of its properties are discussed. An application of this concept to the stress-strength model considered by Ebrahimi (1985) is given. Further results on the subjects are also included.
Let X = (X1, X2,..., Xn) be a random vector in Rn (Euclidean n-space) with density f(x). X or f(x) is said to be multivariate reverse rule of order 2 (MRR2) if f(x [curly logical or] y) f(x [curly logical and] y) <= f(x) f(y) where the lattice operations x [curly logical and] y and x [curly logical or] y refer to the usual ordering of Rn. A density f(x) of X = (X1,...,Xn) is said to be strongly MRR2 if for any set of PF2 functions {[phi]v} (i.e., [phi]v([xi] - [eta]) is totally positive of order 2 on -[infinity] < [xi], [eta] < [infinity]) each marginal g(x[nu]1,x x[nu]2,..., x[nu]k) = [integral operator] ... [integral operator] f(x1,..., xn) [phi]1(x[mu]1)[phi]2(x[mu]2) ... [phi]n - k(x[mu]n - k) dx[mu]1 ... dx[mu]n - k is MRR2 in the variables (x[nu]1, x[nu]2,..., x[nu]k), where ([nu]1,..., [nu]k) and ([mu]1, [mu]2,..., [mu]n - k) are complementary sets of indices. The property of strongly MRR2 prevails for the multinormal, multivariate hypergeometric, Dirichlet, and many other densities. For a strong MRR2 density we establish the reverse generalized correlation inequality P{ai <= Xi <= bi, i [set membership, variant] I, X[nu] <= b[nu], [nu] [set membership, variant] J [union or logical sum] K}P{ai <= Xi <= bi, i [set membership, variant] I} <= P{ai <= Xi <= bi, i [set membership, variant] I, X[nu] <= b[nu], v [set membership, variant] J}P{ai <= Xi <= bi, i [set membership, variant] I, X[nu] <= b[nu], [nu] [set membership, variant] K}, where I, J and K denote the set of indices {1,..., k}, {k + 1,..., k + l}, {k + l + 1,..., n}, respectively. Other inequalities and applications are given.