Max-stable Laws Research Articles

On the convergence of Shannon entropy of distribution functions in the max domain of attraction of max-stable laws

On the convergence of Shannon entropy of distribution functions in the max domain of attraction of max-stable laws

Boolean extremes and Dagum distributions

We study the max-convolution and max-stable laws for Boolean independence and prove that these are Dagum distributions (also known as log-logistical distributions).

Asymptotic Distributions of Record Values under Exponential Normalization

In this paper, we study the limit distribution of the record values under nonlinear normalization of the form $${\cal T}_n(x)=\exp\{u_{n}(| \log |x||)^{v_{n}}\mbox{sign}(\log |x|)\}\mbox{sign}(x),$$ which is called exponential norming ($e-$norming). The corresponding limit laws of the upper extremes are called $e$-max stable laws (denoted by $U(.)$). In this paper, we show that the limit distributions of the record values under exponential norming are of the form $ {\cal N}(-\log (-\log U(x))),$ where ${\cal N}(.)$ is the standard normal distribution. Moreover, we study the domains of attraction for these types of limit laws. Finally, some illustrative examples are given.

The estimations under power normalization for the tail index, with comparison

It is well known that the max-stable laws under power normalization attract more distributions than that under linear normalization. This fact practically means that the classical linear model (L-model) may fail to fit the given extreme data, while the power model (P-model) succeeds to do that. The main object of this paper is developing the modeling of extreme values via P-model by suggesting a simple technique to obtain a parallel estimator of the extreme value index (EVI) in the P-model for every known estimator to the corresponding parameter in L-mode. An application of this technique yields two classes of moment and moment ratio estimators for EVI in the P-model. The performances of these estimators are assessed via a simulation study. Moreover, an efficient criterion for comparing the L and P models is proposed to choose the best model when the two models successfully work.

Implicit extremes and implicit max–stable laws

Let X1, ⋯ , Xn be iid random vectors and f≥0 be a homogeneous non–negative function interpreted as a loss function. Let also k(n)=Argmaxi=1c⋯ , nf(Xi). We are interested in the asymptotic behavior of Xk(n) as n→∞. In other words, what is the distribution of the random vector leading to maximal loss. This question is motivated by a kind of inverse problem where one wants to determine the extremal behavior of X when only explicitly observing f(X). We shall refer to such types of results as implicit extremes. It turns out that, as in the usual case of explicit extremes, all limit implicit extreme value laws are implicit max–stable. We characterize the regularly varying implicit max–stable laws in terms of their spectral and stochastic representations. We also establish the asymptotic behavior of implicit order statistics relative to a given homogeneous loss and conclude with several examples drawing connections to prior work involving regular variation on general cones.

Pseudo-convex mixtures generated by shape-extended stable distributions for extremes

A shape-extended definition of min-stable (and max-stable) laws is provided and applied to obtain pseudo-convex mixtures. The main properties of the obtained class of distributions are derived, and a particular case generated by the exponential distribution is investigated.

Conditional independence among max-stable laws

Let X be a max-stable random vector with positive continuous density. It is proved that the conditional independence of any collection of disjoint subvectors of X given the remaining components implies their joint independence. We conclude that a broad class of tractable max-stable models cannot exhibit an interesting Markov structure.

On sums of independent random variables whose distributions belong to the max domain of attraction of max stable laws

In this article we study the max domains of attraction of distributions of sums of independent random variables belonging to the max domains of attraction of max stable laws under linear normalization. These results lead to the study of the max domains of attraction of distributions of products of independent random variables belonging to the max domain of attraction of max stable laws under power normalization.

On extremes of mixtures of distributions

In this article we study the max domains of attraction of mixtures of distributions belonging to the max domain of attraction of max stable laws.

General max-stable laws

We review the work on max-stable laws and their max domains of attraction introduced by Pancheva (Lect Notes Math 1155:284–309, 1984). We introduce the concept of general max domain of strict attraction of the general max-stable laws, a subclass of the general max domain of attraction and prove new results. Some interesting examples also are discussed.

Conditions based on conditional moments for max-stable limit laws

Let X1, X2, ...Xn be independent and identically distributed random variables with common distribution function F. Necessary and sufficient conditions for F to belong to the domains of attraction of Φα and Ψα are derived in terms of conditional moments.

Free extreme values

Free probability analogs of the basics of extreme-value theory are obtained, based on Ando's spectral order. This includes classification of freely max-stable laws and their domains of attraction, using ``free extremal convolutions'' on the distributions. These laws coincide with the limit laws in the classical peaks-over-threshold approach. A free extremal projection-valued process over a measure-space is constructed, which is related to the free Poisson point process.

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.

On continuation of max-stable laws

We investigate the continuation problem formax-stable laws on the smallest set S such that the distribution of themax-stable type can be uniquely continued from the set S to the whole line and also the restricted convergence problem for themax-stable laws on a finite set of points.

Domains of Attraction of the Max-Semistable Laws under Linear and Power Normalizations

We consider the domains of attraction of the univariate semi-stable laws in the scheme of cumulative maxima of independent identically distributed random variables and obtain necessary and sufficient conditions for a nondegenerate distribution function to belong to the domain of attraction of a max-semistable law under linear and power normalization. These theorems include the well-known criteria for belonging to the domains of attraction of max-stable laws as a special case; examples are given. The relation between the domains of attraction under linear and power normalization is shown.

Max Domains of Attraction of Univariate and Multivariate p-Max Stable Laws

Let F be a distribution function (d.f.) on Rd, d >_ 1. Suppose there exist norming constants an (i) > 0, bn(i) are real, 1 _ 1, and a d.f. K on Rd with ondegenerate univariate marginals such that (1.1) lim Fn(an(i)xi / bn(i), 1 <_ i <_ d) K(x), n.--400 x (xl,..., Xd) E C(K), where C(K) is the set of all continuity points of K. We call K a max stable d.f. under linear normalization or simply l-max stable d.f. In the univariate case it is well known that an l-max stable d.f. can be only one of the three types of extreme value d.f.’s of Gnedenko, namely,