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Journal ArticleDOI

The Asymptotic Theory of Extreme Order Statistics

TL;DR: In this paper, the authors analyze the recent development of the theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i).
Abstract: Abstract. Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.
Citations
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Book
01 Jan 2006
TL;DR: In this paper, the authors provide a comprehensive treatment of the problem of predicting individual sequences using expert advice, a general framework within which many related problems can be cast and discussed, such as repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems.
Abstract: This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

3,615 citations

Journal ArticleDOI
TL;DR: In this article, Modelling Extremal Events for Insurance and Finance is discussed. But the authors focus on the modeling of extreme events for insurance and finance, and do not consider the effects of cyber-attacks.
Abstract: (2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

2,729 citations

Book ChapterDOI
01 Jan 2002
TL;DR: This article deals with the static (nontime- dependent) case and emphasizes the copula representation of dependence for a random vector and the problem of finding multivariate models which are consistent with prespecified marginal distributions and correlations is addressed.
Abstract: Modern risk management calls for an understanding of stochastic dependence going beyond simple linear correlation. This paper deals with the static (non-time-dependent) case and emphasizes the copula representation of dependence for a random vector. Linear correlation is a natural dependence measure for multivariate normally and, more generally, elliptically distributed risks but other dependence concepts like comonotonicity and rank correlation should also be understood by the risk management practitioner. Using counterexamples the falsity of some commonly held views on correlation is demonstrated; in general, these fallacies arise from the naive assumption that dependence properties of the elliptical world also hold in the non-elliptical world. In particular, the problem of finding multivariate models which are consistent with prespecified marginal distributions and correlations is addressed. Pitfalls are highlighted and simulation algorithms avoiding these problems are constructed.

2,052 citations


Cites methods from "The Asymptotic Theory of Extreme Or..."

  • ...dependence, as will be discussed in Section 4. This copula, unlike the Gaussian, is a copula which is consistent with bivariate extreme value theory and could be used to model the limiting dependence structure of component-wise maxima of bivariate random samples (Joe (1997), Galambos (1987) )....

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Journal ArticleDOI
TL;DR: In this article, the authors consider the estimation of a stochastic frontier production function, which is the type introduced by Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977).
Abstract: This article considers estimation of a stochastic frontier production function-the type introduced by Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977). Such a production frontier model consists of a production function of the usual regression type but with an error term equal to the sum of two parts. The first part is typically assumed to be normally distributed and represents the usual statistical noise, such as luck, weather, machine breakdown, and other events beyond the control of the firm. The second part is nonpositive and represents technical inefficiencythat is, failure to produce maximal output, given the set of inputs used. Realized output is bounded from above by a frontier that includes the deterministic part of the regression, plus the part of the error representing noise; so the frontier is stochastic. There also exist socalled deterministic frontier models, whose error term contains only the nonpositive component, but we will not consider them here (e.g., see Greene 1980). Frontier models arise naturally in the problem of efficiency measurement, since one needs a bound on output to measure efficiency. A good survey of such production functions and their relationship to the measurement of productive efficiency was given by F0rsund, Lovell, and Schmidt (1980).

1,518 citations

Journal ArticleDOI
Sidney Redner1
TL;DR: In this paper, the authors examined the distribution of citations for papers published in 1981 in journals which were cataloged by the Institute for Scientific Information (IISI) and 20 years of publications in Physical Review D, vol. 11-50 (24,296 papers).
Abstract: Numerical data for the distribution of citations are examined for: (i) papers published in 1981 in journals which are catalogued by the Institute for Scientific Information (783,339 papers) and (ii) 20 years of publications in Physical Review D, vols. 11-50 (24,296 papers). A Zipf plot of the number of citations to a given paper versus its citation rank appears to be consistent with a power-law dependence for leading rank papers, with exponent close to -1/2. This, in turn, suggests that the number of papers with x citations, N(x), has a large-x power law decay $$N(x) \sim {x^{ - a}}$$ , with $$a \approx 3$$ .

1,476 citations

References
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Journal ArticleDOI
TL;DR: For a wide class of (dependent) random variables, a limit law was proved for the maximum, with suitable normalization, of $X 1, X 2, X 3, X 4, X 5, X n as discussed by the authors.
Abstract: For a wide class of (dependent) random variables $X_1, X_2, \cdots, X_n$, a limit law is proved for the maximum, with suitable normalization, of $X_1, X_2, \cdots, X_n$. The results are more general in two aspects than the ones obtained earlier by several authors, namely, the stationary of the $X$'s is not assumed and secondly, the assumptions on the dependence of the $X$'s are weaker than those occurring in previous papers. A generalization of the method of inclusion and exclusion is one of the main tools.

39 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show that the asymptotic distribution of the maximum of a random number of random variables taken from the model below is the same as when their number is a fixed integer.
Abstract: The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group. ASYMPTOTIC DISTRIBUTION; RANDOM NUMBER OF RANDOM VARIABLES; SERVICE TIME; SYSTEM OF A LARGE NUMBER OF COMPONENTS; DEPENDENT RANDOM VARIABLES 1. The main result and applications Let X1,X2,'"-,X, be random variables with the same distribution function F(x). Let nl 0, if t = io 0 and b, be real numbers such that, for n + o , (4) n[1 F(ax + b,)] -~ w(x). Let further v, be a sequence of positive integer valued random variables such that v,/n converges stochastically to a positive random variable v. Then, as tn -+ + c0, (5) lim P[(W, b,,)/a,, < x] = e-w(x) There are several practical situations when our model, described in (1)-(3) and the result of Theorem 1 are needed. We shall describe one through a concrete example; the similarity of several situations to this example is evident. Consider a system of n components which require regular servicing. The number of components to be serviced at a given time is a random variable, i.e., varies from time to time. If v, is the number of components to be serviced then the service is completed in a time period not exceeding a given number T, if, and only if, Wn = max{X1, X2, ...,X,,} does not exceed T, where X1 is the time period required for servicing the jth component. Thus for large n, the conclusion of Theorem 1 gives a good approximation for the time period needed to complete the service. Our assumptions (1)-(3) were made under the guidance of this specific problem. Namely, the assumption of previous models that the service times are stochastically independent is practically never satisfied, even if as many machines are available as there are components to be serviced. To make clear how our assumptions apply to a practical situation, let us specify our system to be an automobile car. Automatic equipment starts servicing all parts virtually at the same time. Because of the relations of the parts, however, it cannot be assumed that service can continue uninterrupted on all parts. As a matter of fact, some This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 05:47:31 UTC All use subject to http://about.jstor.org/terms

21 citations