Generalized extreme value distribution

About: Generalized extreme value distribution is a research topic. Over the lifetime, 3451 publications have been published within this topic receiving 88078 citations. The topic is also known as: Fisher–Tippett distribution.

An Introduction to Statistical Modeling of Extreme Values

Abstract: 1. Introduction.- 2. Basics of Statistical Modeling.- 3. Classical Extreme Value Theory and Models.- 4. Threshold Models.- 5. Extremes of Dependent Sequences.- 6. Extremes of Non-Stationary Sequences.- 7. A Point Process Characterization of Extremes.- 8. Multivariate Extremes.- 9. Further Topics.- Appendix A: Computational Aspects.- Index.

Statistical Inference Using Extreme Order Statistics

Abstract: A method is presented for making statistical inferences about the upper tail of a distribution function. It is useful for estimating the probabilities of future extremely large observations. The method is applicable if the underlying distribution function satisfies a condition which holds for all common continuous distribution functions.

Limiting forms of the frequency distribution of the largest or smallest member of a sample

Abstract: The limiting distribution, when n is large, of the greatest or least of a sample of n, must satisfy a functional equation which limits its form to one of two main types. Of these one has, apart from size and position, a single parameter h, while the other is the limit to which it tends when h tends to zero.The appropriate limiting distribution in any case may be found from the manner in which the probability of exceeding any value x tends to zero as x is increased. For the normal distribution the limiting distribution has h = 0.From the normal distribution the limiting distribution is approached with extreme slowness; the final series of forms passed through as the ultimate form is approached may be represented by the series of limiting distributions in which h tends to zero in a definite manner as n increases to infinity.Numerical values are given for the comparison of the actual with the penultimate distributions for samples of 60 to 1000, and of the penultimate with the ultimate distributions for larger samples.

The Asymptotic Theory of Extreme Order Statistics

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.

Models for exceedances over high thresholds

Abstract: We discuss the analysis of the extremes of data by modelling the sizes and occurrence of exceedances over high thresholds. The natural distribution for such exceedances, the generalized Pareto distribution, is described and its properties elucidated. Estimation and model-checking procedures for univariate and regression data are developed, and the influence of and information contained in the most extreme observations in a sample are studied. Models for seasonality and serial dependence in the point process of exceedances are described. Sets of data on river flows and wave heights are discussed, and an application to the siting of nuclear installations is described