Skip to main content
SpringerLink
Log in

Extreme Value Statistics

  • Living reference work entry
  • First Online:
Encyclopedia of Complexity and Systems Science

Definition of the Subject

Extreme value theory is concerned with the statistical properties of the extreme events related to a random variable (see Fig. 1), and the understanding and applications of their probability distributions. The methods and the practical use of such a theory have been developed in the last 60 years; though, many complex real-life problems have only recently been tackled. Many disciplines use the tools of extreme value theory including meteorology, hydrology, ocean wave modeling, and finance to name just a few.

Fig. 1
figure 1

We show three samples, 1 to 3, each with N realizations of a random variable \( {X}_i\left(i\in \left\{1,\dots, N\right\}\right) \). Extreme value theory is concerned with the statistical properties of occurrence of the extreme values in those samples, such as the maxima (circled points)

Full size image

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

Random variable:

When a coin is tossed two random outcomes are permitted: head or tail. These outcomes can be mapped to numbers in a process which defines a “random variable”: for instance, “head” and “tail” could be mapped respectively to +1 and −1. More generally, any function mapping the outcomes of a random process to real numbers is defined a random variable (Gnedenko 1998). More technically, a random variable is any function from a probability space to some measurable space, i.e., the space of admitted values of the variable, e.g., real numbers with the Borel σ‑algebra. The amount of rainfall in a day or the daily price variation of a stock are two more examples. It is worth to stress that, formally, the outcome of a given random experiment is not a random variable: the random variable is the function describing all the possible outcomes as numbers. Finally, two random variables are said independent when the outcome of either of them has no influence on the other.

Probability distribution:

The probability of either outcomes, “head” and “tail,” in tossing a coin is 50 %. Similarly, a discrete random variable, X, with values {x 1, x 2, …} has an associate discrete probability distribution of occurrence {p 1, p 2, …}. More generally, for a random variable on real numbers, X, the corresponding probability distribution (Gnedenko 1998) is the function returning the probability to find a value of X within a given interval [x 1, x 2] (where x 1 and x 2 are real numbers): \( \Pr\;\left[{x}_1\le X\le {x}_2\right] \). In particular, the random variable, X, is fully characterized by its cumulative distribution function, F(x), which is: \( F(x)= \Pr\;\left[X<x\right] \) for any x in ℛ. The probability distribution density, f(x), can be often defined as the derivative of \( F(x):f(x)=\mathrm{d}F(x)/\mathrm{d}x \).

The probability distribution of two independent random variables, X and Y, is the product of the distributions, F X and F Y , of X and Y: \( F\left(x,y\right)\equiv \Pr\;\left[X<x;Y<y\right]= \Pr\;\left[X<x\right]\cdot \Pr\;\left[Y<y\right]={F}_X(x)\cdot {F}_Y(y) \).

Expected value:

The expected value (Gnedenko 1998) of a random variable is its average outcome over many independent experiments. Consider, for instance, a discrete random variable, X, with values in the set {x 1, x 2, …} and the corresponding probability for each of these values {p 1, p 2, …}. In probability theory, the expected, or average, value of X (denoted E(X)) is just the sum: \( E(X)={\displaystyle \sum {x}_i}{p}_i \). For instance, if you have an asset which can give two returns {x 1, x 2} with probability {p 1, p 2}, its expected return is \( {x}_1{p}_1+{x}_2{p}_2 \).

In case we have a random variable defined on real numbers and F(x) is its probability distribution function, the expected value of X is: \( E(X)={\displaystyle \int X}\mathrm{d}F \). As for some F(x) the above integral may not exist, the “expected value” of a random variable is not always defined.

Variance and moments:

The variance (Gnedenko 1998) of a probability distribution is a measure of the average deviations from the mean of the related random variable. In probability theory, the variance is usually defined as the mean squared deviation, \( E\left({\left(X-E(X)\right)}^2\right) \), i.e., the expected value of \( {\left(X-E(X)\right)}^2 \). The square root of the variance is named the standard deviation and is a more sensible measure of fluctuations of X around E(X). Alike E(X), for some distributions the variance may not exist. In general, the expected value of the kth power of X, E(X k), is called the kth moment of the distribution.

The central limit theorem:

The central limit theorem (Gnedenko 1998) is a very important result in probability theory stating that the sum of N independent identically distributed random variables, with finite average and variance, has a Gaussian probability distribution in the limit \( N\to \infty \), irrespective of the underlying distributions of the random variables. The domain of attraction of the Gaussian as a limit distribution is, thus, very large and can explain why the Gaussian is so frequently encountered. The theorem is, in practice, very useful since many real random processes have a finite average and variance and are approximately independent.

Bibliography

  • Antal T, Droz M, Gyrgyi G, Racz Z (2001) Phys Rev Lett 87:240601

    Article  ADS  Google Scholar 

  • Berman SM (1964) Ann Math Stat 35:502

    Article  Google Scholar 

  • Bouchaud J-P, Mézard M (1997) J Phys A 30:7997

    Article  ADS  Google Scholar 

  • Bramwell ST, Holdsworth PCW, Pinton J-F (1998) Nature (London) 396:552

    Article  ADS  Google Scholar 

  • Bramwell ST, Christensen K, Fortin J-Y, Holdsworth PCW, Jensen HJ, Lise S, Lopez JM, Nicodemi M, Pinton J-F, Sellitto M (2000) Phys Rev Lett 84:3744

    Article  ADS  Google Scholar 

  • Bunde A, Kropp J, Schellnhuber H-J (eds) (2002) The science of disasters-climate disruptions, heart attacks, and market crashes. Springer, Berlin

    Google Scholar 

  • Castellana M, Decelle A, Zarinelli E (2011) Extreme value statistics distributions in spin glasses. Phys Rev Lett 107:275701

    Article  ADS  Google Scholar 

  • Comtet A, Leboeuf P, Majumdar SN (2007) Phys Rev Lett 98:070404

    Article  ADS  Google Scholar 

  • Dahlstedt K, Jensen HJ (2001) J Phys A 34:11193; [Inspec] [ISI]

    Google Scholar 

  • Dean DS, Majumdar SN (2001) Phys Rev E 64:046121

    Article  ADS  Google Scholar 

  • Eichner JF, Kantelhardt JW, Bunde A, Havlin S (2006) Phys Rev E 73:016130

    Article  ADS  Google Scholar 

  • Embrechts P, Klüppelberg C, Mikosch T, Karatzas I, Yor M (eds) (1997) Modelling extremal events. Springer, Berlin

    MATH  Google Scholar 

  • Finkenstadt B, Rootzen H (2004) Extreme values in finance, telecommunications, and the environment. Chapman and Hall/CRC Press, London

    MATH  Google Scholar 

  • Galambos J (1978) The asymptotic theory of extreme order statistics. Wiley, New York

    MATH  Google Scholar 

  • Galambos J, Lechner J, Simin E (eds) (1994) Extreme value theory and applications. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Gnedenko BV (1998) Theory of probability. CRC, Boca Raton

    MATH  Google Scholar 

  • Guclu H, Korniss G (2004) Phys Rev E 69:065104(R)

    Google Scholar 

  • Gumbel EJ (1958) Statistics of extremes. Columbia University Press, New York

    MATH  Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzen H (1983) Extremes and related properties of random sequences and processes. Springer, New York

    Book  MATH  Google Scholar 

  • Majumdar SN, Comtet A (2004) Phys Rev Lett 92:225501; (2005) J Stat Phys 119:777

    Article  ADS  Google Scholar 

  • Raychaudhuri S, Cranston M, Przybyla C, Shapir Y (2001) Phys Rev Lett 87:136101

    Article  ADS  Google Scholar 

  • Reiss RD, Thomas M (2001) Statistical analysis of extreme values: with applications to insurance, finance, hydrology, and other fields. Birkhäuser, Basel

    MATH  Google Scholar 

  • Smith RL (2003) Statistics of extremes, with applications in environment, insurance and finance, chap 1. In: Statistical analysis of extreme values: with applications to insurance, finance, hydrology, and other fields. Birkhäuser, Basel

    Google Scholar 

  • Smith RL, Tawn JA, Yuen HK (1990) Statistics of multivariate extremes. Int Stat Rev 58:47

    Article  MATH  Google Scholar 

  • von Storch H, Zwiers FW (2001) Statistical analysis in climate research. Cambridge University Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Naples “Federico II”, Complesso Universitario di Monte S. Angelo, 80126, Naples, Italy

    Mario Nicodemi

Authors
  1. Mario Nicodemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Nicodemi .

Editor information

Editors and Affiliations

  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this entry

Cite this entry

Nicodemi, M. (2015). Extreme Value Statistics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_197-3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27737-5_197-3

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Online ISBN: 978-3-642-27737-5

  • eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Search

Navigation