Generalized extreme value

Probability density function
Cumulative distribution function
Parameters	location (real) scale (real) shape (real) Please refer to Real vs. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ...
Support	In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
pdf	where In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
cdf
Mean
Median
Mode
Variance
Skewness
Kurtosis
Entropy
mgf
Char. func.

In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. Its importance arises from the fact that it is the limit distribution of the maxima of a sequence of independent and identically distributed random variables. Because of this, the GEV is used as an approximation to model the maxima of long (finite) sequences of random variables. In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ... Probability theory is the mathematical study of probability. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. ...

Specification

The generalized extreme value distribution has cumulative distribution function

for 1 + ξ(x − μ) / σ > 0, where is the location parameter, σ > 0 the scale parameter and the shape parameter.

The density function is, consequently

again, for 1 + ξ(x − μ) / σ > 0.

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Link to Fréchet, Weibull and Gumbel families

The shape parameter ξ governs the tail behaviour of the distribution, the sub-families defined by , ξ > 0 and ξ < 0 correspond, respectively, to the Gumbel, Fréchet and Weibull families, whose cumulative distribution functions are reminded below.

• Gumbel or type I extreme value distribution

• Fréchet or type II extreme value distribution

• Weibull or type III extreme value distribution

where σ > 0 and α > 0 This article needs cleanup. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ...

Remark I: For reliability issues the Weibull distribution is used with the variable t = μ − x, the time, which is strictly positive. Thus the support is positive - in contrast to the use in extreme value theory.

Remark II: Be aware of an important distinctive feature of the three extreme value distributions: The support is either unlimited, or it has an upper or lower limit.

Extremal types theorem

Credit for the extremal types theorem (or convergence to types theorem) is given to Gnedenko (1948), previous versions were stated by Fisher and Tippett in 1928 and Fréchet in 1927.

Let be a sequence of independent and identically distributed random variables, let if two sequence of real numbers a_n,b_n exist such that a_n > 0 and

then if F is a non degenerate distribution function, it belongs to either the Gumbel, the Fréchet or the Weibull family.

Clearly, the theorem can be reformulated saying that F is a member of the GEV family.

It is worth noting that the result, which is stated for maxima, can be applied to minima by taking the sequence − X_n instead of the sequence X_n.

For the practical application this theorem means: For samples taken from a well behaving, arbitrary distribution X the resulting extreme value distribution M_n can be approximated and parametrised with the extreme value distribution with the appropriate support.

Thus the role of extremal types theorem for maxima is similar to that of central limit theorem for averages. The latter states that the limit distribution of arithmetic mean of a sequence X_n of random variable is the normal distribution no matter what the distribution of the X_n, The extremal types theorem is similar in scope where maxima is substituted for average and GEV distribution is substituted for normal distribution. Central limit theorems are a set of weak-convergence results in probability theory. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ...

References

• Leadbetter, M.R., Lindgreen, G. and Rootzén, H. (1983). Extremes and related properties of random sequences and processes, Springer-Verlag. ISBN 0387907319.
• Resnick, S.I. (1987). Extreme values, regular variation and point processes, Springer-Verlag. ISBN 0387964819.