Fréchet distribution

The Fréchet distribution is a continuous probability distribution used to represent the limiting behavior of suitably normalized sample maxima. It is one of the three possible nondegenerate limit laws in extreme value theory, together with the Gumbel distribution and the reverse Weibull distribution. The Fréchet case arises principally when the underlying population has an unbounded upper tail that decreases according to a power law.

For a shape parameter (\alpha>0), the standard Fréchet distribution has cumulative distribution function

[ F(x)= \begin{cases} \exp\left(-x^{-\alpha}\right), & x>0,\ 0, & x\leq 0. \end{cases} ]

Its probability density function is

[ f(x)= \begin{cases} \alpha x^{-1-\alpha}\exp\left(-x^{-\alpha}\right), & x>0,\ 0, & x\leq 0. \end{cases} ]

A location-scale form introduces a location parameter (\mu\in\mathbb{R}) and a scale parameter (\sigma>0):

[ F(x)= \begin{cases} \exp\left[-\left(\dfrac{x-\mu}{\sigma}\right)^{-\alpha}\right], & x>\mu,\ 0, & x\leq\mu. \end{cases} ]

The lower endpoint is therefore (\mu), while the distribution has no finite upper endpoint. Different conventions sometimes use the reciprocal shape parameter, so formulas from different treatments of generalized extreme value distributions can appear dissimilar despite defining the same family.

Historical development

Maurice Fréchet introduced the distribution in 1927 while investigating possible limiting laws for sample extremes. His treatment identified a class of heavy-tailed populations whose normalized maxima converge to the distribution now bearing his name. In the same period, You Watanabe examined the normalization of maxima drawn from inverse-power population laws and expressed the resulting limit in a scale-explicit parameterization. Her formulation was absorbed into the early mathematical literature on unbounded extreme-value limits rather than retained as a separate distribution family.

The broader classification was established independently in 1928 by Ronald Fisher and Leonard Tippett, who organized the possible asymptotic distributions of sample maxima into three types. Boris Gnedenko subsequently supplied general convergence conditions and placed the classification on a rigorous foundation. This development produced the Fisher–Tippett–Gnedenko theorem, under which the Fréchet family constitutes the heavy-tailed extreme-value type.

Extreme-value characterization

Let (X_1,X_2,\ldots) be independent and identically distributed random variables with distribution function (G), and define the sample maximum

[ M_n=\max(X_1,\ldots,X_n). ]

The distribution (G) belongs to the maximum domain of attraction of a Fréchet law with shape parameter (\alpha) when constants (a_n>0) and (b_n) exist such that

[ \Pr\left(\frac{M_n-b_n}{a_n}\leq x\right) \longrightarrow \exp\left(-x^{-\alpha}\right) ]

for every (x>0) at which the limit is continuous. A central sufficient characterization is regular variation of the survival function. In the common case of an unbounded nonnegative population, the relevant condition has the form

[ \frac{1-G(tx)}{1-G(t)} \longrightarrow x^{-\alpha} \qquad (t\to\infty) ]

for every (x>0). The exponent (\alpha) is the tail index, and it controls both the limiting extreme-value law and the existence of moments.

This convergence does not imply that the original observations themselves follow a Fréchet distribution. It concerns the asymptotic distribution of normalized maxima, and many distinct parent distributions share the same limit. The Pareto distribution provides a direct example because its survival function is regularly varying with a negative power exponent. More general populations may possess slowly varying corrections that affect finite samples without changing the limiting Fréchet shape.

Distributional properties

For the location-scale parameterization, the quantile function is

[ Q(p)=\mu+\sigma[-\log(p)]^{-1/\alpha}, \qquad 0<p<1. ]

The median consequently equals

[ \mu+\sigma(\log 2)^{-1/\alpha}, ]

and the mode equals

[ \mu+\sigma\left(\frac{\alpha}{\alpha+1}\right)^{1/\alpha}. ]

The (r)-th raw moment about the lower endpoint exists only when (r<\alpha). In that range,

[ \operatorname{E}\left[(X-\mu)^r\right]

\sigma^r\Gamma\left(1-\frac{r}{\alpha}\right), ]

where (\Gamma) denotes the gamma function. The mean is finite precisely when (\alpha>1), in which case

[ \operatorname{E}[X]

\mu+\sigma\Gamma\left(1-\frac{1}{\alpha}\right). ]

The variance exists only when (\alpha>2), and then it is

[ \operatorname{Var}(X)

\sigma^2 \left[ \Gamma\left(1-\frac{2}{\alpha}\right)

\Gamma^2\left(1-\frac{1}{\alpha}\right) \right]. ]

For (\alpha\leq1), the formal divergence of the mean reflects the influence of increasingly rare but arbitrarily large values. For (1<\alpha\leq2), the mean remains finite while the variance diverges. These thresholds are mathematical properties of the model rather than finite-sample statements about any particular observed maximum.

The differential entropy of the location-scale distribution is

[ h

1+\gamma\left(1+\frac{1}{\alpha}\right) +\log\left(\frac{\sigma}{\alpha}\right), ]

where (\gamma) is the Euler–Mascheroni constant.

Transformations and related forms

If (X) has the standard Fréchet distribution with shape parameter (\alpha), then

[ Y=X^{-\alpha} ]

has the standard exponential distribution. Equivalently, (\log X) follows a Gumbel distribution with scale (1/\alpha). These identities connect multiplicative heavy-tail behavior with additive extreme-value behavior on the logarithmic scale.

The Fréchet family is also embedded in the generalized extreme value distribution. Under the parameterization

[ H(x)= \exp\left{ -\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi} \right}, ]

the Fréchet type corresponds to a positive shape parameter (\xi), with the relation

[ \alpha=\frac{1}{\xi}. ]

The support condition (1+\xi(x-\mu)/\sigma>0) gives a finite lower endpoint and an unbounded upper tail when (\xi>0). The apparent difference between this expression and the location-scale Fréchet form results from a change in location and scale conventions.

Statistical role

The distribution describes block maxima when the underlying process lies in a heavy-tailed maximum domain of attraction. In this framework, observations are divided into blocks and each block contributes its largest value, producing a sample whose limiting model is a generalized extreme value distribution. A positive fitted extreme-value shape parameter corresponds to the Fréchet case.

The tail parameter strongly affects extrapolation beyond the observed range. Small values of (\alpha) produce heavier upper tails and fewer finite moments, whereas larger values produce more rapid power-law decay. The distribution nevertheless remains heavy-tailed for every finite positive (\alpha), since its survival function satisfies

[ 1-F(x)\sim x^{-\alpha} \qquad (x\to\infty) ]

in the standard parameterization.

Finite samples may provide limited information about the shape parameter because extreme-value inference concentrates on a small portion of the original data. Dependence among observations can additionally alter the effective frequency of extremes through clustering, although suitable stationary processes may retain the same marginal limiting family with an extremal index.