Fréchet distribution explained

The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function

\Pr( X\lex )=

-x-\alpha
e

~if~x>0~.

where is a shape parameter. It can be generalised to include a location parameter (the minimum) and a scale parameter with the cumulative distribution function

\Pr( X\lex )=\exp\left( -\left(\tfrac{ x-m }{s}\right)-\alpha \right)~if~x>m~.

Named for Maurice Fréchet who wrote a related paper in 1927,[1] further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.[2] [3]

Characteristics

The single parameter Fréchet, with parameter

 \alpha ,

has standardized moment

\muk=

infty
\int
0

xkf(x) \operatorname{d}x

infty
=\int
0
-k
\alpha
t

e-t \operatorname{d}t ,

(with

 t=x-\alpha 

) defined only for

 k<\alpha :

 \muk=\Gamma\left(1-

k
\alpha

\right) 

where

 \Gamma\left(z\right) 

is the Gamma function.

In particular:

\alpha>1

the expectation is

E[X]=\Gamma(1-\tfrac{1}{\alpha})

\alpha>2

the variance is

Var(X)=\Gamma(1-\tfrac{2}{\alpha})-(\Gamma(1-\tfrac{1}{\alpha}))2.

qy

of order

y

can be expressed through the inverse of the distribution,
-1
q
y=F

(y)=\left(-logey

-1
\alpha
\right)
.In particular the median is:

q1/2=(loge

-1
\alpha
2)

.

The mode of the distribution is

\left(\alpha
\alpha+1
1
\alpha
\right)

.

Especially for the 3-parameter Fréchet, the first quartile is

q1=m+

s
\sqrt[\alpha]{log(4)
} and the third quartile

q3=m+

s
\sqrt[\alpha]{log(4)
3
}.

Also the quantiles for the mean and mode are:

F(mean)=\exp\left(-\Gamma-\alpha\left(1-

1
\alpha

\right)\right)

F(mode)=\exp\left(-

\alpha+1
\alpha

\right).

Applications

However, in most hydrological applications, the distribution fitting is via the generalized extreme value distribution as this avoids imposing the assumption that the distribution does not have a lower bound (as required by the Frechet distribution).

Zi=-1/logFi(Xi)

and then mapping from Cartesian to pseudo-polar coordinates

(R,W)=(Z1+Z2,Z1/(Z1+Z2))

. Values of

R\gg1

correspond to the extreme data for which at least one component is large while

W

approximately 1 or 0 corresponds to only one component being extreme.

Related distributions

Scaling relations

 X\simU( 0,1 ) 

(continuous uniform distribution) then

 m+s ⋅ l(

-1 
\alpha
-log
e(X) r)

\simsf{Frechet}(\alpha,s,m) 

 X\simsf{Frechet}( \alpha,s,m=0 ) 

then its reciprocal is Weibull-distributed:
  1 
X

\simsf{Weibull}\left( k=\alpha,λ=\tfrac{1}{s} \right) 

 X\simsf{Frechet}(\alpha,s,m) 

then

 k X+b\simrm{Frechet}( \alpha,ks,k m+b ) 

 Xi\simsf{Frechet}( \alpha,s,m ) 

and

 Y=max\{ X1,\ldots,Xn \} 

then

 Y\simsf{Frechet}( \alpha,n\tfrac{1{\alpha}}s,m ) 

Properties

See also

Further reading

External links

Notes and References

  1. Fréchet . M. . Maurice Fréchet . 1927 . Sur la loi de probabilité de l'écart maximum . fr . On the probability distribution of the maximum deviation . . 6 . 93 .
  2. Fisher . R.A. . Ronald Fisher . Tippett . L.H.C. . L. H. C. Tippett . 1928 . Limiting forms of the frequency distribution of the largest and smallest member of a sample . . 24 . 2 . 180–190 . 10.1017/S0305004100015681 . 1928PCPS...24..180F . 123125823 .
  3. Book: Gumbel, E.J. . Emil Julius Gumbel . 1958 . Statistics of Extremes . Columbia University Press . New York, NY . 180577 .
  4. Lee. Se Yoon . Bani . Mallick. Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas. Sankhya B. 2021. 84 . 1–43 . 10.1007/s13571-020-00245-8.

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