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Parameters α ∈ ( 0 , ∞ ) {displaystyle alpha in (0,infty )} shape.(Optionally, two more parameters) s ∈ ( 0 , ∞ ) {displaystyle sin (0,infty )} scale (default: s = 1 {displaystyle s=1,} ) m ∈ ( − ∞ , ∞ ) {displaystyle min (-infty ,infty )} location of minimum (default: m = 0 {displaystyle m=0,} ) Support x > m {displaystyle x>m} PDF α s ( x − m s ) − 1 − α e − ( x − m s ) − α {displaystyle {rac {alpha }{s}};left({rac {x-m}{s}}ight)^{-1-alpha };e^{-({rac {x-m}{s}})^{-alpha }}} CDF e − ( x − m s ) − α {displaystyle e^{-({rac {x-m}{s}})^{-alpha }}} Mean { m + s Γ ( 1 − 1 α ) for α > 1 ∞ otherwise {displaystyle {egin{cases} m+sGamma left(1-{rac {1}{alpha }}ight)&{ ext{for }}alpha >1 infty &{ ext{otherwise}}end{cases}}} Median m + s log e ( 2 ) α {displaystyle m+{rac {s}{sqrt[{alpha }]{log _{e}(2)}}}} |
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
Contents
where α > 0 is a shape parameter. It can be generalised to include a location parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function
Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.
Characteristics
The single parameter Fréchet with parameter
(with
where
In particular:
The quantile
In particular the median is:
The mode of the distribution is
Especially for the 3-parameter Fréchet, the first quartile is
Also the quantiles for the mean and mode are: