Doc

Generalized Pareto Distribution

docs/tutorial:stats:continuous_genpareto

There is one shape parameter $c \neq = 0$ . The support is $x \geq 0$ if $c > 0$ , and $0 \leq x < \frac{1}{∣ c ∣}$ if $c$ is negative.

M (t) = ⎩ ⎨ ⎧ (- \frac{t}{c})^{\frac{1}{c}} e^{- \frac{t}{c}} [Γ (1 - \frac{1}{c}) + (γ (- \frac{1}{c}, - \frac{t}{c}) /Γ (\frac{1}{- c})) - π csc (\frac{π}{c}) /Γ (\frac{1}{c})] (\frac{∣ c ∣}{t})^{1/ ∣ c ∣} Γ (\frac{1}{∣ c ∣}, \frac{t}{∣ c ∣}) \frac{1}{Γ ( \frac{1}{∣ c ∣} )} c > 0 c < 0

μ_{n}^{'} = \frac{( - 1 ) ^{n}}{c ^{n}} k = 0 \sum n (k n) \frac{( - 1 ) ^{k}}{1 - c k} if c n < 1

Thus,

h [X] = 1 + c c > 0

Implementation: scipy.stats.genpareto