Generalized Pareto Distribution¶
Shape parameter c≠0 and defined for x≥0 for all c and x<1|c| if c is negative.
f(x;c)=(1+cx)−1−1cF(x;c)=1−1(1+cx)1/cG(q;c)=1c[(11−q)c−1]
M(t)={(−tc)1ce−tc[Γ(1−1c)+Γ(−1c,−tc)−πcsc(πc)/Γ(1c)]c>0(|c|t)1/|c|Γ[1|c|,t|c|]c<0
μ′n=(−1)ncnn∑k=0(nk)(−1)k1−ckcn<1
μ′1=11−cc<1μ′2=2(1−2c)(1−c)c<12μ′3=6(1−c)(1−2c)(1−3c)c<13μ′4=24(1−c)(1−2c)(1−3c)(1−4c)c<14
Thus,
μ=μ′1μ2=μ′2−μ2γ1=μ′3−3μμ2−μ3μ3/22γ2=μ′4−4μμ3−6μ2μ2−μ4μ22−3
h[X]=1+cc>0
Implementation: scipy.stats.genpareto