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Inverted Gamma Distribution¶
Special case of the generalized Gamma distribution with c=−1 and a>0 , x>0
f(x;a)=x−a−1Γ(a)exp(−1x)F(x;a)=Γ(a,1x)Γ(a)G(q;a)={Γ−1[a,Γ(a)q]}−1
μ′n=Γ(a−n)Γ(a)a>n
μ=1a−1a>1μ2=1(a−2)(a−1)−μ2a>2γ1=1(a−3)(a−2)(a−1)−3μμ2−μ3μ3/22γ2=1(a−4)(a−3)(a−2)(a−1)−4μμ3−6μ2μ2−μ4μ22−3
md=1a+1
h[X]=a−(a+1)Ψ(a)+logΓ(a).
Implementation: scipy.stats.invgamma