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Noncentral t Distribution¶
The distribution of the ratio
U+λχν/√ν
where U and χν are independent and distributed as a standard normal and chi with ν degrees of freedom. Note λ>0 and ν>0 .
f(x;λ,ν)=νν/2Γ(ν+1)2νeλ2/2(ν+x2)ν/2Γ(ν/2)×{√2λx1F1(ν2+1;32;λ2x22(ν+x2))(ν+x2)Γ(ν+12)−1F1(ν+12;12;λ2x22(ν+x2))√ν+x2Γ(ν2+1)}=Γ(ν+1)2(ν−1)/2√πνΓ(ν/2)exp[−νλ2ν+x2]×(νν+x2)(ν−1)/2Hhν(−λx√ν+x2)F(x;λ,ν)=
Implementation: scipy.stats.nct