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Exponentiated Weibull (3-parameter) Parameters
λ
∈
(
0
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+
∞
)
{\displaystyle \lambda \in (0,+\infty )\,}
scale
k
∈
(
0
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+
∞
)
{\displaystyle k\in (0,+\infty )\,}
shape
α
∈
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0
,
+
∞
)
{\displaystyle \alpha \in (0,+\infty )\,}
shape Support
x
∈
[
0
,
+
∞
)
{\displaystyle x\in [0,+\infty )\,}
PDF
f
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{
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{\displaystyle f(x)={\begin{cases}{\frac {\alpha k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{\left(k-1\right)}e^{-\left({\frac {x}{\lambda }}\right)^{k}}\left(1-e^{-\left({\frac {x}{\lambda }}\right)^{k}}\right)^{\left(\alpha -1\right)},&x\geq 0,\\0,&x<0.\end{cases}}}
CDF
{
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1
−
e
−
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/
λ
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k
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α
,
x
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0
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x
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{\displaystyle {\begin{cases}({1-e^{-(x/\lambda )^{k}}})^{\alpha },&x\geq 0,\\0,&x<0.\end{cases}}}
Mean
α
λ
⋅
Γ
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k
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∑
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∞
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Γ
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Γ
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⋅
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−
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{\displaystyle \alpha \lambda \cdot \Gamma \left(1+{\frac {1}{k}}\right)\sum _{n=0}^{\infty }\left({\frac {\Gamma \left(\alpha \right)}{n!\Gamma \left(\alpha -n\right)}}\cdot \left(-1\right)^{n}\left(n+1\right)^{-\left({\frac {1}{k}}+1\right)}\right)}
Median
(
−
ln
(
1
−
(
.5
1
α
)
)
)
1
k
⋅
λ
{\displaystyle \left(-\ln \left(1-\left(.5^{\frac {1}{\alpha }}\right)\right)\right)^{\frac {1}{k}}\cdot \lambda }
Mode
(Not entered) Variance
α
λ
2
⋅
Γ
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)
∑
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−
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Γ
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∑
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Γ
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2
{\displaystyle \alpha \lambda ^{2}\cdot \Gamma \left(1+{\frac {2}{k}}\right)\sum _{n=0}^{\infty }\left({\frac {\Gamma \left(\alpha \right)}{n!\Gamma \left(\alpha -n\right)}}\cdot \left(-1\right)^{n}\left(n+1\right)^{-\left({\frac {2}{k}}+1\right)}\right)-\left(\alpha \lambda \cdot \Gamma \left(1+{\frac {1}{k}}\right)\sum _{n=0}^{\infty }\left({\frac {\Gamma \left(\alpha \right)}{n!\Gamma \left(\alpha -n\right)}}\cdot \left(-1\right)^{n}\left(n+1\right)^{-\left({\frac {1}{k}}+1\right)}\right)\right)^{2}}
Skewness
(Not entered) Excess kurtosis
(Not entered) Entropy
(Not entered) MGF
∑
n
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∞
t
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{\displaystyle \sum _{n=0}^{\infty }{\frac {t^{n}\alpha \lambda ^{n}}{n!}}\cdot \left(\Gamma \left(1+{\frac {n}{k}}\right)\right)\sum _{m=0}^{\infty }\left({\frac {\Gamma \left(\alpha \right)}{m!\Gamma \left(\alpha -m\right)}}\cdot \left(-1\right)^{m}\left(m+1\right)^{-\left({\frac {n}{k}}+1\right)}\right)}
CF
∑
n
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∞
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i
t
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n
α
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{\displaystyle \sum _{n=0}^{\infty }{\frac {\left(it\right)^{n}\alpha \lambda ^{n}}{n!}}\cdot \left(\Gamma \left(1+{\frac {n}{k}}\right)\right)\sum _{m=0}^{\infty }\left({\frac {\Gamma \left(\alpha \right)}{m!\Gamma \left(\alpha -m\right)}}\cdot \left(-1\right)^{m}\left(m+1\right)^{-\left({\frac {n}{k}}+1\right)}\right)}
Kullback–Leibler divergence
(Not entered)
I was able to fill out most of it, but a few areas I cannot find a proper equation. Some of the papers I looked through were also wrong on various areas (the mean's infinite summation can stop at
α
−
1
{\displaystyle \alpha -1}
α
{\displaystyle \alpha }
α
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1
{\displaystyle \alpha -1}
2603:9001:4302:FA1:6942:F8EA:3E3:4DAD (talk ) 05:00, 18 November 2022 (UTC) Reply
