From Wikipedia, the free encyclopedia
This article is within the scope of WikiProject Statistics statistics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.Statistics Wikipedia:WikiProject Statistics Template:WikiProject Statistics Statistics articles Low This article has been rated as Low-importance on the importance scale .
Exponentiated Weibull (3-parameter) Parameters
λ
∈
(
0
,
+
∞
)
{\displaystyle \lambda \in (0,+\infty )\,}
scale
k
∈
(
0
,
+
∞
)
{\displaystyle k\in (0,+\infty )\,}
shape
α
∈
(
0
,
+
∞
)
{\displaystyle \alpha \in (0,+\infty )\,}
shape Support
x
∈
[
0
,
+
∞
)
{\displaystyle x\in [0,+\infty )\,}
PDF
f
(
x
)
=
{
α
k
λ
(
x
λ
)
(
k
−
1
)
e
−
(
x
λ
)
k
(
1
−
e
−
(
x
λ
)
k
)
(
α
−
1
)
,
x
≥
0
,
0
,
x
<
0.
{\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
{
(
1
−
e
−
(
x
/
λ
)
k
)
α
,
x
≥
0
,
0
,
x
<
0.
{\displaystyle {\begin{cases}({1-e^{-(x/\lambda )^{k}}})^{\alpha },&x\geq 0,\\0,&x<0.\end{cases}}}
Mean
α
λ
⋅
Γ
(
1
+
1
k
)
∑
n
=
0
∞
(
Γ
(
α
)
n
!
Γ
(
α
−
n
)
⋅
(
−
1
)
n
(
n
+
1
)
−
(
1
k
+
1
)
)
{\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
(
1
+
2
k
)
∑
n
=
0
∞
(
Γ
(
α
)
n
!
Γ
(
α
−
n
)
⋅
(
−
1
)
n
(
n
+
1
)
−
(
2
k
+
1
)
)
−
(
α
λ
⋅
Γ
(
1
+
1
k
)
∑
n
=
0
∞
(
Γ
(
α
)
n
!
Γ
(
α
−
n
)
⋅
(
−
1
)
n
(
n
+
1
)
−
(
1
k
+
1
)
)
)
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
=
0
∞
t
n
α
λ
n
n
!
⋅
(
Γ
(
1
+
n
k
)
)
∑
m
=
0
∞
(
Γ
(
α
)
m
!
Γ
(
α
−
m
)
⋅
(
−
1
)
m
(
m
+
1
)
−
(
n
k
+
1
)
)
{\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
=
0
∞
(
i
t
)
n
α
λ
n
n
!
⋅
(
Γ
(
1
+
n
k
)
)
∑
m
=
0
∞
(
Γ
(
α
)
m
!
Γ
(
α
−
m
)
⋅
(
−
1
)
m
(
m
+
1
)
−
(
n
k
+
1
)
)
{\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
−
1
{\displaystyle \alpha -1}
2603:9001:4302:FA1:6942:F8EA:3E3:4DAD (talk ) 05:00, 18 November 2022 (UTC) [ reply ]
