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( T o p )
1
C h a r a c t e r i z a t i o n
T o g g l e C h a r a c t e r i z a t i o n s u b s e c t i o n
1 . 1
P r o b a b i l i t y d e n s i t y f u n c t i o n
1 . 2
C u m u l a t i v e d i s t r i b u t i o n f u n c t i o n
2
M e a n
3
R e l a t i o n s h i p t o o t h e r d i s t r i b u t i o n s
4
S e e a l s o
5
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
q - W e i b u l l d i s t r i b u t i o n
1 l a n g u a g e
● C a t a l à
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Characterization
[ edit ]
Probability density function
[ edit ]
The probability density function of a q -Weibull random variable is:[1]
f
(
x
;
q
,
λ
,
κ
)
=
{
(
2
−
q
)
κ
λ
(
x
λ
)
κ
−
1
e
q
(
−
(
x
/
λ
)
κ
)
x
≥
0
,
0
x
<
0
,
{\displaystyle f(x;q,\lambda ,\kappa )={\begin{cases}(2-q){\frac {\kappa }{\lambda }}\left({\frac {x}{\lambda }}\right)^{\kappa -1}e_{q}(-(x/\lambda )^{\kappa })&x\geq 0,\\0&x<0,\end{cases}}}
where q < 2,
κ
{\displaystyle \kappa }
> 0 are shape parameters and λ > 0 is the scale parameter of the distribution and
e
q
(
x
)
=
{
exp
(
x
)
if
q
=
1
,
[
1
+
(
1
−
q
)
x
]
1
/
(
1
−
q
)
if
q
≠
1
and
1
+
(
1
−
q
)
x
>
0
,
0
1
/
(
1
−
q
)
if
q
≠
1
and
1
+
(
1
−
q
)
x
≤
0
,
{\displaystyle e_{q}(x )={\begin{cases}\exp(x )&{\text{if }}q=1,\\[6pt][1+(1-q)x]^{1/(1-q)}&{\text{if }}q\neq 1{\text{ and }}1+(1-q)x>0,\\[6pt]0^{1/(1-q)}&{\text{if }}q\neq 1{\text{ and }}1+(1-q)x\leq 0,\\[6pt]\end{cases}}}
is the q -exponential[1] [2] [3]
Cumulative distribution function
[ edit ]
The cumulative distribution function of a q -Weibull random variable is:
{
1
−
e
q
′
−
(
x
/
λ
′
)
κ
x
≥
0
0
x
<
0
{\displaystyle {\begin{cases}1-e_{q'}^{-(x/\lambda ')^{\kappa }}&x\geq 0\\0&x<0\end{cases}}}
where
λ
′
=
λ
(
2
−
q
)
1
κ
{\displaystyle \lambda '={\lambda \over (2-q)^{1 \over \kappa }}}
q
′
=
1
(
2
−
q
)
{\displaystyle q'={1 \over (2-q)}}
Mean
[ edit ]
The mean of the q -Weibull distribution is
μ
(
q
,
κ
,
λ
)
=
{
λ
(
2
+
1
1
−
q
+
1
κ
)
(
1
−
q
)
−
1
κ
B
[
1
+
1
κ
,
2
+
1
1
−
q
]
q
<
1
λ
Γ
(
1
+
1
κ
)
q
=
1
λ
(
2
−
q
)
(
q
−
1
)
−
1
+
κ
κ
B
[
1
+
1
κ
,
−
(
1
+
1
q
−
1
+
1
κ
)
]
1
<
q
<
1
+
1
+
2
κ
1
+
κ
∞
1
+
κ
κ
+
1
≤
q
<
2
{\displaystyle \mu (q,\kappa ,\lambda )={\begin{cases}\lambda \,\left(2+{\frac {1}{1-q}}+{\frac {1}{\kappa }}\right)(1-q)^{-{\frac {1}{\kappa }}}\,B\left[1+{\frac {1}{\kappa }},2+{\frac {1}{1-q}}\right]&q<1\\\lambda \,\Gamma (1+{\frac {1}{\kappa }})&q=1\\\lambda \,(2-q)(q-1)^{-{\frac {1+\kappa }{\kappa }}}\,B\left[1+{\frac {1}{\kappa }},-\left(1+{\frac {1}{q-1}}+{\frac {1}{\kappa }}\right)\right]&1<q<1+{\frac {1+2\kappa }{1+\kappa }}\\\infty &1+{\frac {\kappa }{\kappa +1}}\leq q<2\end{cases}}}
where
B
(
)
{\displaystyle B()}
is the Beta function and
Γ
(
)
{\displaystyle \Gamma ()}
is the Gamma function . The expression for the mean is a continuous function of q over the range of definition for which it is finite.
Relationship to other distributions
[ edit ]
The q -Weibull is equivalent to the Weibull distribution when q = 1 and equivalent to the q -exponential when
κ
=
1
{\displaystyle \kappa =1}
The q -Weibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support (q <1 ) and to include heavy-tailed distributions
(
q
≥
1
+
κ
κ
+
1
)
{\displaystyle (q\geq 1+{\frac {\kappa }{\kappa +1}})}
.
The q -Weibull is a generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support and adds the
κ
{\displaystyle \kappa }
parameter. The Lomax parameters are:
α
=
2
−
q
q
−
1
,
λ
Lomax
=
1
λ
(
q
−
1
)
{\displaystyle \alpha ={{2-q} \over {q-1}}~,~\lambda _{\text{Lomax}}={1 \over {\lambda (q-1)}}}
As the Lomax distribution is a shifted version of the Pareto distribution , the q -Weibull for
κ
=
1
{\displaystyle \kappa =1}
is a shifted reparameterized generalization of the Pareto. When q > 1, the q -exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically:
If
X
∼
q
-
W
e
i
b
u
l
l
(
q
,
λ
,
κ
=
1
)
and
Y
∼
[
Pareto
(
x
m
=
1
λ
(
q
−
1
)
,
α
=
2
−
q
q
−
1
)
−
x
m
]
,
then
X
∼
Y
{\displaystyle {\text{If }}X\sim \operatorname {{\mathit {q}}-Weibull} (q,\lambda ,\kappa =1){\text{ and }}Y\sim \left[\operatorname {Pareto} \left(x_{m}={1 \over {\lambda (q-1)}},\alpha ={{2-q} \over {q-1}}\right)-x_{m}\right],{\text{ then }}X\sim Y\,}
See also
[ edit ]
References
[ edit ]
^ Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a q -Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF) . Milan Journal of Mathematics . 76 : 307–328. doi :10.1007/s00032-008-0087-y . S2CID 55967725 . Retrieved 9 June 2014 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Q-Weibull_distribution&oldid=1051099719 "
C a t e g o r i e s :
● S t a t i s t i c a l m e c h a n i c s
● C o n t i n u o u s d i s t r i b u t i o n s
● P r o b a b i l i t y d i s t r i b u t i o n s w i t h n o n - f i n i t e v a r i a n c e
● T h i s p a g e w a s l a s t e d i t e d o n 2 1 O c t o b e r 2 0 2 1 , a t 1 5 : 4 4 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
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