- Source: Q-Weibull distribution
In statistics, the q-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution.
Characterization
= Probability density function
=The probability density function of a q-Weibull random variable is:
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
= Cumulative distribution function
=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
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
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
Constantino Tsallis
Tsallis statistics
Tsallis entropy
Tsallis distribution
q-Gaussian
References
Kata Kunci Pencarian:
- Q-Weibull distribution
- Weibull distribution
- Discrete Weibull distribution
- Q distribution
- Fréchet distribution
- Tsallis distribution
- Exponential distribution
- Generalized extreme value distribution
- Weibull
- Poisson distribution
