In statistics, the q -Weibull distributionprobability 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:[ 1]
f
(
x
;
q
,
λ λ -->
,
κ κ -->
)
=
{
(
2
− − -->
q
)
κ κ -->
λ λ -->
(
x
λ λ -->
)
κ κ -->
− − -->
1
e
q
(
− − -->
(
x
/
λ λ -->
)
κ 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 κ -->
{\displaystyle \kappa }
shape parameters scale parameter
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
The cumulative distribution function of a q -Weibull random variable is:
{
1
− − -->
e
q
′
− − -->
(
x
/
λ λ -->
′
)
κ 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 -->
′
=
λ λ -->
(
2
− − -->
q
)
1
κ 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 μ -->
(
q
,
κ κ -->
,
λ λ -->
)
=
{
λ λ -->
(
2
+
1
1
− − -->
q
+
1
κ κ -->
)
(
1
− − -->
q
)
− − -->
1
κ κ -->
B
[
1
+
1
κ κ -->
,
2
+
1
1
− − -->
q
]
q
<
1
λ λ -->
Γ Γ -->
(
1
+
1
κ κ -->
)
q
=
1
λ λ -->
(
2
− − -->
q
)
(
q
− − -->
1
)
− − -->
1
+
κ κ -->
κ κ -->
B
[
1
+
1
κ κ -->
,
− − -->
(
1
+
1
q
− − -->
1
+
1
κ κ -->
)
]
1
<
q
<
1
+
1
+
2
κ κ -->
1
+
κ κ -->
∞ ∞ -->
1
+
κ κ -->
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()}
Beta function and
Γ Γ -->
(
)
{\displaystyle \Gamma ()}
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 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 -->
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 κ -->
{\displaystyle \kappa }
α α -->
=
2
− − -->
q
q
− − -->
1
,
λ λ -->
Lomax
=
1
λ 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 κ -->
=
1
{\displaystyle \kappa =1}
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 λ -->
,
κ κ -->
=
1
)
and
Y
∼ ∼ -->
[
Pareto
-->
(
x
m
=
1
λ λ -->
(
q
− − -->
1
)
,
α 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
References
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bde051d6ff2a26591683f8e2698aad8fd9796f4)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f93a58bf0696cb2e7530ed63b89a818d8cdc6f06)
![{\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\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08518f98bda4cb98bc57fc441e716f351966f504)
