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
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