The unit-Weibull distribution (UW) is a continuous probability distribution with domain on ( 0 , 1 ) {\displaystyle (0,1)} . Useful for indices and rates, or bounded variables with a ( 0 , 1 ) {\displaystyle (0,1)} domain. It was originally proposed by Mazucheli et al using a transformation of the Weibull distribution.
Definitions
Probability density function Its probability density function is defined as:
f ( x ; α , β ) = 1 x α β ( − log x ) β − 1 exp [ − α ( − log x ) β ] {\displaystyle f(x;\alpha ,\beta )={\frac {1}{x}}\,\alpha \,\beta \,(-\log x)^{\beta -1}\exp \left[-\alpha \,(-\log x)^{\beta }\right]}
Cumulative distribution function And its cumulative distribution function is:
F ( x ; α , β ) = exp [ − α ( − log x ) β ] {\displaystyle F(x;\alpha ,\beta )=\exp \left[-\alpha \,(-\log x)^{\beta }\right]}
Quantile function The quantile function of the UW distribution is given by:
Q ( p ) = exp [ − ( − log p α ) 1 β ] , 0 < p < 1. {\displaystyle Q(p)=\exp \left[-\left({\frac {-\log p}{\alpha }}\right)^{\frac {1}{\beta }}\right],\quad 0<p<1.}
Having a closed form expression for the quantile function, may make it a more flexible alternative for a quantile regression model against the classical Beta regression model.
Properties
Moments The r {\displaystyle r} th raw moment of the UW distribution can be obtained through:
μ r ′ = E ( X r ) = E ( e − r Y ) = M Y ( − r ) = ∑ n = 0 ∞ ( − 1 ) n n ! α n / β Γ ( n β + 1 ) . {\displaystyle \mu '_{r}=\mathbb {E} (X^{r})=\mathbb {E} (e^{-rY})=M_{Y}(-r)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\,\alpha ^{n/\beta }}}\,\Gamma \left({\frac {n}{\beta }}+1\right).}
Skewness and kurtosis The skewness and kurtosis measures can be obtained upon substituting the raw moments from the expressions:
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