Unit Weibull distribution

Short description: Continuous probability distribution

Unit Weibull

Probability density function Probability density plots of UW distributions
Cumulative distribution function Cumulative density plots of UW distributions
Parameters	$\alpha >0$ (real) $\beta >0$ (real)
Support	$x\in (0,1)$
PDF	$\frac{1}{x}\alpha \beta (-\log x{)}^{\beta -1}\exp [-\alpha (-\log x{)}^{\beta }]$
CDF	$\exp [-\alpha (-\log x{)}^{\beta }]$
Quantile	$\exp [-{\left(\frac{-\log p}{\alpha }\right)}^{\frac{1}{\beta }}],0<p<1$
Skewness	$\frac{\mu {\text{'}}_3-3\mu {\text{'}}_2\mu +{\mu }^3}{{\sigma }^3}$
Kurtosis	$\frac{\mu {\text{'}}_4-4\mu {\text{'}}_3\mu +6\mu {\text{'}}_2{\mu }^2-3{\mu }^4}{{\sigma }^4}$
MGF	$\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{n!{\alpha }^{n/\beta }}\mathrm{\Gamma }(\frac{n}{\beta }+1)$

The unit-Weibull (UW) distribution is a continuous probability distribution with domain on $(0,1)$. Useful for indices and rates, or bounded variables with a $(0,1)$ domain. It was originally proposed by Mazucheli et al using a transformation of the Weibull distribution.

Definitions

Probability density function

It's probability density function is defined as:

$f(x;\alpha ,\beta )=\frac{1}{x}\alpha \beta (-\log x{)}^{\beta -1}\exp [-\alpha (-\log x{)}^{\beta }]$

Cumulative distribution function

And it's cumulative distribution function is:

$F(x;\alpha ,\beta )=\exp [-\alpha (-\log x{)}^{\beta }]$

Quantile function

The quantile function of the UW distribution is given by:

$Q(p)=\exp [-{\left(\frac{-\log p}{\alpha }\right)}^{\frac{1}{\beta }}],0<p<1.$

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 $r$th raw moment of the UW distribution can be obtained through:

$\mu {\text{'}}_r=\mathbb{𝔼}(X^r)=\mathbb{𝔼}(e^{-rY})=M_Y(-r)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{n!{\alpha }^{n/\beta }}\mathrm{\Gamma }(\frac{n}{\beta }+1).$

Skewness and kurtosis

The skewness and kurtosis measures can be obtained upon substituting the raw moments from the expressions:

${s}{k}{e}{w}{n}{e}{s}{s}=\frac{\mu {\text{'}}_3-3\mu {\text{'}}_2\mu +{\mu }^3}{{\sigma }^3},{k}{u}{r}{t}{o}{s}{i}{s}=\frac{\mu {\text{'}}_4-4\mu {\text{'}}_3\mu +6\mu {\text{'}}_2{\mu }^2-3{\mu }^4}{{\sigma }^4}$

Hazard rate

The hazard rate function of the UW distribution is given by:

$h(x;\alpha ,\beta )=\frac{f(x;\alpha ,\beta )}{1-F(x;\alpha ,\beta )}=\frac{\alpha \beta (-\log x{)}^{\beta -1}\exp [-\alpha (-\log x{)}^{\beta }]}{x(1-\exp [-\alpha (-\log x{)}^{\beta }])},0<x<1.$

Parameter estimation

Let $\mathbf{𝐱}=(x_1,\dots ,x_n)$ be a random sample of size $n$ from the UW distribution with probability density function defined before. Then, the log-likelihood function of $\mathit{\theta }=(\alpha ,\beta )$ is:

$\begin{array}{rl}\ell (\mathit{\theta };\mathbf{𝐱})& =n(\log \alpha +\log \beta )-\sum _{i=1}^n\log x_i+(\beta -1)\sum _{i=1}^n\log (-\log x_i)-\alpha \sum _{i=1}^n(-\log x_i{)}^{\beta }\end{array}$

The likelihood estimate $\hat{\mathit{\theta }}$ of $\mathit{\theta }$ is obtained by solving the non-linear equations

$\frac{\partial \ell }{\partial \alpha }=\frac{n}{\alpha }-\sum _{i=1}^n(-\log x_i{)}^{\beta }=0,$

and

$\frac{\partial \ell }{\partial \beta }=\frac{n}{\beta }+\sum _{i=1}^n\log (-\log x_i)-\alpha \sum _{i=1}^n(-\log x_i{)}^{\beta }\log (-\log x_i)=0.$

The expected Fisher information matrix of $\mathit{\theta }=(\alpha ,\beta )$ based on a single observation is given by

$\mathbf{𝐈}(\mathit{\theta })=[I_{ij}]=\left(\begin{array}{cc}\frac{1}{\alpha }& \frac{1}{\alpha \beta }(1-\gamma -\log \alpha )\\ \frac{1}{\alpha \beta }(1-\gamma -\log \alpha )& \frac{1}{{\beta }^2}[\frac{{\pi }^2}{6}+(1-\gamma -\log \alpha {)}^2]\end{array}\right),$

where $\pi \simeq 3.141593$ and $\gamma \simeq 0.577216$ is the Euler’s constant.

Special cases and related distributions

When $\beta =1$, $x$ follows the power function distribution and the $r$th raw moment of the UW distribution becomes:

$\mu {\text{'}}_r=\mathbb{𝔼}(X^r)=\frac{\alpha }{r+\alpha },r=1,2,\dots .$

In this case, the mean, variance, skewness and kurtosis, are:

$\mu =\frac{\alpha }{1+\alpha },{\sigma }^2=\frac{\alpha }{(1+\alpha {)}^2(2+\alpha )},$

$\text{<mrow data-mjx-texclass="ORD">skewness</mrow>}=\frac{2(1-\alpha )}{(2+\alpha )}\sqrt{1+\frac{2}{\alpha }},\text{<mrow data-mjx-texclass="ORD">kurtosis</mrow>}=\frac{3(2+\alpha )(2-\alpha +3{\alpha }^2)}{\alpha (3+\alpha )(4+\alpha )}.$

The skewness can be negative, zero, or positive when $\alpha <1,\alpha =1,\alpha >1$. And if $\alpha =1$, with $\beta =1$, $x$ follows the standard uniform distribution, and the measures becomes:

$\mu =\frac{1}{2},{\sigma }^2=\frac{1}{12},\text{<mrow data-mjx-texclass="ORD">skewness</mrow>}=0,\text{<mrow data-mjx-texclass="ORD">kurtosis</mrow>}=\frac{9}{5}.$

For the case of $\beta =2$, $x$ follows the unit-Rayleigh distribution, and:

$\mu {\text{'}}_r=\mathbb{𝔼}(X^r)=1-\frac{\sqrt{\pi }}{2\sqrt{\alpha }}r{e}^{r^2/(4\alpha )}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{r}{2\sqrt{\alpha }}\right),r=1,2,\dots ,$

where

$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(z)=\frac{2}{\sqrt{\pi }}\underset{z}{\overset{\mathrm{\infty }}{\int }}{e}^{-x^2}dx,z>0,$

Is the complementary error function. In this case, the measures of the distribution are:

$\mu =1-\frac{\sqrt{\pi }}{2\sqrt{\alpha }}{e}^{1/\alpha }\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{1}{2\sqrt{\alpha }}\right),{\sigma }^2=1-\frac{\sqrt{\pi }}{\sqrt{\alpha }}{e}^{1/\alpha }\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{1}{\sqrt{\alpha }}\right)-{[1-\frac{\sqrt{\pi }}{2\sqrt{\alpha }}{e}^{1/\alpha }\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{1}{2\sqrt{\alpha }}\right)]}^2.$

Applications

It was shown to outperform, against other distributions, like the Beta and Kumaraswamy distributions, in: maximum flood level, petroleum reservoirs, risk management cost effectiveness, and recovery rate of CD34+cells data.

See also

• Weibull distribution