Unit Weibull distribution

The unit-Weibull distribution (UW) 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

Its 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 its 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:

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

Skewness and kurtosis

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

$𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠=\frac{\mu {\text{'}}_3-3\mu {\text{'}}_2\mu +{\mu }^3}{{\sigma }^3},𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠=\frac{\mu {\text{'}}_4-4\mu {\text{'}}_3\mu +6\mu {\text{'}}_2{\mu }^2-3{\mu }^4}{{\sigma }^4}$

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

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

The likelihood estimate $\widehat{𝜽}$ of $𝜽$ 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 $𝜽=(\alpha ,\beta )$ based on a single observation is given by

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

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:

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

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

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

$\mathit{\text{𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠}}=\frac{2(1-\alpha )}{(2+\alpha )}\sqrt{1+\frac{2}{\alpha }},\mathit{\text{𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠}}=\frac{3(2+\alpha )(2-\alpha +3{\alpha }^2)}{\alpha (3+\alpha )(4+\alpha )}.$

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:

$\mu =\frac{1}{2},{\sigma }^2=\frac{1}{12},\mathit{\text{𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠}}=0,\mathit{\text{𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠}}=\frac{9}{5}.$

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

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

where

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

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