Unit Gompertz distribution

Short description: Continuous probability distribution

unit-Gompertz Distribution

Probability density function
Cumulative distribution function
Parameters	$\alpha >0$ (real) $\beta >0$ (real)
Support	$x\in (0,1)$
PDF	$\alpha \beta x^{-(\beta +1)}\exp [-\alpha (x^{-\beta }-1)]$
CDF	$\exp [-\alpha (x^{-\beta }-1)]$
Quantile	$\exp [-\frac{1}{\beta }\log (\alpha -\log p)-\log \alpha ]$
Mean	${\alpha }^{\frac{1}{\beta }}{e}^{\alpha }\mathrm{\Gamma }(1-\frac{1}{\beta },\alpha )$
Mode	$x_0={\left(\frac{\alpha \beta }{\beta +1}\right)}^{\frac{1}{\beta }}$
Variance	${\alpha }^{\frac{2}{\beta }}{e}^{2\alpha }\{\mathrm{\Gamma }(1-\frac{2}{\beta },\alpha )-{\left[\mathrm{\Gamma }(1-\frac{1}{\beta },\alpha )\right]}^2\}$
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	${\alpha }^{\frac{r}{\beta }}{e}^{\alpha }\mathrm{\Gamma }(1-\frac{r}{\beta },\alpha )$

The unit-Gompertz distribution (UGo) is a continuous probability distribution with domain on $(0,1)$. Useful for bounded variables with a $(0,1)$ domain. It was originally proposed by Mazucheli et al using a transformation of the Gompertz distribution. It is a suitable alternative for fitting skewed data. A quantile regression model has also been derived from the distribution.

Definitions

Probability density function

Its probability density function is defined as:

$f(x\mid \alpha ,\beta )=\alpha \beta x^{-(\beta +1)}\exp [-\alpha (x^{-\beta }-1)].$

Cumulative distribution function

And its cumulative distribution function is:

$F(x\mid \alpha ,\beta )=\exp [-\alpha (x^{-\beta }-1)].$

Quantile function

The quantile function of the UGo distribution is given by:

$Q(p\mid \alpha ,\beta )=\exp [-\frac{1}{\beta }\log (\alpha -\log p)-\log \alpha ].$

Properties

Moments

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

$\mu {\text{'}}_r=\mathbb{𝔼}(X^r)=\underset{0}{\overset{1}{\int }}{x}^r\alpha \beta x^{-(\beta +1)}\exp [-\alpha (x^{-\beta }-1)]dx={\alpha }^{\frac{r}{\beta }}{e}^{\alpha }\mathrm{\Gamma }(1-\frac{r}{\beta },\alpha ),$

where $\mathrm{\Gamma }(\cdot ,\cdot )$ is the upper incomplete gamma function. The moments exists only when $\frac{r}{\beta }<1$.

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 UGo distribution is given by:

$h(x\mid \alpha ,\beta )=\frac{\alpha \beta x^{-(\beta +1)}\exp [-\alpha (x^{-\beta }-1)]}{1-\exp [-\alpha (x^{-\beta }-1)]},0<x<1.$

Parameter estimation

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

$\begin{array}{r}\ell (\mathbf{𝐱}\mid \mathit{\theta })=n\log \alpha +n\log \beta -(\beta +1)\sum _{i=1}^n\log x_i-\alpha \sum _{i=1}^n{x}_i^{-\beta }+n\alpha .\end{array}$

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

$\frac{\partial \ell }{\partial \alpha }=n(1+\frac{1}{\alpha })-\sum _{i=1}^n{x}_i^{-\beta },$

and

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

The Hessian matrix of the log-likelihood with respect to the unknown parameters is given by

$\mathbf{𝐇}=\left[\begin{array}{ccc}-\frac{n}{{\alpha }^2}& ⋮& \sum _{i=1}^n\frac{\log x_i}{x_i^{\beta }}\\ \cdots \cdots \cdots \cdots & \cdot & \cdots \cdots \cdots \cdots \cdots \cdots \\ \sum _{i=1}^n\frac{\log x_i}{x_i^{\beta }}& ⋮& -\alpha \sum _{i=1}^n\frac{(\log x_i{)}^2}{x_i^{\beta }}+\frac{n}{{\beta }^2}\end{array}\right]$

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

$\mathbf{𝐈}(\mathit{\theta })=n\left[\begin{array}{ccc}\frac{1}{{\alpha }^2}& ⋮& \frac{e^{\alpha }{E}_1(\alpha )+1}{\beta \alpha }\\ \cdots \cdots \cdots \cdots \cdots & \cdot & \cdots \cdots \cdots \cdots \cdots \cdots \\ \frac{e^{\alpha }{E}_1(\alpha )+1}{\beta \alpha }& ⋮& {\alpha }^2\beta I_{22}+\frac{1}{{\beta }^2}\end{array}\right],$

where $E_n(\cdot )$ represents the exponential integral function defined as

$E_k(x)=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{{\mathrm{e}}^{-xt}}{t^k}dt=x^{k-1}\mathrm{\Gamma }(1-k,x),$

and

$I_{22}=\underset{0}{\overset{1}{\int }}\left[\frac{(\log x{)}^2}{x^{\beta }}\right]x^{-(\beta +1)}\exp [-\alpha (x^{-\beta }-1)]dx.$

Applications

It was shown to outperform, against other distributions, like the beta and Kumaraswamy distributions, in: maximum flood level, literacy rates, and stress-strength reliability estimation data.

See also

• Gompertz distribution