Chen distribution

Short description: Continuous probability distribution

Chen Distribution

Probability density function
Cumulative distribution function
Parameters	$\lambda >0$ (real) $\beta >0$ (real)
Support	$x>0$
PDF	$f(x)=a\beta x^{\beta -1}\exp \{x^{\beta }+a(1-{\mathrm{e}}^{x^{\beta }})\}\cdot {\mathbf{𝟏}}_{{\mathbb{ℝ}}_{+}}(x)$
CDF	$F(x)=1-{\mathrm{e}}^{\lambda (1-{\mathrm{e}}^{x^{\beta }})}$
Quantile	$Q(p;a,\beta )={[\ln \{1-\ln (1-p)a^{-1}\}]}^{{\beta }^{-1}}\cdot {\mathbf{𝟏}}_{(0,1)}(p)$
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	${\mathbb{𝕄}}_T(s)=a{\mathrm{e}}^a\underset{1}{\overset{\mathrm{\infty }}{\int }}{\mathrm{e}}^{-s(\ln x{)}^{{\beta }^{-1}}-ax}\mathrm{d}x$

The Chen Distribution is a continuous probability distribution with domain on $(0,\inf )$. It was originally proposed by Zhenmin Chen as an alternate two-parameter lifetime distribution. The motivation was the need to model bathtub-shaped failure rates, useful for reliability and survival analysis. The failure rate function can also show an increasing shape.

Definitions

Probability density function

Its probability density function is defined as:

$f(x;\lambda ,\beta )=\lambda \beta x^{\beta -1}\exp \{x^{\beta }+\lambda (1-{\mathrm{e}}^{x^{\beta }})\}$

where $x>0$, and $\lambda >0$ and $\beta >0$ are shape parameters.

Cumulative distribution function

And its cumulative distribution function is:

$F(x;\lambda ,\beta )=1-{\mathrm{e}}^{\lambda (1-{\mathrm{e}}^{x^{\beta }})}$

where $x>0$.

Quantile function

The quantile function of the Chen distribution is given by:

$Q(p;\lambda ,\beta )={[\ln \{1-\frac{\ln (1-p)}{\lambda }\}]}^{\frac{1}{\beta }}$

where $0<p<1$.

Properties

Moments

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

$\mu {\text{'}}_r=\mathbb{𝔼}[X^r]=\lambda {\mathrm{e}}^{\lambda }\underset{1}{\overset{\mathrm{\infty }}{\int }}{\mathrm{e}}^{-\lambda x}(\ln x{)}^{\frac{r}{\beta }}\mathrm{d}x=\lambda {\mathrm{e}}^{\lambda }\mathrm{\Gamma }(\frac{r}{\beta }+1)E_0^{\frac{r}{\beta }}(\lambda )$

for $r>0$.

Skewness and kurtosis

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

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

Hazard rate

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

$h(x;\lambda ,\beta )=\lambda \beta x^{\beta -1}{\mathrm{e}}^{x^{\beta }}$

where $x>0$.

Parameter estimation

Let $\mathbf{𝐱}=(x_1,\dots ,x_n)$ be a random sample of size $n$ from the Chen distribution with cumulative probability function defined before. Then, the likelihood function of $(\lambda ,\beta )$ based on the first $k$ order statistics is given by

$L(\lambda ,\beta )=\frac{n!}{(n-k)!}{\lambda }^k{\beta }^k\prod _{i=1}^k{x}_{(i)}^{\beta -1}{\mathrm{e}}^{\sum _{i=1}^k{x}_{(i)}^{\beta }+k\lambda -\lambda \sum _{i=1}^k{\mathrm{e}}^{x_{(i)}^{\beta }}+(n-k)\lambda -(n-k)\lambda {\mathrm{e}}^{x_{(k)}^{\beta }}}.$

The components of the score vector $\mathit{U}\left(\mathit{\theta }\right)={[\frac{\partial \ell (\mathit{\theta })}{\partial \lambda },\frac{\partial \ell (\mathit{\theta })}{\partial \beta }]}^{\mathrm{\top }}$ are

$\begin{array}{r}\frac{\partial \ell (\mathit{\theta })}{\partial \lambda }=\frac{k}{\lambda }-\sum _{i=1}^k{\mathrm{e}}^{x_{(i)}^{\beta }}+n+(n-k){\mathrm{e}}^{x_{(k)}^{\beta }}\end{array}$

and

$\begin{array}{r}\frac{\partial \ell (\mathit{\theta })}{\partial \beta }=\frac{k}{\beta }+\sum _{i=1}^k\log x_{(i)}+\sum _{i=1}^k{x}_{(i)}^{\beta }\log x_{(i)}-\lambda [\sum _{i=1}^k{\mathrm{e}}^{x_{(i)}^{\beta }}{x}_{(i)}^{\beta }\log x_{(i)}+(n-k){\mathrm{e}}^{x_{(k)}^{\beta }}\log x_{(k)}]\end{array}$

The MLEs of $\mathit{\theta }$, denoted by $\hat{\mathit{\theta }}={(\hat{\lambda },\hat{\beta })}^{\mathrm{\top }}$, are obtained as the simultaneous solution of $\mathit{U}(\mathit{\theta })=\mathit{0}$, where $\mathit{0}$ is a two-dimensional null vector.

Specifically, $\hat{\beta }$ is obtained by solving the non-linear equation:

$\frac{k}{\beta }+\sum _{i=1}^k\log x_{(i)}+\sum _{i=1}^k{x}_{(i)}^{\beta }\log x_{(i)}-\frac{k[\sum _{i=1}^k{\mathrm{e}}^{x_{(i)}^{\beta }}{x}_{(i)}^{\beta }\log x_{(i)}+(n-k){\mathrm{e}}^{x_{(k)}^{\beta }}{x}_{(k)}^{\beta }\log x_{(k)}]}{\sum _{i=1}^k{\mathrm{e}}^{x_{(i)}^{\beta }}-n-(n-k){\mathrm{e}}^{x_{(k)}^{\beta }}}=0$

and $\hat{\lambda }$ is subsequently given in closed form by:

$\hat{\lambda }=\frac{k}{\sum _{i=1}^k{\mathrm{e}}^{x_{(i)}^{\hat{\beta }}}-n-(n-k){\mathrm{e}}^{x_{(k)}^{\hat{\beta }}}}$

Applications

The Chen distribution was utilized in:

• Reliability inference for lifetime data in oil breakdown and cancer patient survival data.
• Graft survival times of renal transplant patients.