The analysis and modeling of lifetime data are crucial in almost all applied sciences including medicine, insurance, engineering, behavioral sciences and finance, amongst others. The main objective of this paper is to have a comparative study of two-parameter gamma and Weibull distributions for modeling lifetime data from various fields of knowledge. Since exponential distribution is a particular case of both gamma and Weibull distributions and the exponential distribution is a classical distribution for modeling lifetime data, the goodness of fit of both gamma and Weibull distributions are compared with exponential distribution.

Keywords: gamma distribution, weibull distribution, exponential distribution, lifetime data, estimation of parameter, goodness of fit

The lifetime or survival time or failure time in reliability analysis is the time to the occurrence of event of interest. The event may be failure of a piece of equipment, death of a person, development (or remission) of symptoms of disease, health code violation (or compliance). The modeling and statistical analysis of lifetime data are crucial for statisticians and research workers in almost all applied sciences including behavioral sciences, engineering, medical science/biological science, insurance and finance, amongst others.

The statistics literature is flooded with lifetime distributions including exponential distribution, gamma distribution, Lindley distribution, Weibull distribution and their generalizations, some amongst others.

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of two-parameter gamma distribution (GD) having parameters $\theta$ and $\alpha$ are given by

$f_1\left(x;\theta ,\alpha \right)=\frac{{\theta }^{\alpha }}{\Gamma \left(\alpha \right)}{x}^{\alpha -1}{e}^{-\theta x};x>0,\theta >0,\alpha >0$ (2.1)

$F_2\left(x;\theta ,\alpha \right)=1-\frac{\Gamma \left(\alpha ,\theta x\right)}{\Gamma \left(\alpha \right)};x>0,\theta >0,\alpha >0$ (2.2)

Where $\Gamma \left(\alpha ,z\right)$ is the upper incomplete gamma function defined as

$\Gamma \left(\alpha ,z\right)={\displaystyle \underset{z}{\overset{\infty }{\int }}{e}^{-y}{y}^{\alpha -1}dy;\alpha >0,z\ge 0}$ (2.3)

It can be easily shown that the gamma distribution reduces to classical exponential distribution for $\alpha =1$ having p.d.f. and c.d.f.

$f_2\left(x;\theta \right)=\theta e^{-\theta x};x>0,\theta >0$ (2.4)

$F_2\left(x;\theta \right)=1-e^{-\theta x};x>0,\theta >0$ (2.5)

It should be noted that the gamma distribution is the weighted exponential distribution. Stacy¹ obtained the generalization of the gamma distribution. Stacy & Mihram² have detailed discussion about parametric estimation of generalized gamma distribution.

The p.d.f. and the c.d.f. of two-parameter Weibull distribution having parameters $\theta$ and $\alpha$ are given by

$f_3\left(x;\theta ,\alpha \right)=\theta \alpha x^{\alpha -1}{e}^{-\theta x^{\alpha }};x>0,\theta >0,\alpha >0$ (3.1)

$F_3\left(x;\theta ,\alpha \right)=1-e^{-\theta x^{\alpha }};x>0,\theta >0,\alpha >0$ (3.2)

It can be easily shown that the Weibull distribution reduces to classical exponential distribution at $\alpha =1$ . It should be noted that Weibull distribution is nothing but the power exponential distribution.

Taking $x=y^{\frac{1}{\alpha }}$ and thus $y=x^{\alpha }$ in (2.4), we have

$$g\left(y;\theta ,\alpha \right)=f\left(y^{\alpha }\right)f^{\prime }\left(y\right)=\theta e^{-\theta y^{\alpha }}\alpha y^{\alpha -1}=\theta \alpha y^{\alpha -1}{e}^{-\theta y^{\alpha }}$$

Which is the p.d.f. of Weibull distribution defined in (3.1)

Maximum likelihood estimates of the parameters of gamma distribution (GD): Assuming $\left(x_1,x_2,x_3,\mathrm{...},x_n\right)$ be a random sample of size $n$ from Gamma distribution (2.1), the likelihood function is given by

$L={\left(\frac{{\theta }^{\alpha }}{\Gamma \left(\alpha \right)}\right)}^n{\left({\displaystyle \prod _{i=1}^n{x}_i}\right)}^{\alpha -1}{e}^{-n\theta \overline{x}}$ , $\overline{x}$ being the sample mean

The natural log likelihood function, ln L of Gamma distribution is thus given by

$$\ln L=n\left[\alpha \ln \theta -\ln \left(\Gamma \left(\alpha \right)\right)\right]+\left(\alpha -1\right){\displaystyle \sum _{i=1}^n\ln x_i}-n\theta \overline{x}$$

The maximum likelihood estimate (MLE) $\widehat{\theta }\text{and}\widehat{\alpha }$ of parameters $\theta \text{and}\alpha$ of gamma distribution can be obtained by solving the natural log likelihood equation using R software (Package Stat 4).

Maximum likelihood estimates of the parameters of weibull distribution

Assuming $\left(x_1,x_2,x_3,\mathrm{...},x_n\right)$ be a random sample of size $n$ from GD (3.1), the natural log likelihood function, ln of Weibull distribution is given by

$\ln L={\displaystyle \sum _{i=1}^n\ln f\left(x_i;\theta ,\alpha \right)}$ $=n\left(\ln \theta +\ln \alpha \right)+\left(\alpha -1\right){\displaystyle \sum _{i=1}^n\ln x_i}-\theta {\displaystyle \sum _{i=1}^n{x}_i{}^{\alpha }}$

The maximum likelihood estimate (MLE) $\widehat{\theta }\text{and}\widehat{\alpha }$ of parameters $\theta \text{and}\alpha$ of Weibull distribution can be obtained by solving the natural log likelihood equation using R software (Package Stat 4).

None.

Author declares that there are no conflicts of interest.

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