The analysis and modeling of lifetime data are crucial in almost all applied sciences including behavioral sciences, medicine, insurance, engineering, and finance, amongst others. In this paper an attempt has been made for comparative study of generalized Lindley distribution (GLD) introduced by Zakerzadeh & Dolati¹ and generalized gamma distribution (GGD) introduced by Stacy² for modeling lifetime data from different fields of knowledge. The goodness of fit for both GLD and GGD , based on maximum likelihood estimates, shows that GGD gives much closer fit than GLD in majority of data sets and hence GGD can be considered as an important tool for modeling lifetime data over GLD.

Keywords: generalized lindley distribution, generalized gamma distribution, lifetime data, estimation of parameter, goodness of fit

In reliability analysis the time to the occurrence of event of interest is the lifetime or survival time or failure time. 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 statistical analysis and the modeling 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.

A number of lifetime distributions are available for modeling lifetime data in statistics including exponential distribution, gamma distribution, Lindley distribution, Weibull distribution and their generalizations, some amongst others.

In this paper various lifetime data have been considered for modeling using three- parameter generalized Lindley distribution (GLD) and generalized gamma distribution (GGD) because gamma, Lindley and exponential distributions are particular cases of GLD whereas gamma, Weibull and exponential distributions are particular cases of GGD.

The probability density function of three-parameter generalized Lindley distribution (GLD) introduced by Zakerzadeh & Dolati¹ having parameters $\alpha ,\beta ,\text{and}\theta$  is given by

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

Clearly the gamma distribution, the Lindley³ distribution and the exponential distribution are particular cases of (2.1) for $\left(\beta =0\right)$ , $\left(\alpha =\beta =1\right)$  and $\left(\alpha =1,\beta =0\right)$  respectively. The discussion about its properties, estimation of parameters and applications are available in Zakerzadeh & Dolati.¹ Ghitany et al.⁴ have detailed study about various properties of Lindley distribution, estimation of parameter and application for modeling waiting time data in a bank. Shanker et al.⁵ have detailed and comparative study about modeling of lifetime data using one parameter Lindley and exponential distributions.

 The corresponding distribution function of the GLD can be     obtained as

$$F_1\left(x;\alpha ,\beta ,\theta \right)=1-\frac{\alpha \left(\beta +\theta \right)\Gamma \left(\alpha ,\theta x\right)+\beta {\left(\theta x\right)}^{\alpha }{e}^{-\theta x}}{\left(\beta +\theta \right)\Gamma \left(\alpha +1\right)}$$ ; $x>0,\alpha >0,\beta >0,\theta >0$ (2.2)

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

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

Recently Shanker⁶ has detailed study about GLD and obtained expressions for coefficient of variation, skewness, kurtosis and index of dispersion. Shanker⁶ has also studied its hazard rate function and the mean residual life function.

The probability density function of three-parameter generalized gamma distribution (GGD) introduced by Stacy² having parameters $\alpha ,\beta ,\text{and}\theta$  is given by

$f_2\left(x;\alpha ,\beta ,\theta \right)=\frac{\beta {\theta }^{\alpha }}{\Gamma \left(\alpha \right)}{x}^{\beta \alpha -1}{e}^{-\theta x^{\beta }};x>0,\alpha >0,\beta >0,\theta >0$ (3.1)

 Where $\alpha$ and $\beta$  are the shape parameter and $\theta$  is the scale parameter. Clearly the gamma distribution, the Weibull distribution and the exponential distribution are particular cases of (3.1) for $\left(\beta =1\right)$ , $\left(\alpha =1\right)$  and $\left(\alpha =\beta =1\right)$  respectively. Detailed discussion about GGD is available in Stacy² and parametric estimation for the GGD is available in Stacy & Mihram.⁷

 The cumulative distribution function of the GGD is thus given by

$$F_2\left(x;\alpha ,\beta ,\theta \right)=1-\frac{\Gamma \left(\alpha ,\theta x^{\beta }\right)}{\Gamma \left(\alpha \right)}$$ (3.2)

Where $\Gamma \left(\alpha ,z\right)$ is the upper incomplete gamma function defined in (2.3)

Maximum likelihood estimates of the parameters of GLD

Assuming $\left(x_1,x_2,x_3,\mathrm{...},x_n\right)$  be a random sample of size $n$  from GLD (2.1), the likelihood function, L of GLD is given by

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

The natural log likelihood function is thus obtained as

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

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

Maximum likelihood estimates of the parameters of GGD

Let $\left(x_1,x_2,x_3,\mathrm{...},x_n\right)$  be a random sample of size $n$  from GGD (3.1). Then

$f\left(x_i\right)=\frac{\beta {\theta }^{\alpha }}{\Gamma \left(\alpha \right)}{\left(x_i\right)}^{\beta \alpha -1}{e}^{-\theta {\left(x_i\right)}^{\beta }}$

This gives

$\ln f\left(x_i\right)=\ln \beta +\alpha \ln \theta -\ln \left(\Gamma \left(\alpha \right)\right)+\left(\beta \alpha -1\right)\ln \left(x_i\right)-\theta {\left(x_i\right)}^{\beta }$

Thus the natural log likelihood function of the GGD is given by

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

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

In this sectioNewn, the goodness of fit and applications of GLD and GGD have been discussed for several lifetime data In order to compare GLD and GGD, $-2\ln L$  and K-S Statistics Kolmogorov-Smirnov Statistics) for eighteen data sets have been computed and presented in Table 1.

The formula for K-S Statistics is defined as follow:

$K\text{-}S=\underset{x}{\text{Sup}}\left|F_n\left(x\right)-F_0\left(x\right)\right|$ , where $F_n\left(x\right)$ is the empirical distribution function. The best distribution corresponds to lower values of $-2\ln L$ and K-S statistics and higher p-values. It is clear from the goodness of fit of GLD and GGD that in most of the data sets except Data sets (1-18) GGD gives much closer fit than GLD for modeling lifetime data.

The modeling and analysis of lifetime data using lifetime distributions are crucial in almost all applied sciences including behavioral sciences, medicine, insurance, engineering, and finance, amongst others. In this paper an attempt has been made to have a comparative study on modeling of lifetime data on eighteen data sets using three parameters generalized Lindley distribution (GLD) introduced by Zakerzadeh & Dolati² and generalized gamma distribution (GGD) introduced by Stacy.² Maximum likelihood estimates have been used for fitting both GLD and GGD. The goodness of fit for both GLD and GGD shows that GGD gives much closer fit than GLD in majority of data sets and hence GGD can be considered as an important tool for modeling lifetime data over GLD.

None.

Author declares that there are no conflicts of interest.

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