We propose a new lifetime model called the odd log-logistic generalized gamma distri­bution that can be easily interpreted. Some of its special models are discussed. We obtain general mathematical properties of this distribution including the ordinary moments, and quantile functions. We discuss parameter estimation by the maximum likelihood method and a Bayesian approach, where Gibbs algorithms along with metropolis steps are used to obtain the posterior summaries of interest for survival data with right censoring. Further, for different parameter settings, sample sizes and censoring percentages, we perform various simulations and evaluate the behavior of the estimators. The potentiality of the new distri­bution is proved by means of two real data sets. In fact, the new distribution can produce better fits than some well-known distributions.

Keywords:censored data, exponentiated distribution, generalized gamma distribution, moments, survival analysis

The statistics literature is filled with hundreds of continuous univariate distributions. Recent developments focus on new techniques for building meaningful models. More recently, seve­ral methods of introducing one or more parameters to generate new distributions have been proposed. Among these methods, the compounding of some discrete and important lifetime distributions has been in the vanguard of lifetime modeling. So, several families of distributions were investigated by compounding some useful lifetime and truncated discrete distributions. The log-logistic (LL) distribution with a shape parameter $\lambda >0$ is a useful model for survival analysis and it is an alternative to the log-normal distribution. Unlike the more commonly used Weibull distribution, the LL distribution has a non-monotonic hazard rate function (hrf), which makes it suitable for modeling cancer survival data. For $\lambda >1$ , the hrf is unimodal and when $\lambda =1$ , the hazard decreases monotonically. The fact that its cumulative distribution function (cdf) has a closed-form is particularly useful for analysis of survival data with censoring.

The odd log-logistic (OLL) family of distributions was pioneered by Gleaton and Lynch;¹ they called this family the generalized log-logistic (GLL) family. Recently, Braga et al.² studied the odd log-logistic normal distribution, da Cruz et al.³ proposed the odd log-logistic Weibull distribution and Cordeiro et al.² proposed the beta odd log-logistic generalized family. We develop a similar methodology to propose a new model based on the generalized gamma (GG) distribution. The GG distribution plays a very important role in sta­tistical inferential problems. When modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes. However, the Weibull distribution does not provide a reasonable parametric fit for modeling phenomenon with bathtub shaped and unimodal failure rates, which are common in biological and reliability studies. Alternatively, other extensions of the GG distribution were developed for modeling lifetime data. For example, Cordeiro et al.⁴ defined the exponentiated generalized gamma with applications, Pascoa et al.⁵ introduced the Kumaraswamy generalized gamma distri­bution, Ortega et al.⁶ proposed the generalized gamma geometric distribution, Cordeiro et al.⁷ studied the beta generalized gamma distribution and, more recently, Lucena et al.⁸ defines the transmuted generalized gamma distribution and Silva et al.⁹ proposed the generalized gamma power series class.

Given a continuous baseline cdf $G\left(t;\xi \right)$ with a parameter vector $\xi$ , the cdf of the odd log-logistic-G (“OLL-G” for short) distribution with an extra shape parameter $\lambda >0$ is defined by

$F\left(t\right) = {\int }_0{}^{\frac{G\left(t;\xi \right)}{\overline{G}\left(t;\xi \right)}}\frac{\lambda x^{\lambda -1}}{{\left(1+x^{\lambda }\right)}^2}dx=\frac{G{\left(t;\xi \right)}^{\lambda }}{G{\left(t;\xi \right)}^{\lambda }+\overline{G}{\left(t;\xi \right)}^{\lambda }}.$ (1)

We can write

$\lambda =\frac{\log \left[\frac{F\left(t\right)}{\overline{F}\left(t\right)}\right]}{\log \left[\frac{G\left(t\right)}{\overline{G}\left(t\right)}\right]}$ and $\overline{G}\left(t;\xi \right) = 1-G\left(t;\xi \right)$

So, the parameter $\lambda$ represents the quotient of the log odds ratio for the generated and baseline distributions. We note that there is no complicated function in equation (1) in contrast with the beta generalized family (Eugene et al.,¹⁰), which includes two extra parameters and also involves the beta incomplete function. The baseline cdf $G\text{(}t\text{;}\xi \text{)}$ is clearly a special case of (1) when $G\text{(}t\text{;}\xi \text{)}$ . If $G\text{(}t\text{;}\xi \text{) = }t\text{/(1+}t\text{)}$ , it becomes the LL distribution. Several distributions can be generated from equation (1). For example, the odd log-logistic Fréchet and odd log-logistic gamma distributions are obtained by taking $G\text{(}t\text{;}\xi \text{)}$ to be the Fréchet and gamma cumulative distributions, respectively. The probability density function (pdf) of the new family is given by

$f\text{(}t\text{) =}\frac{\lambda g\left(t;\xi \right){\left\{G\left(t;\xi \right)\left[1-G\left(t;\xi \right)\right]\right\}}^{\lambda -1}}{{\left\{G{\left(t;\xi \right)}^{\lambda }+\left[1+G{\left(t;\xi \right)}^{\lambda }\right]\right\}}^2}$ (2)

The OLL-G family of densities (2) allows for greater flexibility of its tails and can be widely applied in many areas of engineering and biology. We can study some of its mathematical properties because it extends several well-known distributions.

The inferential part of this model is carried out using the asymptotic distribution of the maximum likelihood estimators (MLEs), which in situations when the sample size is small or moderate, might lead to poor inference on the model parameters. Hence, in this paper, we also explore the Markov Chain Monte Carlo (MCMC) techniques to develop a Bayesian inference as an alternative analysis for the model. So, we discuss the inference aspects of the OLL-G model following both a classical and a Bayesian approach.

The rest of the paper is organized as follows. In Section 2, we define the odd log-logistic generalized gamma (OLLGG) distribution and present some special cases. Section 3 provides a useful linear representation for the OLLGG density function. We derive in Section 4 some structural properties of the new distribution. Considering censored data, we adopt a classic analysis for the parameters of the model in Section 5. In Section 6, the Bayesian approach is considered using MCMC with Metropolis-Hasting algorithms steps to obtain the posterior summaries of interest. In Section 7, we present results from various simulation studies displayed graphically and commented. Two applications to real data are performed in Section 8. Some concluding remarks are given in Section 9.

First, we define the exponentiated-generalized gamma (“Exp-GG”) distribution, say $W~Ex{p}^c\left(GG\right)$ with power parameter $c>0$ , if $W$ has cdf and pdf given by

$H_c\left(t\right)=G{\left(t;\alpha ,\tau ,k\right)}^c$ and  $h_c\left(t\right)=\frac{c\tau }{\alpha \Gamma \left(k\right)}{\left(\frac{t}{\alpha }\right)}^{\tau k-1}G{\left(x;\alpha ,\tau ,k\right)}^{c-1},$

respectively. In a general context, the properties of the exponentiated-G (Exp-G) distributions have been studied by several authors for some baseline G models, see Mudholkar and Srivastava¹⁴ and Mudholkar et al.¹⁵ for exponentiated Weibull, Nadarajah¹⁶ for exponentiated Gumbel, Shirke and Kakade.¹⁷ for exponentiated log-normal and Nadarajah and Gupta¹⁸ for exponentiated gamma distributions. See, also, Nadarajah and Kotz,¹⁹ among others.

First, we obtain an expansion for $F\left(t;\alpha ,\tau ,k,\lambda \right)$ using a power series for $G{\left(t;\alpha ,\tau ,k\right)}^{\lambda }$  ( $\lambda >0$ real)

$G{\left(t;\alpha ,\tau ,k\right)}^{\lambda }={{\displaystyle \sum _{j=0}^{\infty }{a}_{j}G\left(t;\alpha ,\tau ,k\right)}}^j,$  (9)

where

$a_j=a_j\left(\lambda \right){{\displaystyle \sum _{\upsilon =j}^{\infty }\left(-1\right)}}^{j+\upsilon }\left({}_{\upsilon }^{\lambda }\right)\left({}_j^{\upsilon }\right).$

For any real $\lambda >0$ , we consider the generalized binomial expansion

${\left[1-G\left(t;\alpha ,\tau ,k\right)\right]}^{\lambda }={{\displaystyle \sum _{j=0}^{\infty }\left(-1\right)}}^j\left(\begin{array}{c}\lambda \\ j\end{array}\right)G{\left(t;\alpha ,\tau ,k\right)}^j.$   (10)

Inserting (9) and (10) in equation (1), we obtain

$F\left(t;\alpha ,\tau ,k,\lambda \right)=\frac{{\displaystyle {\sum }_{j=0}^{\infty }{a}_{j}G{\left(t;\alpha ,\tau ,k\right)}^j}}{{\displaystyle {\sum }_{j=0}^{\infty }{b}_{j}G{\left(t;\alpha ,\tau ,k\right)}^j}},$

where $b_j=a_j+{\left(-1\right)}^j\left(\begin{array}{c}\lambda \\ j\end{array}\right)$ for $j\ge 0.$

The ratio of the two power series can be expressed as

$F\left(t;\alpha ,\tau ,k,\lambda \right)={{\displaystyle \sum _{j=0}^{\infty }{c}_{j}G\left(t;\alpha ,\tau ,k\right)}}^j,$               (11)

where $c_0={\text{a}}_0/b_0$ and the coefficients $c_j$ ’s (for $j\ge 1$ ) are determined from the recurrence equation

$c_j=b_0^{-1}\left(a_j-{\displaystyle \sum _{r=1}^j{b}_r{c}_{j-r}}\right).$

The pdf of ${\rm T}$ is obtaining by differentiating (11) as

$f\left(t;\alpha ,\tau ,k,\lambda \right)={\displaystyle \sum _{j=0}^{\infty }{c}_{j+1}{h}_{j+1}\left(t\right),}$        (12)

where

$h_{j+1}\left(t\right)=\frac{\left(j+1\right)\tau }{\alpha \Gamma \left(k\right)}{\left(\frac{t}{\alpha }\right)}^{\tau k-1}\exp \left[{\left(\frac{t}{\alpha }\right)}^{\tau }\right]G{\left(t;\alpha ,\tau ,k\right)}^j$

is the Exp-GG density function with power parameter $j+1$ .

For $j\ge 0$ , we can write

$h_{j+1}\left(t\right)=\frac{\left(j+1\right)\tau }{\alpha \Gamma \left(k\right)}{\left(\frac{t}{\alpha }\right)}^{\tau k-1}\exp \left[-{\left(\frac{t}{\alpha }\right)}^{\tau }\right]{\gamma }_1{\left(k,{\left(t/\alpha \right)}^{\tau }\right)}^j,$             (13)

where ${\gamma }_1\left(k,{\left(t/\alpha \right)}^{\tau }\right)=\gamma \left(k,{\left(t/\alpha \right)}^{\tau }\right)/\Gamma \left(k\right).$

By application of an equation in Section 0.314 of Gradshteyn and Ryzhik²⁰ for a power series raised to a power, we obtain for any $j$ positive integer

${\left({\displaystyle \sum _{i-0}^{\infty }{a}_i{x}^i}\right)}^j={\displaystyle \sum _{i=0}^{\infty }{d}_{j,i}{x}^i,}$ (14)

where the coefficients $d_{j,i}\left(for i=1,2,\mathrm{.....}\right)$  satisfy the recurrence relation

$d_{j,i}={\left(i{a}_0\right)}^{-1}{\displaystyle \sum _{p=1}^i\left[j\left(p+1\right)-i\right]}{a}_p{d}_{j,i-p}$    (15)

and $d_{j,0}=a_0^j$ . The coefficient $d_{j,i}$ can be expressed explicitly from $d_{j,0,\mathrm{...},}{d}_{j,i-1}$ and then from $a_{0,\mathrm{..},}{a}_{i,}$ , although it is not necessary for programming numerically our expansions using any software with numerical facilities.

Further, using equation (14), we can write (for $j\ge 1$ )

${\gamma }_1{\left(k,{\left(t/\alpha \right)}^{\tau }\right)}^j=\frac{{\left(t/\alpha \right)}^{jk\tau }}{\Gamma {\left(k\right)}^j}{\displaystyle \sum _{i=0}^{\infty }{d}_{j,i}{\left(t/\alpha \right)}^{i\tau }},$ (16)

where the coefficients $d_{j,i}\left(for i\ge 1\right)$ are determined from (15) with $a_p={\left(-1\right)}^p/\left[\left(k+p\right)p!\right].$ Based upon equation (16), we can write the Exp-GG density (for $j\ge 1$ ) from (13) as

$h_{j+1}\left(t\right)=\frac{\left(j+1\right)\tau }{\alpha \Gamma {\left(k\right)}^{j+1}}\exp \left[-{\left(\frac{t}{\alpha }\right)}^{\tau }\right]{{\displaystyle \sum _{i=0}^{\infty }\left(\frac{t}{\alpha }\right)}}^{\left[i+\left(j+1\right)k\right]\tau -1}.$

The last density can be expressed in terms of the GG density functions. By noting the form of (4), we can write (for $j\ge 1$ )

$h_{j+1}\left(t\right)={\displaystyle \sum _{i=0}^{\infty }{e}_{j,i}g\left(t;\alpha ,\tau ,\left[i+\left(j+1\right)k\right]\right)},$                 (17)

where $g\left(t;\alpha ,\tau ,\left[i+\left(j+1\right)k\right]\right)$  is the GG density function with parameters $\alpha ,\tau$ and $i+\left(j+1\right)k$ and

$e_{j,i}=\frac{\left(j+1\right)\Gamma \left(\left[i+\left(j+1\right)k\right]\right)}{\Gamma {\left(k\right)}^{j+1}}{d}_{j,i}.$ (18)

For $j=0$ , we have from (13) $h_1\left(t\right)=\frac{\tau }{\alpha \Gamma \left(k\right)}{\left(t/\alpha \right)}^{\tau k-1}\exp \left[{\left(\frac{t}{\alpha }\right)}^{\tau }\right]=g\left(t;\alpha ,\tau ,k\right).$ Combining the result (17) (for $j\ge 1$ ) and that one for $j=0$ , we can write $f\left(t\right)=f\left(t;\alpha ,\tau ,k,\lambda \right)$  in (12) as

$f\left(t\right)=c_{1}g\left(t;\alpha ,\tau ,k\right)+{\displaystyle \sum _{j=1}^{\infty }{\displaystyle \sum _{i=0}^{\infty }{e}_{j,i}}}{c}_{j+1}g\left(t;\alpha ,\tau ,\left[i+\left(j+1\right)k\right]\right).$  (19)

Equation (19) reveals that the OLLGG density function is a linear combination of Exp-GG densities. Hence, some mathematical properties of the OLLGG distribution can follow directly from those properties of the GG distribution. For example, the ordinary, central, fac¬torial moments and the moment generating function (mgf) of the proposed distribution can be obtained from the same weighted infinite linear combination of the corresponding quantities for the GG distribution. This equation is the main result of this section.

Some of the most important features and characteristics of a distribution can be studied through moments (e.g., tendency, dispersion, skewness and kurtosis). In this section, we give two different expansions for calculating the moments of the EGG distribution.

First, we obtain an infinite sum representation for the $r$ th ordinary moment ${\mu }_r^{\text{'}}$ of the EGG distribution based on the equation (19). The $r$ th moment of the $GG\left(\alpha ,\tau ,k\right)$ distribution is well known to be

${\mu }_{r,GG}^{\text{'}}=\frac{{\alpha }^r\Gamma \left(k+r/\tau \right)}{\Gamma \left(k\right)}$

Equation (19) then immediately gives

${\mu }_r^{\text{'}}=\frac{c_1{\alpha }^r\Gamma \left(k+r/\tau \right)}{\Gamma \left(k\right)}+\frac{{\alpha }^r}{\Gamma \left(k\right)}{\displaystyle \sum _{j=1}^{\infty }{\displaystyle \sum _{i=0}^{\infty }{e}_{j,i}{c}_{j+1}\Gamma \left(\left[i+\left(j+1\right)k\right]+r/\tau \right)}}.$          (20)

Equation (20) reveals that the moment ${\mu }_r^{\text{'}}$ does have the inconvenient of depending on the quantities $e_{j,i}$ given by (18).

We now derive another infinite sum representation for ${\mu }_r^{\text{'}}$ by computing the $r$ th moment directly without requiring the quantities $e_{j,i}$ . We readily obtain

${\mu }_r^{\text{'}}=\frac{\beta {\alpha }^{r-1}}{\Gamma \left(k\right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\left(\frac{t}{\alpha }\right)}^{\beta k+r-1}}\exp \left\{-{\left(\frac{t}{\alpha }\right)}^{\beta }\right\}{\left\{{\gamma }_{{}_1}\left[k,{\left(\frac{t}{\alpha }\right)}^{\beta }\right]\right\}}^{\lambda -1}dt$

and then $x={\left(t/\alpha \right)}^{\beta }$ gives

${\mu }_r^{\text{'}}=\frac{\text{λ}{\alpha }^r}{\Gamma {\left(k\right)}^{\lambda }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{x}^{k+r/\beta -1}}{e}^{-x}\gamma {\left(k,x\right)}^{\lambda -1}dx.$

Using expansion (16) for $\gamma \left(k,x\right)$ leads to

$\gamma {\left(k,x\right)}^{\lambda -1}={{\displaystyle \sum _{j=0}^{\infty }{\displaystyle \sum _{m=0}^j\left(-1\right)}}}^{j+m}\left(\stackrel{\lambda -1}{j}\right)\left(\stackrel{j}{m}\right)\gamma {\left(k,x\right)}^m.$

Inserting the last equation in the expression for ${\mu }_r^{\text{'}}$ and interchanging terms, we obtain

${\mu }_r^{\text{'}}=\frac{\lambda {\alpha }^r}{\Gamma {\left(k\right)}^r}{{\displaystyle \sum _{j=0}^{\infty }{\displaystyle \sum _{m=0}^j\left(-1\right)}}}^{j+m}\left(\stackrel{\lambda -1}{j}\right)\left(\stackrel{j}{m}\right)I\left(k,r/\beta ,m\right),$  (21)

where

$I\left(k,r/\beta ,m\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{x}^{k+r/\beta -1_{e^{-x_{\gamma \left(k,x\right)}}}{}^m}}dx$ .

For calculating the last integral, the series expansion (16) for the incomplete gamma function gives

$I\left(k,r/\beta ,m\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{{}^{x^{k+r/\beta -1}}}^{{{}_{{{}^e}^{{}^{-x}}{}^{\left\{x^k{\displaystyle \sum _{p=0}^{\infty }\frac{{\left(-x\right)}^p}{\left(k+p\right)p!}}\right\}}}}^m}dx}.$

Now this integral can be obtained from equations (24) and (25) of Nadarajah²¹ in terms of the Lauricella function of type A (Exton,²² Aarts,²³) defined by

$F_A^{\left(n\right)}\left(a;b_1,...,b_n;c_1,...,c_n;x_1,...,x_n\right)=$

${\displaystyle \sum _{m_1=0}^{\infty }\mathrm{...}}{\displaystyle \sum _{m_n=0}^{\infty }{\frac{{\left(a\right)}_m{}_{{}_1+\mathrm{...}+m_n}{\left(b_1\right)}_m{}_{{}_1}\mathrm{...}{\left(b_n\right)}_m{}_{{}_n}}{{\left(c_1\right)}_{m_1}\mathrm{...}{\left(c_n\right)}_{m_n}}}_{}}\frac{x_{{}_1}^{m_1}\mathrm{...}{x}_{{}_n}^{m_n}}{m_1!\mathrm{...}{m}_n!},$

where ${\left(a\right)}_i$ is the ascending factorial defined by (with the convention that ${\left(a\right)}_0=1$ )

${\left(a\right)}_i=a\left(a+1\right)\mathrm{...}\left(a+i-1\right).$

Numerical routines for the direct computation of the Lauricella function of type A are available, see Exton²² and Mathematica (Trott,²⁴). We obtain

$I\left(k,r/\beta ,m\right)=k^{-m}\Gamma \left(r/\beta +k\left(m+1\right)\right)F_A^{\left(m\right)}\left(r/\beta +k\left(m+1\right);k,\mathrm{...}k;k+1,\mathrm{...},k+1;-1,\mathrm{...}-1\right).$

(22) Hence, as an alternative way to equation (20), the rth moment of the EGG distribution follows from both formulae (21) and (22) as an infinite weighted sum of the Lauricella functions of type A. In Figures 3 and 3, we display plots of the skewness and kurtosis the OLGG distribution for some parameter values.

We evaluate some properties of the MLEs using the classical and Bayesian analysis by means of a simulation study. We simulate the OLLGG distribution considering modality form from equation (8) by using a random variable U having a uniform distribution in (0, 1).

We take n=50, 150 and 350 and, for each replication, we calculate the MLEs $\widehat{\alpha }, \widehat{\tau }, \widehat{k} and \widehat{\lambda }$ . We repeat this process 1, 000 times and determine the average estimates (AEs), biases and means squared errors (MSEs). In this study, we consider two scenarios. In the first scenario, we take $\alpha = 2, \tau = 5, k = 10, \lambda = \mathrm{0.5.}$ In the second scenario, we use the values fitted in the adjustment to the temperature data set in Section 8 $(\alpha = 21.2911, \tau = 13.0661, k = 2.8755, \lambda = 0.2882)$ . The estimates of $\alpha ,\tau ,k, and\lambda$ are determined by solving the nonlinear equations $U_{\alpha }(\theta ) = 0, U_T(\theta ) = 0, U_k(\theta ) = 0, U_{\lambda }(\theta ) = 0$ . The results of the Monte Carlo study under maximum likelihood and Bayesian estimation are given in Tables 2 and 3, respectively. They indicate that the MSEs of the MLEs of $\alpha , \tau , k, and \lambda$ decay toward zero as the sample size increases, as expected under first-order asymptotic theory. The same results are obtained using the Bayesian approach. In Figures 5 and 6, we present the estimated densities based on 1,000 samples of the AEs of the parameters $\alpha ,\tau ,k, and\lambda$ , respectively and n = 50, 150 and 350 for both scenarios. These plots are in agreement with the first-order asymptotic theory for the MLEs and reveal a fast convergence even for small sample sizes.

Simulation study of random censored values

Similarly, we also consider a simulation study in the presence of censored data. The censoring times $C_i$  are sampled from the uniform distribution in the interval $(0,v), where v$ denotes the proportion of censored observations. In this study, the proportions of censored observations are approximately equal to 10% and 30%. In this scenario, we take the values of the parameters as $\alpha = 2, \tau = 5, k=10, \lambda =0.15$ . Table 4 lists the averages of the MLEs (Mean) and the MSEs. The figures in this table indicate that the MSEs increase when the censoring percentage increases. Further, the MSEs of the MLEs of $\alpha ,\tau ,k, and\lambda$ decay toward zero as the sample size increases, as expected under first-order asymptotic theory.

Table 5 lists the posterior means (Mean) and the MSEs. We can note that increasing the sample size and decreasing the percentage of censure, the estimates are closer to the true values with lower MSEs.

In Figures 7 and 8, we present the estimated densities based on 1,000 samples of the AEs of the parameters $\alpha ,\tau ,k, and\lambda$ respectively, and n = 50, 150 and 350 for both scenarios with 10% and 30% of censored. These plots are in agreement with the first-order asymptotic theory for the MLEs and indicate a fast convergence even for small sample sizes and considering censored data.

In this section, we provide two applications to real data to prove empirically the flexibility of the OLLG model. The computations are performed using the R software and NLMixed procedure in SAS. In the first application, we give an application for bimodal data comparing the OLLGG, GG and Weibull models. In the second application, we prove the usefulness of the new distribution for censored data.

Temperature data

The first data set refers to daily temperatures $\left({}^{0}C\right)$ $\left(n=365\right)$ in the period from January 1 to December 31, 2011 in the city of Piracicaba obtained from the Department of Biosystems Engi-neering of the Luiz de Queiroz Superior School of Agriculture (ESALQ), part of the University of São Paulo (USP).

We show the superiority of the OLLGG distribution as compared to some of its sub-mo¬dels and also to the following non-nested models: the exponentiated generalized gamma (EGG) proposed by Cordeiro et al.²⁸ and beta Weibull (BW) distributions. The BW cdf (Famoye et al.,²⁹) is given by

$F\left(t\right)=\frac{1}{B\left(a,b\right)}{\displaystyle \int {}_0^{\left\{1-\exp \left[-\left(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.\right)\gamma \right]\right\}}}{w}^{a-1}{\left(1-w\right)}^{b-1}dw.$

The Kumaraswamy generalized gamma (KumGG) distribution (for t > 0) is defined by Pascoa et al.⁵ Its density function with five positive parameters $\alpha ,\tau ,k,\lambda and \phi$  is given by

$f\left(t\right)=\frac{\lambda \phi \tau }{\alpha \Gamma \left(k\right)}{\left(\frac{t}{\alpha }\right)}^{\tau k-1}\exp \left[-{\left(\frac{t}{\alpha }\right)}^{\tau }\right]{\left\{{\gamma }_1\left[k{\left(\frac{t}{\alpha }\right)}^{\tau }\right]\right\}}^{\lambda -1}{\left(1-{\left\{{\gamma }_1\left[k,{\left(\frac{t}{\alpha }\right)}^{\tau }\right]\right\}}^{\lambda }\right)}^{\phi -1}$ , (26)

where ${\gamma }_1\left(k,x\right)=\raisebox{1ex}{$\gamma \left(k,x\right)$}\!\left/ \!\raisebox{-1ex}{$\Gamma \left(k\right)$}\right.$ is the incomplete gamma function ratio, $\alpha$ is a scale parameter and the other positive parameters $\tau ,k,\phi and \lambda$ are shape parameters.

Next, we report the MLEs and their corresponding standard errors (SEs) in parentheses of the parameters and the values of the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC) and Bayesian Information Criterion (BIC). The lower the values of these criteria, the better the fit. In each case, the parameters are estimated by maximum likelihood using the NLMixed procedure in SAS.

We compute the MLEs of the model parameters and the AIC, CAIC and BIC statistics for each fitted model to these data. The OLLGG model was fitted and compared with the fits from two sub-models cited before. The results are reported in Table 6. The three information

criteria agree on the model’s ranking. The lowest values of these criteria correspond to the OLLGG distribution, which could be preferred in this case.

We perform the LR tests to verify if the extra shape parameter $\lambda$  is really necessary. We provide the histogram of the data and the fitted density functions. Formal tests for the skewness parameter in the generated distribution can be based on LR statistics. The LR statistics for comparing the fitted models are listed in Table 7. We reject the null hypotheses in the two tests in favor of the wider distribution. The rejection is extremely highly significant and it gives clear evidence of the potential need for the shape parameter $\lambda$ when modeling real data. More information is provided by a visual comparison of the histogram of the data and the fitted density functions. The plots of the fitted OLLGG, GG and Weibull densities are displayed in Figure 9a. The estimated OLLGG density provides the closest fit to the histogram of the data.

In order to assess if the model is appropriate, plots of the fitted OLLGG, GG and Weibull cumulative distributions and the empirical cdf are displayed in Figure 9b. They indicate that the OLLGG distribution gives a good fit to these data.

Under a Bayesian approach, we also fit the OLLGG model and some models described above. For each fitted model to these data, the Bayesian estimates of the model parameters and the DIC, EAIC and EBIC statistics are shown in the Tables 8 and 9, respectively. According to the three Bayesian information criteria, the OLLGG model stands out as the best one.

Survival data

Aids is a pathology that mobilizes its sufferers because of the implications for their interpersonal relationships and reproduction. Therapeutic advances have enabled seropositive women to bear children safely. In this respect, the pediatric immunology outpatient service and social service of Hospital das Cl´ınicas have a special program for care of newborns of seropositive mothers to provide orientation and support for antiretroviral therapy to allow these women and their babies to live as normally as possible. Here, we analyze a data set on the time to serum reversal of 148 children exposed to HIV by vertical transmission, born at Hospital das Cl´ınicas (associated with the Ribeirão Preto School of Medicine) from 1995 to 2001, where the mothers were not treated (Silva,³⁰; Perdoná,³¹). Vertical HIV transmission can occur during gestation in around 35% of cases, during labor and birth itself in some 65% of cases, or during breast feeding, varying from 7% to 22% of cases. Serum reversal or serological reversal can occur in children of HIV-contaminated mothers. It is the process by which HIV antibodies disappear from the blood in an individual who tested positive for HIV infection. As the months pass, the maternal antibodies are eliminated and the child ceases to be HIV positive. The exposed newborns were monitored until definition of their serological condition, after administration of Zidovudin (AZT) in the first 24 hours and for the following 6 weeks. We assume that the lifetimes are independently distributed, and also independent from the censoring mechanism.

Tables 10-12 list, respectively, the MLEs and their corresponding SEs in paren¬theses and posterior mean (standard deviation) and 95% highest posterior density (HPD) interval for the parameters and the values of the model selection statistics. These results indicate that the OLLGG model has the lowest AIC, BIC, CAIC, DIC, EAIC e EBIC values among those of all fitted models, and hence it could be chosen as the best model.

Note that the KumGG model is competitive with the model OLLGG. However, the model KumGG has two disadvantages:

It does not model bimodal data.

It has five parameters, i.e. is less parsimonious.

A comparison of the proposed distribution with some of its sub-models using LR statis¬tics is performed in Table 13. The figures in this table, specially the p-values, suggest that the OLLGG model yields a better fit to these data than the other three distributions. In order to assess if the model is appropriate, plots of the estimated survival functions of the KumGG, EGG, GG, Weibull and BW distributions and the empirical survival function are given in Figure 10. We conclude that the OLLGG distribution provides a good fit for these data.