The Transmuted Generalized Gamma Distribution : Properties and Application

Abstract: The generalized gamma model has been used in several applied areas such as engineering, economics and survival analysis. We provide an extension of this model called the transmuted generalized gamma distribution, which includes as special cases some lifetime distributions. The proposed density function can be represented as a mixture of generalized gamma densities. Some mathematical properties of the new model such as the moments, generating function, mean deviations and Bonferroni and Lorenz curves are provided. We estimate the model parameters using maximum likelihood. We prove that the proposed distribution can be a competitive model in lifetime applications by means of a real data set.


Introduction
Standard lifetime distributions usually present very strong restrictions to produce bathtub curves, and thus appear to be unappropriate for analyzing data with this characteristic.The three-parameter generalized gamma ("GG" for short) distribution (Stacy, 1962) includes as special models the exponential, Weibull, gamma and Rayleigh distributions, among others.It is suitable for modeling data with hazard rate function (hrf) of different forms (increasing, decreasing, bathtub and unimodal) and also useful for estimating individual hazard functions and both relative hazards and relative times (Cox, 2008).The GG distribution has been used in several research areas such as engineering, hydrology and survival analysis.Its probability density function (pdf) and cumulative distribution function (cdf) are given by (for x > 0) and G(x) = γ(ν, [x/a]p )/Γ(ν ), respectively, where Γ(ν ) = R ∞ ων −1 e−ω dω (for ν > 0) is the gamma function and γ(x, ν ) = R x ων −1e−ω dω is the incomplete gamma function.For the density function (1), a > 0 is a scale parameter and p > 0 and ν > 0 are shape parameters.The Weibull and gamma distributions are special cases of (1) when ν = 1 and p = 1, respectively.The GG distribution approaches the log-normal distribution when a = 1 and ν → ∞.We denote by a random variable having density function (1).
The GG distribution includes all four more common types of the hrf: monotonically increasing and decreasing, bathtub and unimodal (Cox et al., 2007).This property is useful in reliability and survival analysis.This model has been used in several applied areas such as engineering, economics and survival analysis.Yamaguchi (1992) used it for the analysis of permanent employment in Japan, Allenby (1999)  Another family of distributions arises from the general rank transmutation (GRT) defined by Shaw and Buckley (2007).Suppose we have two cdfs F (x) and G(x) with common sample space, then the GRT is defined as (3) Note that the pair in (3) takes the unit interval [0, 1] and under suitable assumptions are mutual inverses and satisfy PRij (0) = 0, PRij (1) = 1.A quadratic rank transmutation map (QRTM) is obtained by considering PR12 (t) = t + λt(1 − t).Then, it follows that the cdfs are related by and the corresponding pdf is given by Here, G(x) and g(x) can be understood as the cdf and pdf of the baseline distribution, respectively.Note that this generator, called the transmuted class (TC) of distributions, is a linear combination of 9 Sadraque E.F.Lucena, Ana Herm´ınia A. Silva, Gauss M. Cordeiro 189 the baseline and Exp-G distributions with power parameter equal to two.Also, note that λ = 0 in (5) gives the baseline distribution.Further details can be seen in Shaw and Buckley (2007).
Some distributions belonging to the TC class have been proposed recently.Aryal and Tsokos (2009) studied the transmuted Gumbel distribution and its application to climate data.Aryal and Tsokos (2011) pioneered the transmuted Weibull distribution and used it for modelling the tensile fatigue characteristics of a polyester/viscose yarn.Khan and King (2013) intro-duced the transmuted modified Weibull distribution and Ashour and Eltehiwy (2013) defined the transmuted exponentiated Lomax model.In the present study, we provide some mathe-matical properties of a new lifetime model named the transmuted generalized gamma (TGG) distribution.
The sections are organized as follow.In Section 2, we define the TGG model.Some of its mathematical properties are investigated in Section 3.An application to a real data set is reported in Section 4. Section 5 ends with some conclusions.

The TGG distribution
The cdf of the TGG distribution can be obtained from (4) as where |λ| ≤ 1, a > 0, ν > 0 and p = 0. Henceforth, a random variable X having the cdf (6) is denoted by X ∼ T GG(λ, a, ν, p).We can prove that the TGG distribution is a linear combination of the GG and the exponentiated generalized gamma (EGG) distributions, the last one with power parameter two.The pdf of X is given by The TGG family model has many distributions as special cases.Some of the sub-models encompassed by the TGG family are listed in Table 1.
The hazard rate function (hrf ) of X is given by Some possible shapes of the density function (7) and the hrf are plotted in Figure 1.We conclude that the hrf r(x) is very flexible, assuming different shapes.

Properties of the TGG distribution
Let g(x) and G(x) are the pdf and cdf of a baseline model.A random variable is Exp-G distributed with power parameter α > 0 if its cdf and pdf are given by Hα (x) = G(x)α and hα (x) = αg(x)G(x)α−1, respectively.We obtain some properties of the TGG distribution.If the baseline is taken to be the GG distribution, the random variable Y is said to be Exp-GG distributed, say Y ∼ Exp-GG(α, a, ν, p) distribution.The density function of X can be expressed as  9) reveals that the TGG distribution is a mixture of GG distributions.This result enable us to derive some mathematical properties of the TGG distribution.

Moments
Some of the most important features and characteristics of a distribution can be studied through moments like tendency, dispersion, skewness and kurtosis.For a random variable Z having the GG(a, ν, p) distribution, the sth moment of Z becomes IE(Z s ) = as Γ(ν + s/p)/Γ(ν ).Based on equation (9), the sth moment of X is given by These moments can be computed numerically by standard statistical softwares.

Generating Function
The moment generating function (mgf) of Z, say Ma,ν,p (s) = IE(esZ ), has an explicit expression using the Wright function (Wright, 1935).It can be expressed as  9) and (10), the mgf of X reduces to where Ma,ν * ,p (x) is the mgf of Z with parameters a, ν * and p.This result reveals that the TGG mgf is a mixture of GG mgfs.

Skewness and Kurtosis
The qf of X, say x = Q(u), follows by inverting the cdf (6).It is given by where QGG denotes the qf of the GG distribution with parameters a, ν and p.
There are several robust measures in the literature for location and dispersion.The median, for example, can be used for location and the interquartile range.Both the median and the interquartile range are based on quantiles.From this fact, Bowley (1920) proposed a coefficient of skewness based on quantiles given by where Q(•) here is the qf of the TGG distribution given by (12).Moors (1988) demonstrated that the conventional measure of kurtosis may be interpreted as a dispersion around the values µ + σ and µ − σ, where µ is the mean of the distribution and σ is its standard error.Thus, the probability mass focuses around µ or on the tails of the distribution.Therefore, based on this interpretation, Moors (1988) proposed, as an alternative to the conventional coefficient of kurtosis, a robust measure based on octiles given by These measures are less sensitive to outliers and they exist even for distributions without moments.Figure 3 displays the plots of the Bowleys skewness and Moors kurtosis for the TGG distribution, respectively.

Application
We use a real data set to show that the TGG distribution provide a better fit than that one based on the GG distribution.We also emphasize that the fitted TGG distribution is better than the fitted EGG (Cordeiro et al., 2011), BGG (Cordeiro et al., 2012) and Marshall-Olkin generalized gamma (MOGG) distributions to these data.The corresponding cdf 's are given by where α > 0, λ > 0, Ga,ν,p (x) denotes the GG cdf and  ̅ ,, () is the survival function.The data represent the times between successive failures (in thousands of hours) in events of secondary reactor pumps studied by Salman et al. (1999).Table 2 gives some statistic measures for these data, which indicate that the empirical distribution is skewed to the left and platycurtic.
proposed a dynamic model based on it and Cox The Transmuted Generalized Gamma Distribution: Properties and Application et al. (2007) presented a parametric survival analysis and taxonomy of its hrf.Some extensions of the GG distribution has emerged recently.For example, Pascoa et al. (2011) proposed the Kumaraswamy generalized gamma (KwGG) distribution, Ortega et al. (2011) proposed the generalized gamma geometric distribution and Cordeiro et al. (2012) defined the beta generalized gamma (BGG) distribution.Now, we de_ne an extended form of the density function (1) (for x > 0) given by , , () Where p is not zero and the other parameters are positive.The cdf corresponding to (2) becomes Several continuous univariate distributions have been extensively used in the literature for modelling data in many areas such as engineering, economics, biological studies and environmental sciences.However, applied areas such as lifetime analysis, finance and insurance clearly require extended forms of these distributions.Thereby, classes of distributions have been pro-posed in the literature by extending and creating new families of continuous distributions.These extensions generalize distributions giving more flexibility by adding one or more param-eters to the baseline model.They were pioneered by Gupta et al. (1998), who proposed the exponentiated-G ("Exp-G") distribution, by raising the cdf G(x) to a positive power parameter.Many other classes can be found in the literature such as the beta generalized (BG) family of distributions proposed by Eugene et al. (2002), the Kumaraswamy (Kw-G) family of distri-butions introduced by Cordeiro and de Castro (2011) and the exponentiated generalized (EG) family defined by Cordeiro et al. (2013).

) 1 Figure 1 :
Figure 1: Plots of the TGG pdf and hrf for some parameter values Setting u = x/a, we have By expanding the first exponential in power series and the result∫  −1 ∞ 0 exp(−  ) =  −1 Γ( + /) , we obtain The above equation holds for p = 0. Additionally, for a given p > 1, it can be expressed in terms of the Wright generalized hypergeometric function (Wright, 1935) defined by This function exists if 1 +∑    =1 − ∑    =1 > 0. By combining the last two equations, we obtain Then, from equations (

Figure 2 :
Figure 2: Bonferroni and Lorenz curves for some parameter values.

Figure 3 :
Figure 3: Skewness and Kurtosis of the TGG distribution for different values of λ.

Figure 4 :
Figure 4: Histogram and estimated densities of the BGG, TGG, MOGG and GG models (left panel) and empirical cumulative function (right panel) of times between successive failures (in thousands of hours) of secondary reactor pumps

Table 2 :
Descriptive statistics for the times between successive failures (in thousands of hours) of secondary reactor pumps