A STUDY ON THE PROPERTIES AND APPLICATIONS OF LOMAX-GOMPERTZ DISTRIBUTION

The Gompertz distribution can be skewed to the right or to the left. This dissertation introduces a new positively skewed Gompertz model known as Lomax-Gompertz Distribution (LGD). This extension was possible with the aid of a Lomax generator.Some basic statistical propertiesof the new distribution such as moments, moment generating function, characteristics function, reliability analysis, quantile function and distribution of order statistics were derived. A plot of the probability density function (pdf)of the distribution revealed that it ispositively skewed. The model parameters have been estimated using the method of maximum likelihood estimation.The plot for the survival function  indicates that  the  Lomax-GompertzDistribution could  be used to  model time or  age- dependent variables, where probability of survival decreases with time or age.The performance of the Lomax-GompertzDistribution has been compared to the Generalized Gompertz, Transmuted Gompertz,  Odd  Generalized  Exponential  Gompertz  and  the  Gompertz  distributions  by  some applications  to  three  real-life  data  sets.  The  results  show  that  the  proposed  distribution outperformed the  Generalized Gompertz,  Transmuted Gompertz,  Odd  Generalized  Exponential Gompertz and the Gompertz distributions in two of the datasets. The model should be used to modelpositively skewed datasets with various peaks where the sample size is large.

1.1 Introduction

Lomax (1954) pioneered the study of a distribution used for modeling business failure data called the Lomax or Pareto II distribution. This distribution has found wide application in a variety of fields such as income and wealth inequality, size of cities, actuarial science, medical and biological sciences, engineering, lifetime and reliability modeling. It has been applied to model data obtained from income and wealth (Harris, 1968), firm size (Corbellini et al., 2007), size distribution of computer files on servers (Holland et al.,1989), reliability and life testing (Hassan and Al-Ghamdi, 2009), receiver operating characteristic curve analysis (Campbell and Ratnaparkhi, 1993) and Hirsch-related statistics (Gl‟anzel, 2008). It is known as a special form of Pearson type VI distribution. In the lifetime context, the Lomax model belongs to the family of decreasing failure rate (Chahkandi and Ganjali, 2009) and arises as a limiting distribution of residual  lifetimes  at  great  age  (Balkema  and  de  Hann,  1974).  This  distribution  has  been suggested as  heavy-tailed  alternative  to the exponential,  Weibull and gamma  distributions (Bryson, 1974). Further, it is related to the Burr family of distributions (Tadikamalla, 1980) and as a special case obtained from compound gamma distributions (Durbey, 1970). Some details about the Lomax distribution and Pareto family are given in (Arnold, 1983) as well as (Johnson et al., 1994). In record value theory, some properties and moments for the Lomax distribution have been discussed in (Balakrishnan and Ahsanullah, 1994) as well as (Amin, 2011). The moments and inference for the order statistics and generalized order statistics are given in (Saran and Pushkarna. 1999) as well as (Moghadam et al., 2012). The estimation of parameters in  case  of progressive  and  hybrid  censoring  have  been  investigated  in  (Asgharzadan  and

Valiollahi, 2011) as well as(Ashour et al., 2011).The problem of Bayesian prediction bounds for future observation based on uncensored and type-I censored sample from the Lomax model are dealt with in (Abd-Ellah, 2003) and (Al-Hussaini et al., 2001). Furthermore, the Bayesain and non-Bayesian estimators of the sample size in case of type-I censored samples for the Lomax distribution are obtained in (Abd-Elfattah et al., 2007), and the estimation under step- stress accelerated life testing  for the Lomax distribution is considered  in  (Hassan and  Al- Ghamdi, 2009). The parameter estimation through generalized probability weighted moments (PWMs) is addressed in (Abd-Elfattah and Alharby, 2010). More recently, the second-order bias and bias-correction for the maximum likelihood estimators (MLEs) of the parameters of the Lomax distribution are determined in (Giles et al., 2013). Cordeiro et al. (2014d) introduced a new family of distributions based on the Lomax distribution, called the Lomax-G generator. The Lomax-G generator adds two additional positive parameters to an existing continuous distribution. It allows for greater flexibility of its tails and can be widely applied to many areas of Engineering and Biology. This study takes advantage of this generator to introduce Lomax- Gompertz  Distribution,  by  generalizing  the  Gompertz  Distribution.  The  resultant  Lomax- Gompertz Distribution (LGD) will have four parameters, two from the  baseline Gompertz Distribution and two additional positive parameters from the Lomax-G generator. This will increase the flexibility of the Gompertz distribution and also widen its areas of applications in Engineering and Biology.

The Gompertz distribution (GD) can be skewed to the right or to the left. It is a generalization of the exponential distribution (ED) and is commonly used in many applied  lifetime data analysis (Johnson et al., 1995). The GD is applied in the analysis of survival, in some sciences such as Gerontology (Brown and  Forbes,  1974),  Computer  (Ohishi et  al.,  2009), Biology (Economos,  1982),  and  Marketing  science  (Bemmaor  and  Glady,  2012).  The  hazard  rate function  of  Gompertz  distribution  is  an  increasing  function,which  makes  it  applicable

2todescribe the distribution of adult life spans by actuaries and demographers (Willemse and Koppelaar, 2000).