On Odds Generalized Exponential-Power Lomax Distribution

Email: salwaassar@yahoo.com Abstract: A new distribution, called odds generalized exponential-power Lomax distribution is suggested for modeling lifetime data. Some structural properties of the new distribution including; quantiles, Rényi entropy, moments and distribution of order statistics are provided. The model parameters of the new distribution are estimated by the maximum likelihood method. Finally, applications to two real data sets are analyzed to illustrate the importance of the new distribution compared with some known distributions.


Introduction
The Lomax or Pareto II distribution was introduced originally for modeling business failure data by Lomax (1954). Moreover it has been widely applied in a variety of contexts such as; income and wealth inequality, size of cities, actuarial science, medical and biological sciences. Hassan and Al-Ghamdi (2009) mentioned that it can be applied in reliability modeling and life testing. In the literature, some extensions of the Lomax distribution are available such as Marshall-Olkin extended-Lomax (Ghitany et al., 2007;Gupta et al., 2010), the exponentiated Lomax (Abdul-Moniem, 2012), beta-Lomax, Kumaraswamy-Lomax and McDonald-Lomax (Lemonte and Cordeiro, 2013) and gamma-Lomax , the transmuted exponentiated Lomax (Ashour and Eltehiwy, 2013), the extended Poisson- Lomax (Al-Zahrani, 2015), exponential Lomax (El-Bassiouny et al., 2015), Weibull Lomax (Tahir et al., 2015a) and exponentiated Weibull-Lomax (Hassan and Abd-Allah, 2018). Rady et al. (2016) proposed one of these extensions of the Lomax distribution and referred to as a Power Lomax (PL) distribution with three-parameter α, β and γ. The probability density function (pdf) and cumulative distribution function (cdf) of PL distribution are given, respectively by: where, α and β are shape parameters and γ is a scale parameter.
In recent years, there has been a great interest among statisticians and applied researchers in constructing flexible distribution to facilitate better modeling of lifetime data in various situations. Several methods have been developed for generating new family of lifetime distributions. The T-X family is one of the generalized family of distributions which was proposed by Alzaatreh et al. (2013) with the following cdf: where, the random variable T called the transformer has pdf r (t) defined on [a, b]-∞≤a<b≤∞ and W(G(x)) be a function of G(x). As a result, many new distributions have come up and studied. Tahir et al. (2015b) proposed a new class of distributions called the odd generalized exponential (OGE) family and study each of the OGE-Weibull distribution, the OGE-Fréchet distribution and the OGEnormal distribution. These models are flexible because of the hazard shapes: increasing, decreasing, bathtub and upside subset of down bathtub. Rosaiah et al. (2016) proposed a new lifetime model, called the odd generalized exponential log logistic distribution and  suggested a new distribution, called odds generalized exponential-inverse Weibull distribution for modeling lifetime data. Silva et al. (2017) proposed a new generator of continuous distributions with one extra positive parameter called the odd Lindley-G family. They discussed estimation of the model parameters by maximum likelihood and provided an application to a real data set. Haq and Elgarhy (2018) proposed a new generator from Frѐchet random variable that is known as the odd Frѐchet-G (OFr-G) family of distributions and the maximum likelihood equations are derived for OFr-G family parameters. Also, Hosseini et al. (2018) introduced the generalized odd gamma-G distribution. In particular, G has been considered as the uniform distribution and some statistical properties have been calculated.
The main motivation of this study is to introduce a more flexible model with increasing, decreasing, constant and reversed-J shaped hazard rate and at the same time provide consistently better fits than other well known-distributions. We provide and study a new form for Power Lomax distribution with three parameters. We call the new distribution; the odds generalized exponential-Power Lomax (OGE-PL) distribution, which is a particular case of T-X family of distributions. The rest of the paper contains the following sections. The new distribution is provided in Section 2. Some statistical properties are given in Section 3. Then, in Section 4, maximum likelihood estimators are obtained. An application of the OGE-PL model to real data set is presented in Section 5. At the end, concluding remarks are addressed in Section 6.

The OGE-PL Distribution
In this section, the pdf, cdf, reliabilty function, hazard rate function (hrf), reversed-hazard rate function and cumulative hazard rate function of OGE-PL distribution are derived. Expansions for its pdf and cdf are also provided.

Useful Expansions for OGE-PL Distribution
Firstly, an expansion for pdf is derived. The power series for the exponential function in pdf (5) can be written as follows: where, ξ = (α, β, γ). Then, by inserting expansion (6) in pdf (5), we have: By using the generalized binomial expansion, then: Inserting (8) in (7), the OGE-PL density function can be expressed as follows: So, the expansion form of pdf is as follows: Again, by using the binomial expansion for the last term in (9) where, is the pdf of PL distribution defined in (1) with parameters a (k-1), β and γ.

Reliability Analysis
This subsection gives expressions for the reliability function, hazard function, reversed hazard function and cumulative hazard rate function for the proposed model.

Some Mathematical Properties
In this section, some mathematical properties of the OGE-PL distribution including, quantiles, moments, incomplete moments, order statistics and Rényi entropy measure are derived.

Quantiles of the OGE-PL Distribution
The quantile function of the OGE-PL is given by: where, q is a uniform (0,1) distribution. We can generate random numbers from our model by using (11) and special cases can be obtained using (11) such as the second quartile (median), when q = 0.5: and the Inter-Quantile Range (IQR) which defined as the difference between the third quartile and the first quartile can be expressed as:

∑∑∑
In particular, the mean and variance of the OGE-PL distribution are given respectively by:

∑∑∑ ∑∑∑
The skewness (g 1 ) of the OGE-PL distribution is given by: The kurtosis (g 2 ) of the OGE-PL is given by: Furthermore, the moment generating function of OGE-PL is obtained through the following relation as follows:

Incomplete Moments
The main application of the first incomplete moment refers to the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The incomplete moments, say ϕ s (t), is given by: where, Ψ = (α, β, γ, λ) is the set of parameters. Using (10), then ϕ s (t) can be written as follows: where, , , is the incomplete beta function. In particular, the first incomplete moment of the OGE-PL distribution can be obtained by putting s = 1 in (12), as follows: Additionally, the mean deviations of X about the mean µ and about the median m can be calculated by the following relations:

Rényi Entropy
The entropy is a measure of the uncertainty variation. So, the Rényi entropy of a random variable X with density function ( ; ), ( , , , ) f x α β γ λ Ψ Ψ = is defined as: , (14), then we obtain: ( 1) Applying the binomial and exponential expansions in (15), then we have: Therefore, the Rényi entropy of OGE-PL distribution is given by: ... n n n n X X X < < < denote the order statistics for a random sample 1 2 , ,..., n X X X from OGE-PL distribution with cdf (4) and pdf (5). The pdf of r th order statistics is defined by: Hence, the pdf of r th order statistics will be as follows: In particular, the pdf of the smallest order statistics is obtained by substituting r = 1 in (17) as follows: Also, the pdf of largest order statistics is obtained by substituting r = n in (17) as follows: ( )

Parameter Estimation
In this section, the estimators of unknown parameters of the OGE-PL model are obtained based on maximum likelihood (ML) method. Let X 1 , X 2 ,….,X n , be observed values from the OGE-PL distribution. The total loglikelihood function, denoted by lnL, for the parameters λ, α, β and γ for a complete sample is as follows: The partial derivatives of the log-likelihood function with respect to λ, α, β and γ components of the score vector can be obtained as follows: Then the maximum likelihood estimates (MLEs) of the parameters, denoted byˆ, λ, α β and γ are obtained by setting U λ , U α , U β and U γ to be zero and solving them numerically. It is clear that, there is no closed solution for the above non-linear equations, so an extensive numerical solution will be applied via iterative technique.
The MLEs for OGE-PL, Ext.PL, Exp Lomax, McLomax, TE-Lomax, L and EL models are given in Table 1. Also, to compare the distribution models, we consider criteria like; Akaike information criterion (AIC), the Correct Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC) and Hannan-Quinn information criterion (HQIC). However, the better distribution corresponds to the smaller values of AIC, BIC, CAIC, HQIC criteria. These criteria are provided in Table 2.
The required numerical evaluations are carried out using the package of Mathcad software. The MLEs for OGE-PL, ELP, EL, McLomax, L and E models and the values of the AIC, BIC, CAIC and HQIC are listed in Table 3.
It is clear from Table 3 that the proposed OGE-PL distribution fits to this data is better than the other models according to the AIC, BIC, CAIC and HQIC.

Conclusion
In this article, we propose a new model, called the odds generalized exponential-Power Lomax distribution based on T-X family presented by Alzaatreh et al. (2013). Some statistical properties of current distribution have been derived and discussed. The estimation of the model parameters is approached by maximum likelihood method. We fit the OGE-PL distribution to two real life data sets as two examples, the OGE-PL distribution provides consistently a better fit than some other known models.