Statistical properties of Odd Frѐchet Lomax Distribution

A new lifetime distribution with three parameters, called odd Frѐchet Lomax (OFrL), is introduced. Some statistical properties of the OFrL are provided. Explicit expressions for the quntile, moments, moment generating function, probability weighted moments and order statistics are studied. Maximum likelihood estimation technique is employed to estimate the model parameters are studied. In addition, the superiority of the OFrL distribution is illustrated with applications to one real data set.


Introduction
introduced The Lomax (L) distribution. The L distribution has found wide applications such as the analysis of the business failure life time data, income and wealth inequality, medical and biological sciences, engineering, lifetime and reliability modeling. The L distribution is used for reliability modelling and life testing by Hassan and Al-Ghamdi (2009). Corbelini et al. (2007) proposed it to model firm size and queuing problems.
Many researchers introduced several generalizations of the L distribution. Ghitany et al. (2007) investigated the Marshal-Olkin extended L distribution, Abdul-Moniem and Abdel-Hameed (2012) introduced the exponentiated L distribution, Lemonte and Cordeiro (2013) proposed the McDonald L, Cordeiro et al. (2013) investigated the gamma L distribution. The exponential L distribution is studied by ElBassiouny et al. (2015). Al-Weighted L introduced by Kilany (2016), and Tahir et al. (2015) introduced Weibull L distribution. The L distribution it has the following cumulative distribution function (cdf) and probability density function (pdf) as Where α is a shape parameters and λ is a scale parameter.
The corresponding pdf to (3) is given by where ( : ) considers a pdf of baseline distribution. Hereafter, a random variable with density function (4) is denoted by ~− ( , ).
The rest of the paper is arranged as follows: In Section 2, we define the OFrL distribution. In Section 3, we derive a very useful expansion for the OFrL density and distribution functions. Further, we derive some Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, No.1, 2019 95 mathematical properties of the new distribution. The maximum likelihood (ML) method is used to estimate the model parameters in Section 4. Simulation study is carried out to estimate the model parameters of OFrL distribution in Section 5. In Section 6, we using one real data set to show the importance of the OFrL distribution. Finally, summary in Section 7.

The OFrL distribution
In this section,we introduce the new three-parameter OFrL distribution, the cdf and pdf of the OFrL distribution is given by and Where λ is scale parameter and α, θ are two shape parameters.  The survival function (sf), hazard rate function (hrf), reversed hrf and cumulative hrf of X are given, respectively, as follows:

Fundamental properties
In this section, we study some fundamental statistical properties for OFrL distribution.

Quantile and Median
The quantile function, say Where, u is considered as a uniform random variable on the unit interval ( ) 0,1 .
The median can be calculated by setting 0.5 u = in (7). Then, the median (M) is given by

Moments
In this subsection, we intend to derive the moments and the moment generating function of the OFrL model. If X has the pdf (11), then its rth moment is given by By inserting (11) into (12), we get Hence, the rth moment of OFrL distribution takes the following form The moment generating function (mgf) of the OFrL distribution is then, .
be the order statistics of a random sample of size n following the OFrL distribution, with parameters  ,  and , then, the pdf of the th k order statistic, can be written as follows 1 : is the beta function. By substituting (5) and (6) in (13), then When we put k=1 in (14) we get the pdf of the smallest order statistics as ( 1) ln 1 ( 1) ln 1 1 1 1 .
Then the ML estimators of the parameters α, λ and θ are obtained by setting U  , U  and U  to be zero and solving them. Clearly, it is difficult to solve them, therefore applying the Newton-Raphson's iteration method and using the computer package such as Maple or R or other software.

Simulation Study
It is very difficult to compare the theoretical performances of the different estimators (MLE) for the OFrL distribution. A numerical study is performed using Mathematica 9 software. Different sample sizes are considered through the experiments at size n = 30, 50 and 100. In addition, the different values of parameters α, λ and θ.
The experiment will be repeated 3000 times. In each experiment, the estimates of the parameters will be obtained by ML methods of estimation. The means, MSEs and biases for the different estimators will be reported from these experiments.
The ML estimates along with their standard errors (SEs) of the model parameters are provided in Tables 2 and 3 Table 3.    Figure 2. Figure 3 shows the estimated cdf and sf for the OFrL model. From these plots it is evident that the new model provides close fit to the data.

Summary
In this paper, we study a three-parameter distribution, called the odd Frѐchet Lomax (OFrL) distribution. The OFrL pdf can be expressed as a mixture of L densities. We derive explicit expressions for the quantile function, moments, moment generating function, probability weighted moments, and order statistics. The ML estimation method is used to estimate the model parameters. We provide some numerical results to assess the performance of the proposed model. The practical importance of the OFrL distribution is demonstrated by means of one real data set.