Marshall-Olkin Power Lomax Distribution: Properties and Estimation Based on Complete and Censored Samples

We study a new distribution called the Marshall-Olkin Power Lomax distribution. A comprehensive account of its mathematical properties including explicit expressions for the ordinary moments, moment generating function, order statistics, Renyi entropy, and probability weighted moments are derived. The model parameters are estimated by the method of maximum likelihood. Monte Carlo simulation study is carried out to estimate the parameters and the performance of the estimates is judged via the average biases and mean squared error values. The usefulness of the proposed model is illustrated via real-life data set.


Introduction
The Lomax distribution introduced by Lomax (1954), (also known as Pareto Type II distribution) is one of the well-known distributions used for modeling of actuarial sciences, business failure, size of cities, medical and biological sciences, income and wealth inequality, engineering, lifetime and reliability datasets. Lomax distribution has been considered as a heavy tailed alternative to the exponential, Weibull and gamma distributions (Bryson, 1974). It is also associated with Burr family of distributions (Tadikamalla, 1980). In the lifetime, the Lomax model belongs to the family of decreasing failure rate (Chahkandi & Ganjali, 2009). Lomax distribution has been used for modeling income and wealth data (Atkinson & Harrison, 1978), for firm size data (Corbellini et al. 2010), for reliability and life testing (Harris, 1968), receiver operating characteristic (ROC) curve analysis (Campbell & Ratnaparkhi, 1993) and Hirsch-related statistics (Glä nzel, 1987).
The cumulative distribution function of PLx distribution is given by The corresponding probability density function is     Power Lomax (MOPLx). Additionally, we also discuss its theoretical properties. The maximum likelihood method is considered to estimate the parameters using complete and type II censored sampling. We considered real-life data to illustrate the usefulness of the proposed model. The real data application shows that the suggested distribution performs superior when compared with other important distributions.
The cumulative distribution function (cdf) of Marshall and Olkin (MO) family is defined by with probability density function Where   Gx and   gxare cdf and pdf of the baseline distribution. Using this approach an additional shape parameter (γ) is added which is responsible for the skewness, kurtosis and tail weights. Moreover, this new model can be used as an alternative to gamma and Weibull distributions.

The MOPLx Distribution
We define the four parameter Marshall-Olkin Power Lomax distribution by inserting (1) The corresponding pdf is where , , ,     are positive parameters.

Mathematical Properties of MOPLx Distribution
The mathematical properties of MOPLx distribution including shapes of the pdf and hrf, linear representations of the cdf and pdf, quintile function (qf), random number generator, ordinary moments, incomplete moments, moment generating function, mean residual life, probability weighted moments and reversed residual life are investigated in this section.

Shape Characteristics of the Pdf and Hrf of MOPLx Distribution
In this subsection, the limiting behavior of the pdf and hrf of MOPLx distribution at the origin are calculated.

Useful Representations
Here we present two linear representations of the pdf and cdf of MOPLx distribution. Consider the following well-known binomial expansions (for 0 < α < 1), Using (7), the numerator of (6) can be expressed as Therefore, from (6) and (8)

Quantile Function and Random Number Generator
The quantile function, say Upon some simplifications, it reduces to the following form where uis obtained from a uniform random variable on the unit interval   0,1 .

Moments
In this subsection, we derive the expressions for ordinary moments, incomplete moments and the moment generating function.

Ordinary Moments
If X is a continuous random variable with the MOPLx distribution, then the rth moment of X is given by a.= 0.5 p = 0.5 J. = 0.5 y = 0.5 a.= 0.5 p = 1.0 J. = 0.5 y= 1.5 a.= 0.5 p = 1.5 J. = 0.5 y= 1.5 u. = 1.5 p = 1.5 ), = 1.5 y = 1.5 u. = 2.5 p = 1.5 J. = 2.5 y = 1.5 a.= 3.5 p = The coefficients of skewness and kurtosis for the MOPLx distribution can also be obtained from the third and the fourth standardized cumulants by using formulae

Incomplete Moments
If X is a continuous random variable with the MOPLx distribution, then the incomplete moment of X is given by

Ré nyi Entropy
If X is a continuous random variable with the MOPLx distribution, then the Ré nyi Entropy of X is given by     1 1 1 1 To obtain such result note that Hence, we have that Substitution completes the proof.

The Probability Weighted Moments
The probability weighted moments can be obtained from the following relation Substituting (9) and (10) into (12) and replacing h with s , leads to: Hence, the PWM of MOPLx distribution takes the following form

Stress Strength Reliability
In this subsection, we derive the stress strength reliability R when X 1 and X 2 are independent random variables, X 1 follows MOPLx(α 1 , λ, β, γ 1 ) and X 2 follows MOPL(α 2 , λ, β, γ 2 ), then R=P(X 1 <X 2 ) is the measured as; Applying the binomial theory, we can rewrite the previous equation as

Order Statistics
The kth order statistics of MOPLx distribution is 1 : where, (.,.) B is the beta function. Substituting (9) and (10) in (13) and replacing (14), we obtain the pdfs of the first and largest order statistics of the MOPLx distribution.
Further, the th r moment of th k order statistics for MOPLx distribution is defined by: Lorenz and Bonferroni curves are the most widely used inequality measures in income and wealth distribution ( see Kleiber 1999). The Lorenz, Bonferroni and Zenga curves are obtained, respectively, as follows after some algebraic manipulation, we have that

Characterization
This section deals with the characterizations of the MOPLx distribution based on: (i) a simple relation between two truncated moments; (ii) hazard function and (iii) reverse hazard function. It is worth mentioning that the characterization (i) can be employed when the cdf does not have a closed form. We present the characterizations (i) -(iii) in three subsections.

Characterizations (i)
This subsection deals with the characterizations of MOPLx distribution based on two truncated moments. For the first characterization we use Theorem 1 of (Glä nzel, 1987). The result holds when the interval H is not closed.
Corollary 7.1.1. The continuous random variable   : 0, X    has the pdf (6) if and only if there exist function 2 q and  defined in theorem 1 of (Glä nzel, 1987)satisfying the following differential equation The general solution of the differential equation in Corollary 9.1.1 is For the functions given in Proposition 7.1.1 with D = 0.

Characterization (ii)
For the hazard function, , .
The following proposition presents a non-trivial characterization of MOPL distribution based on the hazard function.

The ML Estimators for Complete Samples
Here, we discuss the estimation of the unknown parameters of the MOPLx distribution by the maximum likelihood method. The log-likelihood function for the vector of parameters     can be estimated by equating the derived simultaneous equations to zero. Since the structure of the equations is complex therefore, it is hard to get unique solution without iterative procedure. Therefore, we use statistical software to get the estimates numerically and used Newton-Raphson algorithm for this purpose.

The ML Estimators for Type-II Censored Samples
The occurrence of type-II censoring is very common in many applications in survival analysis, for instance, the lifetime of electronic devices are usually finished after a fixed number of failures r, hence, − are be the number of censored data. In this case the logarithm of the likelihood function is given by

Simulation Study
In this section, we present a simulation study for the purpose of estimation of unknown parameters of MOPLx distribution via maximum likelihood method for complete and censored samples. The algorithm for estimation used here is designed as follows;  Random samples of sizes 30,50,100 and 300 n  are generated form MOPLx distribution under complete and type II censored samples.
.5 In this section, a real life data set is used to illustrate the usefulness of the derived model. For this purpose, we utilize the data set belongs to the remission time of cancer patients. The data consist of 128 observations and earlier studied by Lee and Wang (2003).
We utilize the MLE method to observe the goodness of fit for the MOPLx distribution and compare the proposed distribution with Weibull-Lomax (WLx), transmuted Lomax (TLx), exponential-Lomax (ELx), beta-Lomax (BLx), Lomax (Lx), beta-exponential (BE), Marshall-Olkin length exponential (MOLBE) and Marshall-Olkin extended exponential (MOEE) distribution. The selection of model is based on AIC (Akaike information criterion), the BIC (Bayesian information criterion). Furthermore, we also consider Anderson and Darling (A*), Cramé r-von Mises (W*) and Kolmogorov-Smirnov (D) where L is log-likelihood function and q is the number of parameters.
The Maximum Likelihood Estimates along with the goodness of fit measures are presented in Table (5). The numerical results are computed using R software.  Table 4 which shows that data set is positively skewed. The MLEs and goodness of fit for considered distributions are reported in Table 5. The MOPLx distribution provided the best fit among the chosen models. Figure 2 provides the estimated pdf and cdf superimposed on the histogram of dataset. This fitted densities support the findings presented in Table 5. Figure 3 illustrate the plots of profile-log likelihood functions of the fitted MOPLx distribution.

Conclusion
In this work, we derive and study a four parameter distribution called Marshall-Olkin Power Lomax (MOPLx) distribution. The proposed distribution is derived using the generator approach by Marshall and Olkin (1997). We study some of its statistical properties including, moments, moment generating function, incomplete moments, mean residual life, mean activity time, expressions of the order statistics. The estimation of parameters is computed by the method of maximum likelihood for complete and type II censored data. A comprehensive simulation study is used to evaluate the proposed estimators. A real life data set is used to illustrate the usefulness of the proposed distribution. In conclusion, the MOPLx distribution provides a better fit and can be consider a good model for skewed dataset.