POWER LOMAX POISSON DISTRIBUTION : PROPERTIES AND ESTIMATION

A new four-parameter lifetime distribution named as the power Lomax Poisson is introduced and studied. The subject distribution is obtained by combining the power Lomax and Poisson distributions. Structural properties of the power Lomax Poisson model are implemented. Estimation of the model parameters are performed using the maximum likelihood, least squares and weighted least squares techniques. An intensive simulation study is performed for evaluating the performance of different estimators based on their relative biases, standard errors and mean square errors. Eventually, the superiority of the new compounding distribution over some existing distribution is illustrated by means of two real data sets. The results showed the fact that, the suggested model can produce better fits than some well-known distributions.


Introduction
suggested an important model for lifetime analysis called Lomax (Pareto type II) distribution.Its widely applied in some areas, such as, analysis of income and wealth data, modeling business failure data, biological sciences, model firm size and queuing problems, reliability modeling and life testing (see Harris (1968), Atkinson and Harrison (1978) A random variable T has the Lomax distribution with shape parameter  and scale parameter  if it has the probability density function (pdf) given by 1 ( ; , ) ; , , 0.
The cumulative distribution function (cdf) corresponding to (1) 2013) introduced the gamma-Lomax using the gamma generator.The Weibull Lomax using Weibull generator has been proposed by Tahir et al. (2015).Also, the Gumbel-Lomax using Gumbel-X generator has been introduced by Tahir et The cdf of the power Lomax distribution is as follows ( ; , , ) G y y 4, the estimation of the model parameters is carried out using the maximum likelihood, least squares and weighted least squares methods.In Section 5, a simulation study is achieved to illustrate the theoretical results.Section 6 gives the applicability of the proposed model and compares with other competing probability models.At the end, concluding remakes are presented in Section 7.

Power Lomax-Poisson Model
In this section, we introduce and study the power Lomax Poisson (PLP) distribution.The probability density, cumulative distribution, reliability and hazard rate functions are obtained.Suppose that  = {  } =1  be independently and identically distributed (iid) failure times of Z component connected in series and each Y has the power Lomax distribution with pdf (3) and cdf (4).Let the random variable Z has zero-truncated Poisson distribution with probability mass function given by Assume that the variables  ' and  are independent, then the conditional density function of | =  is given by  | (|) =    −1 ( +   ) −(+1) ; , , ,  > 0.
The joint distribution of the random variables  and , denoted by   (; ) is given by The marginal pdf of  is as follows ; , , , , 0, which defines the PLP distribution, where Φ ≡ (, , , ).Or it can be written as follows ; , , , , 0.
The distribution function of PLP is as follows 1 ( ; ) .( 1) Or it can be written as follows A random variable  with density function (5) will be denoted by ~(, , , ).

Lemma
The power Lomax Poisson distribution reduces to the power Lomax distribution as  → 0.

Proof
If  approaches zero, then which is the cdf of power Lomax distribution as defined in (4).Furthermore, the reliability and hazard rate functions are as follows Figures 1 and 2 illustrate plots of the PLP densities and hazard rate functions for some selected values of the parameters.Figure 1 shows that the density of PLP takes different shapes as symmetrical, right skewed, reversed-J and unimodel.From Figure 2, it can be observe that the shapes of the hazard rate function are increasing, decreasing and constant at some selected values of parameters.

3.1.Some Structural Properties
Here, some statistical properties of the PLP distribution including, quantile function, th moment, Re'nyi entropy, order statistics and moments of the residual life are obtained.

Quantile Function
The quantile function of PLP distribution, denoted by, () =  −1 () of  has the following form where  is a uniform random variable on the unit interval (0,1).In particular the median of the PLP distribution, denoted by , is obtained by substituting  = 0.5 in (7) as follows

Moments
Many of the important characteristics and features of a distribution can be obtained using ordinary moments.The th moment of  can be easily obtained from pdf (5) as follows Using the exponential expansion for () , then (8) can be written as follows After simplification, the th moment of PLP distribution takes the following form where Γ(. ) stands for gamma function.
In particular, the mean and variance of PLP distribution are obtained, respectively, as follows Furthermore, the moment generating function can be obtained from moments as follows,   where,   ́ is the th moment about the origin.

Re'nyi Entropy
The entropy of a random variable  is a measure of uncertainty variation.If  is a random variable distributed as PLP, then the Re'nyi entropy, for  > 0 and  ≠ 1 is defined by: 1 Then by using pdf (5), the Re'nyi entropy of PLP distribution can be written as follows: ( 1)


Using the exponential expansion and after simplification, then the Renyi entropy of PLP distribution takes the following form !

Moments of the Residual Life
The th moment of the residual life of PLP distribution is obtained by inserting the pdf (5) and binomial expansion in   () as follows By using the exponential expansion, then   () takes the following form .
( ; ) !( 1) , then the th moment of the PLP distribution takes the following form


which is incomplete beta.In particular, the mean residual life of the PLP distribution is obtained by substituting  = 1 in (10) as follows ) Inserting cdf (6) and pdf ( 5) in (11), then . ( Using the power series for the exponential function will be Then, substituting (13) in (12), the pdf of the th order statistics takes the following form .
Hence, the pdf of the th order statistics of  is as follows; , , 0 0 ( ) (1 ) , ( 14) In particular, the pdf of the smallest order statistics  1: is obtained from (14), by substituting  = 1, as follows , . ! ( 1)  Also, the pdf of the largest order statistics  : is obtained from (14), by substituting  = .

Parameter Estimation
In this section, the estimates of the PLP model parameters are obtained using maximum likelihood, least squares and weighted least squares methods.

Maximum Likelihood Estimators
Let  1 ,  2 , … ,   be a simple random sample from the PLP distribution with set of parameters Φ ≡ (, , , ) .The log-likelihood function, denoted by ln , l based on the observed random sample of size  from density (5) is given by: ii Sx 

 
The partial derivatives of the log-likelihood function with respect to the unknown parameters are given by:

Least Squares and Weighted Least Squares Estimators
Suppose  1 ,  2 , … ,   is a random sample of size  from PLP distribution and suppose  Weighted least squares (WLS) estimators can be obtained by minimizing the sum of squares errors with respect to the unknown parameters , ,  and .Therefore, the weighted least squares estimators  ⃛,  ⃛ ,  ⃛ and  ⃛ can be obtained by minimizing the following quantity with respect to the parameters , ,  and .

Simulation Study
In any estimation problems, it is required to study the properties of the derived estimators.The derived expressions for the estimators are too complicated to study analytically.Consequently, a numerical study will be set up, treating separately the sampling distribution of the estimators.A numerical study is performed to compare the different estimators discussed in the previous section.The performances of the different estimators are compared in terms of their relative bias (RB), mean square error (MSE) and standard error (SE).The numerical procedures will be described below: Step 1: 1000 random samples of size 10, 20, 30, 40, 50 and 100 are generated from the power Lomax Poisson distribution.
Step 3: the MLEs, LS estimators and WLS estimators of the unknown parameters are obtained.
Step 4: The biases, MSEs and SEs of different estimators of unknown parameters are computed.
Simulation results are reported in Tables ( 1) and (2) (at the end of the article) and represented through some Figures from ( 3) to (10).From these tables and figures, the following conclusions can be observed on the performance of different estimators.
1.For all methods of estimations, it is clear that MSEs and SEs decrease as sample size increases (see Tables (1) and ( 2)). 2. The MSEs of MLEs, for all parameters values, are the smallest among the other estimators in almost all cases (see for example Table (1) and Figures (3 -5)).For fixed value of ( = 0.5,  = 1) and as the value of shape parameters (, ) increases, the MSEs for estimators based on maximum likelihood, least squares and weighted least squares methods are increasing (see Table (1)).8.For fixed value of ( = 0.5,  = 1) and as the value of shape parameters (, ) decrease, the MSEs for estimators based on maximum likelihood, least squares and weighted least squares methods are decreasing (see Table (1)).9.The MSEs of  for LS estimators and WLS estimators are approximately constant, when the shape parameters (, ) increase.

Applications To Real Data
In this section, an application of the proposed PLP model to two real data sets is provided to show the flexibility and applicability of the new model in practice.For the first data, the PLP distribution is compared with Kumuerswmay Lomax (KL), power Lomax, beta Lomax (BL), exponentiated Lomax (EL) and Lomax (L) distributions.While, for the second data, the PLP distribution is compared with Weibull Lomax (WL), KL, BL, EL and L distributions Example 6.1:The first data set represents 84 observations of failure time for particular windshield model given in To compare the fitted models, some criteria measures like; Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), Hannan-Quinn information criterion (HQIC) and the Kolmogorov-Smirnov (K-S) statistics are considered.Generally, the smaller values of these statistics are corresponding to the better fit model to the data.The mathematical form of these measures is as follows , where k is the number of models parameter,  is the sample size and ln l is the maximized value of the log-likelihood function under the fitted models.Table 3 lists the numerical values of the statistics measures.As seen from the above two figures that, the PLP distribution provides a closer fit to the histogram and then it is the best model among the other models to analyze these data.

Example 6.2:
The second data set represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal (1960).The data are as follows:  The values in Table 4 indicate that the most fitted distribution to the data is PLP distribution compared to other distributions.Also, it gives the best fit to these data.

Concluding Remarks
In this paper we have introduced a new four-parameter compounding distribution, called the power Lomax Poisson distribution.Statistical properties of the new distribution such as, moments, mean residual life, order statistics, quantile measures and Re'nyi entropy are obtained.Three methods of estimation, namely; maximum likelihood, least squares, and weighted least squares are proposed to estimate the model parameters and simulation results are provided to assess the model performance.The PLP model is fitted to two real life data sets to illustrate the usefulness of the proposed distribution.The new model provides consistently a better fit than the other competitive models.

Figure 1 :Figure 2 :
Figure 1: Pdfs of the PLP distribution for some parameter values Several functions, namely; the failure rate function, mean residual life function and the left censored mean function are related to the residual life.These three functions uniquely determine ().The  th moment of residual life denoted by   () = [( − )  |  > ],  = 1, 2, 3, . .., is derived.The th moment of the residual life of a random variable is defined as follows The maximum likelihood estimators (MLEs) of the model parameters are determined by solving numerically the non-linear equations

𝑋 1 :
<  2: < … <  : denotes the corresponding ordered sample.According toJohnson et al. (1995), the expectation and the variance of distribution are independent of the unknown parameter and are given by : ( : ) is cdf for any distribution and  : is the  th order statistic.Then the least squares (LS) estimators can be obtained by minimizing the sum of squares errors the unknown parameters.So the least squares estimators of the unknown parameters , ,  and , denoted by ̈,  ̈,  ̈ and  ̈, of the PLP model can be obtained by minimizing , ,  and .

Figure 3 :Figure 4 :Figure 5 :Figure 6 : 3 .
Figure 3: The MSE for  of the first case based on MLE, LS, WLS methods

Figure 7 :Figure 8 :
Figure 7: MSE for LSs for the third case of parameters

Figure 9a : 6 .
Figure 9a: MSEs of  for all the cases based on different methods

Figure 10 a
Figure 10 a: MSEs of the estimate  for all the cases based on different methods

Figure. 11 Figure 12 .
Figure.11 Estimated cumulative densities of models for the first data set Generalizations of the Lomax distribution have been formulated by several authors.For example, Ghitany et al. (2007) proposed a new distribution using the Marshall-Olkin generator.Abdul-Moniem and Abdel-Hameed (2012) introduced the exponentiated Lomax by adding a new shape parameter to the Lomax distribution.Lemonte and Cordeiro (2013) investigated beta Lomax, Kumaraswamy Lomax and McDonald Lomax.Cordeiro et al. ( al. (2016).Rady et al. (2016) proposed a recent development called the power Lomax (PL) distribution as a new extension of the Lomax distribution by considering the power transformation , where the random variable T follows Lomax distribution with parameters  and  .

6. Order Statistics
Suppose  1 ,  2 , … ,   is a random sample from PLP distribution.Let  1: ,  2: , … ,  : denote the corresponding order statistics.It is well known that the probability density function of the th order statistics is given by   1 : 0

Table 1 :
Results of simulation study of MSEs, RBs and SEs of estimates for different values of parameters (, , , ) for the power Lomax Poisson distribution

Table 2 :
Results of simulation study of MSEs, RBs and SEs of estimates for different values of parameters (, , , ) for the power Lomax Poisson distribution