A New Flexible Version of the Lomax Distribution with Applications

A new version of the Lomax model is introduced and studied. The major justification for the practicality of the new model is based on the wider use of the Lomax model. We are also motivated to introduce the new model since the density of the new distribution exhibits various important shapes such as the unimodal, the right skewed and the left skewed. The new model can be viewed as a mixture of the exponentiated Lomax distribution. It can also be considered as a suitable model for fitting the symmetric, left skewed, right skewed, and unimodal data sets. The maximum likelihood estimation method is used to estimate the model parameters. We prove empirically the importance and flexibility of the new model in modeling two types of aircraft windshield lifetime data sets. The proposed lifetime model is much better than gamma Lomax, exponentiated Lomax, Lomax and beta Lomax models so the new distribution is a good alternative to these models in modeling aircraft windshield data.


Introduction
A random variable (rv) W has the Lomax (Lx) distribution with two parameters λ and β if it has cumulative distribution function (CDF) (for w > 0) given by G λ,β (w) = 1 − where λ > 0 and β > 0 are the shape and scale parameters, respectively.Then the corresponding PDF of (1) is In the literature, the Lomax (Lx) or Pareto type II (PaII) model (see Lomax (1954)) was originally pioneered for modeling business failure data.The Lx distribution has found a wide application in many fields such as biological sciences, ctuarial science, engineering, size of cities, income and wealth inequality, amedical and reliability modeling.It has been applied to model data obtained from income and wealth (Harris (1968) and Atkinson and Harrison (1978)), firm size (Corbellini et al., (2007)), reliability and life testing (Hassan Al-Ghamdi (2009)), Hirschrelated statistics (Glanzel (2008)), for modeling gauge lengths data (Afify et al., (2015)), for modeling bladder cancer patients data and remission times data (Yousof et al., (2016) and Yousof et al., (2018)).According to Yousof et al. (2016) the CDF of the Burr X generator (BrX-G) is The PDF of the BrX-G is given by where θ is the shape parameter, g(x; ξ) and G(x; ξ) denote the PDF and CDF of the baseline model with parameter vector ξ and 1 − G(x; ξ) = G(x; ξ).Inserting (1) in to (3) we get the the CDF of the Burr X Lomax (BrXLx) as The PDF of the BrXLx is given by We are motivated to introduce the new model since the PDF of the new distribution exhibits various important shapes such as the unimodal, the right skewed and the left skewed (see figurue 1).The new model can be viewed as a mixture of the exponentiated Lx distribution (see Subsection 2.1).It can also be considered as a suitable model for fitting the symmetric, left skewed, right skewed, and unimodal data sets (see aplications Section).The maximum likelihood estimation method is used to estimate the model parameters.We prove empirically the importance and flexibility of the new model in modeling two types of aircraft windshield lifetime data sets.The proposed lifetime model is much better than gamma Lx, beta Lx, exponentiated Lx and Lx models so the new model is a good alternative to these models in modeling aircraft windshield data.

Linear Representation
In this section, we provide a very useful linear representation for the BX-G density function.If |s| < 1 and b > 0 is a real non-integer, the power series holds Applying ( 7) to (6) we have Applying the power series to the term exp Consider the series expansion Applying ( 10) to (9) for the term .
This can be written as where .
The CDF of the BrXLx , similarly, can also be expressed as a mixture of exp-Lx CDFs as where is the CDF of the exp-Lx model with power parameter 2 j + k + 2.

Moments and Generating Function
The r−th ordinary moment of X is given by Then we obtain where B(•, •) is the complete beta function, setting r = 1 in (13), we have the mean of X Setting r = 2.3 and 4 in ( 13), we have the 2-nd, 3-rd and the 4-th moments about the origin which can be used to obtain the central moments.We proved, numerically, that the BrXLx model provides better fits than other four competitive extensions of the Lx models (see Section 6) so the BrXLx model is a exemplary alternative to these mosels.The skewness (Skew(X)) of the BrXLx distribution can range in the interval (35.23, 0.17), whereas the kurtosis (Kur(X)) of the BrXLx distribution varies only in the interval (1685.7,2.61) also the mean of X (Mean (X)) increases as θ increases, the skewness is always positive (see Table 1 and 2).
Clearly, the first one can be derived from equation (10) as

Incomplete Moments and Mean Deviations
The s−th incomplete moment, say I s (t), of X can be expressed from (10) as The mean deviations about the mean and about the median of X , F(E (X)) is easily calculated from (5) and I 1 (t) is the first incomplete moment given by ( 14) with s = 1.Now, we provide two ways to determine MD (mean) and MD (median) .The I 1 (t) can be derived from ( 14) as

Probability Weighted Moments (PWM)
The (s, r)−th PWM of X following the BrXLx model, say ρ s,r , is formally defined by Using ( 5), ( 6) we can write where Then, the (s, r)−th PWM of X can be expressed as

Moments of the Reversed Residual Life
The n−th moment of the reversed residual life, say uniquely determines F θ,λ,β (x).We obtain Then, the n−th moment of the reversed residual life of X becomes where and is the incomplete beta function.

Reliability Estimation
The most widely approach used for reliability estimation is the stress-strength model (SSM), this model is used in many applications of physics and engineering such as strength failure and system collapse.In SSM, we have is a measure of reliability of the system when it is subjected to random stress T 2 and has strength T 1 .The system fails if and only if the applied stress is greater than its strength (T 1 < T 2 ).Other interpretation can be given as the reliability ) of a system is the probability that the system is strong enough to overcome the stress imposed on it.Let T 1 and T 2 be two independent rvs with BrXLx(θ 1 , λ, β) and BrXLx(θ 2 , λ, β) distributions, respectively.The PDF of T 1 and the CDF of T 2 can be written from Equations ( 6) and ( 5), respectively as and .
Then, the reliability is defined by We can write where .
Thus, the reliability, R ( , can be expressed as r j,k,w,m .

Order Statistics
Let X 1 , . . ., X n be a random sample (RS) from the BrXLx distribution and let X 1:n , . . ., X n:n be the corresponding order statistics.The PDF of i−th order statistic, say X i:n , can be expressed as using ( 5), ( 6) and ( 15) we get where The PDF of X i:n can be written as Then, the density function of the BrXLx order statistics is a mixture of eponentiated Lomax (ELx).The q−th moments of X i:n can be expressed as where a w,k (−1) j+m (2

Quantile Spread (QS) Ordering
The QS of the rv U ∼BrXLx(θ, λ, β) having CDF ( 5) is given by where is the survival function.The QS of a any distribution describes how the probability mass is placed symmetrically about its median and hence it can be used to formalize concepts such as peakedness and tail weight traditionally associated with the kurtosis.So, it allows use to separate concepts of the kurtosis and peakedness for asymmetric models.Let U 1 and U 2 be two rvs following the BrXLx model with QS U 1 and QS U 2 .Then U 1 is called smaller than U 2 in quantile spread order, denoted as Following are some properties of the QS order which can be obtained.
The order ≤ {QS } is a location-free The order Let F U 1 and F U 2 be symmetric, then The order ≤ {QS } implies ordering of the mean absolute deviation around the median, say π(

Entropies
The Rényi entropy is defined by Using PDF (4), we can write where Then, the Rényi entropy of the BrXLx is given by The δ-entropy, say E δ (X), can be obtained as The Shannon entropy of a rv X, say S E, is defined by follows by taking the limit of I δ (X) as δ tends to 1.
For determining the maximum likelihood estimation (MLE) of Θ, we have the log-likelihood (Log L) function The components of the score vector, U (Θ) = ) , are availble if needed, Via setting the nonlinear system of equations U θ = 0, U γ = 0 and U β = 0 and solving them simultaneously yields the MLE Θ = ( θ, λ, β, ) .To solve those equations, it is usually more convenient to use the nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ(Θ).

Applications
In this section, we provide two applications to two real data sets to prove the importance and flexibility of the BrXLx distribution.We compare the fit of the BrXLx with competitve models namely: ELx model (Gupta et al., 1998), gamma Lomax (GLx) model (Cordeiro et al., 2015), beta Lomax (BLX) model (Lemonte and Cordeiro, 2013) and Lx model.The CDFs of these distributions are, respectively, given by (for x > 0 and α, β, λ, a > 0): where Γ ( •; •) is the incomplete gamma function.
In order to make a real comparison among the distributions, the estimated Log L values ( ℓ), Akaike Information Criteria (AIC), Cramer von Mises (W * ) and Anderson-Darling (A * ) goodness of-fit statistics were calculated for all competitive models.The statistics W * and A * were defined in Chen and Balakrishnan (1995) with details.In general, it can be chosen as the best model which has the smaller values of the W * , A * and AIC statistics and the larger values of ( ℓ).The below computations are obtained via the "maxLik" and "goftest" sub-routines using the R-software.The analysis results are listed in Tables 3, 4, 5 and 6.These results show obviously that the new distribution has the lowest W * , AIC and A * values and has the biggest estimated − ℓ among all the fitted models.Hence, it could be chosen as the best model under these criteria.From tables 3 and 5, the new model is much better than all competitive (BLx, ELx, GLx and Lx) models so the new model is a adequate alternative to these models in modeling aircraft windshield data.

Conclusions
In this work, a new lifetime model called the Burr X Lomax (BrXLx) is introduced and studied.The major justification for introducing and studying the BrXLx model is based on the wider use of the Lx model in applied fields.We are also motivated to introduce and study the BrXLx model since the density of the BrXLx distribution displays various important shapes such as the unimodal, the right skewed and the left skewed.The new model can be viewed as a mixture of the exponentiated Lx distribution.It can also be considered as a convenient model for fitting the symmetric, the left skewed, the right skewed, and the unimodal data sets.The maximum likelihood estimation method is used to estimate the BrXLx parameters.We prove empirically the importance and flexibility of the BrXLx in modeling two types of aircraft windshield lifetime data.The proposed BrXLx lifetime model is much better than gamma Lomax, beta Lomax, exponentiated Lomax and Lomax models so the exponentiated Lomax, model is a good alternative to these models in modeling aircraft windshield data.

Figure 1 .
Figure 1.Plots of the BrXLx PDF

Table 1 .
Mean, variance, skewness and kurtosis of the BrXLx distribution with λ = β = 0.5 and different values of θ

Table 2 .
Mean, variance, skewness and kurtosis of the BrXLx distribution with λ = β = 3.5 and different values of θ

Table 3 .
MLEs, standard erros of the estimates (in parentheses) for the first data set

Table 4 .
− l and goodness-of-fits statistics for the first data set Figure 3.The fitted PDF and PP plot for the first data set

Table 5 .
MLEs, standard erros of the estimates (in parentheses) for the second data setFigure 4. The fitted PDF and PP plot for the second data set

Table 6 .
− l and goodness-of-fits statistics for the second data set BrXLx 98.10294 202.206 0.08763 0.527784 * A *