Generalized Bur X Lomax Distribution: Properties, Inference and Application to Aircraft Data

We proposed and studied a flexible distribution with wider applications called Generalized Burr X Lomax (GBX-L) distribution. Some well-known mathematical properties such as ordinary moments, incomplete moment probability weighted moments, stress-strength model, mean residual lifetime, characteristic function, quantile function, order statistics and Renyi entropy of GBX-L distribution are investigated. The expressions of order statistics are derived. Parameters of the derived distribution are obtained using the maximum likelihood method and simulation studied is carried out to examine the validity of the method of estimation. The applicability of the proposed distribution is exemplified using aircraft data.

Such advantage includes the ability of the new distribution to model data of any shape of the hazard function which further extend its scope of applications.Based on this, we are proposing a new extension of the Lomax distribution developed by using Generalized Bur X family of distributions which added two extra shape parameters to the Lomax distribution that induces flexibility into the Lomax distribution and also improve its fits by controlling its skewness and kurtosis for a better modeling capability.The new model is called Generalized Burr X Lomax (GBX-L) distribution.
The rest of the paper is organized as follows.In Section 2, we introduce the GBX-L model.Section 3 focuses on its properties.In Section 4, we carried out the estimation procedures and also simulation of data set and finally in Section 5, we concluded.

Generalized Burr X Lomax (GBX-L) Distribution
Let (; ) and (; ) represents the  and  of the baseline distribution with parameter vector .Then, according to Aldahlan and Khalil (2021), the CDF of the  −  family is given by () = 1 −  − (; ) 1 − () (3) And its associated PDF is given by By taking F(x) as the CDF of the Lomax distribution in (1), we obtain the CDF OF GBX-L distribution with a wide range of applications including ecology, medicine, and reliability.The four-parameter GBX-L distribution is given by for  > 0; , , ,  > 0. The corresponding PDF is given by for  > 0, , , ,  > 0.

Reliability, Hazard, Cumulative Hazard Function of GBX-L Model
The Reliability function [()] and the hazard function [ℎ()] for the GBX-L will be obtained in this sub-section.
Using some values of , , , and , some plots of the survival and the hazard function are presented.The Reliability and the hazard function of the GBX-L are respectively given by for  > 0,  > 0,  > 0,  > 0, and  > 0, where, The graph the hazard function is drawn below in figure 2 for various values of , , , and .This graph indicates that new GBX-L model is capable of modeling upside-down bathtub (unimodal), increasing, decreasing hazard rate functions which are widely used in engineering for repairable systems.

Some Statistical Properties
This section provides some statistical properties of GBX-L distribution.

Quantile Function GBX-L Distribution
Let X denotes a random variable has the pdf (6), the quantile function, say () of X is given by

Important Representation
In this subsection, a useful expansion of the pdf for GBX-L is provided.
Since the generalized binomial series is And || < 1 and  is a positive real non-integer.Then, by applying the binomial theorem ( 10) in ( 11), the density function of GBX-L distribution becomes () =  , , (; , ([ + 1] + 1)) where and (; , ([ + 1] + 1)) is the Lomax PDF with parameters with positive shape parameter ([ + 1] + 1) and positive scale parameter .This shows that the GBX-L model can be written as a linear combination of Lomas density functions.Hence mathematical properties of the GBX-L can be obtained from the Lomax properties.
Hence, the pdf (6) can be written as (12)
Finally, we obtained an expression for the  moment of  −  given as

Incomplete Moment of GBX-L Distribution
More importantly, the first incomplete moment can be used to obtain the mean and the Bonferroni and Lorenz curves.The curves are very important in reliability, economics, demography, medicine, insurance and many others.The   incomplete moment, say   () can be expressed from (12) as Taking  =   ⁄ ,  = , then (17) we transform to By letting,  = , 1 +  = (1 − ) ,  = (1 − )  and putting it in (18), we have Finally, we have

The Probability Weighted Moments
Considering the class of moments, called the probability-weighted moments (PWMs), has been proposed by (Greenwood et al. (1979)).This class is used to derive estimators of the parameters and quantiles of distributions expressible in inverse form.For a random variable , X the PWMs, denoted by  , , can be calculated through the following relation The PWMs of  −  distribution is obtained by substituting ( 5) and ( 6) into (19), and using the binomial series given in ( 10) and ( 11 () =  −   ∑  , ,  ( + 1), ( + 1) −  ;

Stress-Strength Model of 𝑮𝑩𝑿 − 𝑳 Distribution
Stress-Strength model is the most commonly used approach in reliability estimation.This model can be applied in engineering and physics as system collapse and strength failure.In stress-strength modeling, Ɽ = (  <   ) is a measure used in determining the reliability of a system when it is subjected to random stress   and has strength   .The system fails if the applied stress is greater than its strength and function satisfactorily whenever   >   .Ɽ can be taken a measure of system performance is commonly encountered in electrical and electronic systems.Where,  =  +  +  +  +  and  =   ( +  + ) +   ( + ) −

Order Statistics of 𝑮𝑩𝑿 − 𝑳 Distribution
Order statistics have been extensively studied and found applications in many applied fields of statistics, such as reliability and life testing.Let  ,  , … ,  be independent and identically distributed (i.i.d) random variables with their corresponding continuous distribution function ( ) F x .Let  : <  : < ⋯ <  : the corresponding ordered random sample from a population of size .According to (David, 1981), the pdf of the  order statistic, is defined as Putting ( 5) and ( 6) in ( 25), we have Putting equation ( 5) and ( 6) in ( 26) and thereafter applying ( 10) and ( 11), the  order statistics of GBXL is given by (27)

Renyi Entropy of 𝑮𝑩𝑿 − 𝑳 Distribution
The entropy of a random variable X is a measure of variation of uncertainty and has been used in many fields such as physics, engineering and economics.As mentioned by (Renyi 1961), the Renyi entropy is defined by By putting ( 6) in ( 28) and applying the binomial theory given ( 10) and ( 11), then the pdf () can be expressed as follows Finally, an expression for the entropy of  −  distribution is given by

Simulation Study
To conduct a simulation study, ( 9) is used to generate random data from the Generalized Bur X-Lomax distribution.The simulation experiment is repeated for 1000 times each with sample of size n = 50, 100, 150 and 200 for parameter values of  = 0.5,  = 0.5,  = 1.3, and  = 1.3.

Maximum Likelihood Method
This section deals with the maximum likelihood estimators of the unknown parameters for the GBXL distributions based on the principle of complete samples.Let  ,  , … ,  represent the observed values from the GBXL distribution with set of parameter  = (, , , ) .The log-likelihood function  =  for parameter vector  = (, , , ) is obtained as follows The maximum likelihood (ML) method and its procedures exist in the literature with details.

Conclusion
In the present paper, the new Generalized Burr X Lomax distribution is proposed and studied.Some characteristics of the GBX-L distribution, such as, expressions for the density function, moments, incomplete moment, probability weighted moments, characteristic function, quantile function, mean residual life, stress-strength model, orders statistics and Renyl entropy are discussed.The maximum likelihood estimation technique is employed for estimating the model parameters.Aircraft data is employed to validate the relevance of GBL-X Model when compared to other models such as Harris power Lomax, Harris Lomax, Power Lomax, and Lomax models.we also carried out data simulation to validate the method of estimation.https://doi.org/10.1080/00401706.1974.10489150 Cordeiro, G., Ortega, E., & Popovic, B. (2013).The gamma Lomax distribution.Journal of Statistical Computing and

Figure 1 .
Figure 1.Plot of the CDF and the PDF of GBX-L distribution for different values of , , , and

Figure 2 .
Figure 2. Plot of the hazard function of GBX-L model Function of GBX-LThe characteristic function of a distribution is always unique and is related to the moments of the distribution by  () = ( ) =  () = (in (21), we obtained the characteristics function of  −  distribution as integral () is the incomplete moment of GBX-L and  is the first moment obtained by taking  = 1 in (16) and () is given in (7).
= −2 + 2,  = −2 + (),  = −2 + 2[log ()] and  = −2 + 2/( −  − 1)Where,  is the maximized likelihood function,  stands for the number of the model parameters and  is the sample size of the data considered.The model with minimum AIC (or CAIC, BIC, and HQIC) value is chosen as the best model to fit the data.Finally, we provide a representation of the histograms of the data sets and plot the fitted density functions to obtain a visual representation of the data set.The data set represents the failure times of 84 Aircraft windshields recently studied byRamos et al. (2013).The data set values are 0.040, 1

Figure 3 .
Figure 3. Graph of TTT plot (Diagram I) and Box plot (Diagram II) (LR) statistic was obtained for testing the hypotheses  : =1   =    e, that is to compare the  −  model with the  model.The LR statistic  = -2{130.063-200.268}= 140.41(−  < 0.01), sufficient to show that the  −  model is a better model that can be used to fit the data.

Figure 4 .
Figure 4. Plots of estimated CDF and histogram fitted PDF of the fitted models for the aircraft data
Table2represents the maximum likelihood estimates of the GBX-L model for aircraft data.Table3represents the goodness of fit criteria including AIC, CAIC, BIC, and HQIC.The numerical values in Table3indicate that the GBX-L model has the minimum value of the information criterion than all other models considered.Hence, we conclude that the GBX-L distribution perform better as compared to Harris Power Lomax, Power Lomax, Harris Lomax and Lomax distribution.