On Generalized Log Burr Xii Distribution

In this paper, we present flexible generalized log Burr XII (GLBXII) distribution developed on the basis of generalized log Pearson differential equation. GLBXII distribution is also obtained from compounding mixture of distributions. Some structural and mathematical properties including moments, inequality measures, uncertainty measures and reliability measures are theoretically established. Characterizations of GLBXII distribution are also studied through different techniques. Parameters of GLBXII distribution are estimated using maximum likelihood method. Goodness of fit of probability distribution through different methods is studied.


Introduction
suggested 12 distributions as Burr family to fit cumulative frequency functions on frequency data.The Burr-XII (BXII) distribution is widely used in modeling lifetime, finance and insurance data analysis.The Burr-XII (BXII) distribution also has wide applications in reliability and acceptance sampling plans.In many areas of applications, the Burr distribution of any type is inadequate and as such various generalizations and extensions of the Burr distribution have been proposed in the statistical literature.
The cumulative distribution function (cdf) for random variable Y having Burr XII distribution is ( ) ( ) The probability density function (pdf) of Burr XII distribution for random variable Y is ( ) 1 1 1 , y 0, 0, 0. g y y y Recently, new generated families of continuous distributions have attracted several statisticians to develop new models.These families are obtained by introducing one or more additional threshold/shape/scale parameter(s) or using some transformation to the baseline distribution.
Many authors like Zimmer and Burr (1963), Takahasi (1965), Tadikamalla (1980), Saran and Pushkarna (1999), Begum and Parvin (2002), Shao et al. (2004), Olapade (2008) In this paper, a new family of distribution is studied, that is more flexible probability model in fitting lifetime data, annual maximum wave height and strengths of materials, describe special types of hazard functions, study positively skewed and heavy tailed data sets and provide better fits for survival data and waiting times than other competing models.
The article is composed of the following sections.In section 2, the pdf for GLBXII distribution is developed on the basis of the generalized log Pearson differential equation (GLPE).In section 3, GLBXII distribution is studied in terms of some structural properties, plots, sub-models and stochastic ordering.In section 4, moments, negative moments, central moments, incomplete moments, inequality measures, residual life functions and some other properties are presented.In section 5, reliability measures and uncertainty measures are studied.In section 6, GLBXII distribution is obtained from some compound scale mixture of generalized log-Weibull (GLW) distribution and gamma distribution and sized biased Erlang distribution.In section 7, characterizations of GLBXII distribution is studied through (i) Truncated moment of the log of the random variable; (ii) Truncated moment of a function (not log) of the random variable; (iii) the hazard function and (iv) certain functions of the random variable.In section 8, the potentiality of GLBXII distribution is demonstrated by its application to real data sets; parameters of GLBXII are estimated using maximum likelihood method.Goodness of fit of probability distribution through different methods is studied.Finally, in Section 9, we provide some concluding remarks.

Development of Glbxii Distribution
In this section, we derive the pdf of GLBXII using GLPE given by mn For a b a b = − = − , after simplification and integration of both sides of (4), we have ( where ( ) , B the beta function.The normalizing constant c is ( ) The cdf of the random variable X with GLBXII distribution and parameters ,p a b and is ( )

Mathematical Properties Of Glbxii Distribution
In this section, the GLBXII distribution is studied in terms of some structural properties, plots, sub-models and stochastic ordering.

Structural Properties of GLBXII Distribution
The survival, hazard, cumulative hazard and reverse hazard function of a random variable X with GLBXII distribution are given, respectively, by ( ) The mills ratio of GLBXII distribution is The elasticity of GLBXII distribution is given by ( ) ( ) The mode of GLBXII distribution with pdf ( 6) is obtained by solving , for 1 2 a = , we obtain mode

Quantile function of GLBXII Distribution
The quantile function of GLBXII distribution is ( ) and random number generator for GLBXII distribution is , where the random variable U has the uniform distribution on (0,1).

Plots of the GLBXII Density and Hazard Rate Function
The following graphs show that shapes of GLBXII density are arc, exponential, positively skewed and symmetrical.The GLBXII distribution has increasing, decreasing, upside-down bathtub and constant after some time.
then GLBXII distribution is ordered strongly according to likelihood ratio ordering.
Proof: For ( ) ( ) then GLBXII distribution is ordered strongly according to hazard rate ordering.
Proof: For ( ) ( ) Therefore for GLBXII, random variable X is said to be smaller than a random variable Y in hazard rate order

Moments
Moments, negative moments, central moments, incomplete moments, inequality measures, residual life functions and some other properties are achieved.

Moments of GLBXII Distribution
The r th moment about the origin of X with GLBXII distribution is The mean and variance of GLBXII distribution are The fractional positive moments of X with GLBXII distribution are The r th negative moment about the origin of X with GLBXII distribution is ( ) Negative moments help to harmonic mean and many other measures.The fractional negative moment of X with GLBXII distribution is The k th moment about mean of X is determined from the relationship The kth moment about mean of X with GLBXII distribution is determined from the relationship  is given by ( )  is given by ( ) The moment generating function for the random variable X having GLBXII distribution is The cumulants are obtained from the recurrence relation The cumulants of GLBXII distribution are obtained from the recurrence relation The r th moment about the origin of ln(X) with GLBXII distribution is The factorial moments for GLBXII distribution are given by

The Mellin Transform of GLBXII Distribution
The Mellin transform helps to determine moments for a probability distribution.By definition, the Mellin transform is The Mellin transform of X with GLBXII distribution is written as

Moments of Order Statistics
Moments of order statistics have wide applications in reliability and life testing.Moments of order statistics also design to replacement policy with the prediction of failure of future items determined from few early failures.
The pdf for m th order statistic m:n X is The pdf of m th order statistic m:n X for GLBXII distribution is ( ) The r th moments about the origin of m th order statistic m:n X for GLBXII distribution is given by ( ) Mean of m th order statistic m:n X for GLBXII distribution is given as

L-Moments and TL-Moments
L-moments (Hosking;1990) are used to estimate the parameters.The r th L-moment for a probability distribution is ( ) ) The Legendre polynomial is and the r th shifted polynomial is According to Hosking (1990), the r th L-moments for a probability distribution are ( ) ( )( ) ( ) ( ) The r th L-moments for a GLBXII distribution are ( ) ( )( ) ( ) The probability weighted moments (PWM) for GLBXII distribution are ( ) TL-moments (Elamir and Seheult 2003) are stronger than L-moments due to trimming of outliers.TL-moments provide best estimates of the parameters for the probability distributions.
The r th TL-moment for a probability distribution is defined as ) The r th TL-moments for GLBXII distribution is given by

Incomplete Moments
Incomplete moments are used in mean inactivity life, mean residual life function, and other inequality measures.The lower incomplete moments for random variable X having GLBXII distribution are where ( ) The upper incomplete moments for random variable X having GLBXII distribution are ( ) 14) and (36), we have ( ) The mean deviation about mean is ( ) ( ) , and mean deviation about median is ( ) ( ) q Q p = .

Residual Life functions
The residual life, say, ( ) n mz of X with GLBXII distribution has the following n th moment From above equation and (37), we have The average remaining lifetime of a component at time z, say, ( ) mzor life expectancy is known as mean residual life (MRL) function is given by The reverse residual life, say, ( ) From above equation and (36), we have The waiting time z for failure of a component has passed with condition that this had happened in the interval [0, z] is called mean waiting time (MWT) or mean inactivity time.The waiting time z for failure of a component of X having GLBXII distribution is defined by The median inactivity time function in terms of cdf of a continuous life time distribution is ( ) ( ) .
The median inactivity time function in terms of cdf of GLBXII distribution is written as ( )

Reliability and Uncertainty Measures
In this section, reliability and uncertainty measures are studied.

Stress-strength Reliability for GLBXII Distribution
If Therefore (i) R is independent of a and b .(ii) for 12 pp = , R=0.

Shannon Entropy and Awad Entropy
According to Shannon (1948), the measurement of expected information in a message is called entropy.Shannon entropy for random variable X with pdf ( 6) is given as Shannon entropy for GLBXII random variable X with pdf ( 6) is given by ( ) where ( )   and ( ) where  is maximum value of ordinate of GLBXII distribution in the domain of X.
If random variable X has GLBXII distribution, then Awad entropy is given by

Rényi Entropy, Q-Entropy, Havrda and Chavrat Entropy and Tsallis-Entropy
Rényi entropy ( 1961) is an extension of Shannon entropy.Rényi entropy for GLBXII random variable X with pdf ( 6) is theoretically computed as ( ) The Q-entropy for GLBXII distribution is ( ) ( ) The Havrda and Chavrat entropy (1967) for GLBXII distribution is The Tsallis-entropy (1988) for GLBXII distribution is Shannon entropy, Collision entropy (quadratic entropy), Hartley entropy and Min entropy can be obtained from Rényi entropy.For 1 v → , Rényi entropy tends to Shannon entropy. 2 For v → , Rényi entropy tends to quadratic entropy.Entropies are applied to study heart beat intervals cardiac autonomic neuropathy (CAN), DNA sequences, anomalous diffusion, daily temperature fluctuations (climatic), and information content signals.

Compound Probability Distribution
In this section, GLBXII distribution is obtained from some compound scale mixture of GLW distribution and gamma distribution and sized biased Erlang distribution., GLBXII distribution will become GLW distribution with the pdf .Then X has pdf (6).

Proof:
For compounding ( is pdf of GLBXII distribution.

Compound Scale
Then X has pdf (6).
is pdf of GLBXII distribution.

Characterization
In this section, GLBXII distribution is characterized through: (i) Truncated moment of the log of random variable (ii) Truncated moment of a function (not log) of random variable; (ii) Truncated moment of a function (not log) of the random variable; (iii) the hazard function and (iv) certain functions of the random variable.One of the advantages of characterization (ii) is that the cdf is not required to have a closed form.We present our characterizations (i) -(iv) in four subsections.

Characterization Based on Truncated Moment of Log of the Random Variable
Here is our first characterization of GLBXII distribution.
Upon integration by parts and simplification, we arrive at .
After simplification and integration we arrive at ( )

Characterizations Based on Truncated Moment of A Function of The Random Variable
In this subsection we first present a characterization of GLBXII distribution in terms of a simple relationship between truncated moment of function of X and another function.This characterization result employs a version of the theorem due to Glänzel (1987); see Theorem 1 of Appendix A. Note that the result holds also when the interval H is not closed.Moreover, as mentioned above, it could be also applied when the cdf F does not have a closed form.As shown in Glänzel (1990), this characterization is stable in the sense of weak convergence.: 1, X  →  be a continuous random variable.The pdf of X is (6) if and only if there exist functions ( ) ( ) x and q x  defined in Theorem 1satisfying the differential equation .
The general solution of differential equation in corollary is where D is constant.Note that a set of functions satisfying above differential equation is given in Proposition 7.2.1 with D=0.However, it should also be noted that are other pairs ( ) , q  satisfying conditions in Theorem 1.

Characterization Based on Hazard Function
It is well known that hazard function, F h of a twice differential distribution function, F, satisfies the first order differential equation For many univariate continuous distributions, this is the only characterization available in terms of the hazard function.The following characterizations establish a non-trivial characterization of GLBXII distribution which is not of the above trivial form.

Characterization based on certain functions of the random variable
The following proposition has already appeared in Hamedani (2013).So we will just state it here which can be used to characterize GLBXII distribution.( ) Let X: 1,  →  be continuous random variable with cdf (7).Let

( )
x  be differentiable function ( ) x  , we chose the case for simplicity.

Maximum Likelihood Estimation
In this section, parameters estimates are derived using maximum likelihood Method.Generalized Log-Burr XII (GLBXII), Log-Burr XII (LBXII), Log-Lomax (LL), Generalized Burr XII (GBXII), Burr XII, probability distribution are fitted to real data sets of survival times of patients (Harter and Moore;1965) and eruptions data for comparison purpose.The likelihood function for GLBXII distribution with the vector of parameters ( ) In order to compute the estimates of parameters of GLBXII distribution; the following nonlinear equations must be solved simultaneously.
The maximum likelihood estimates of parameters and values of goodness of fit measures are computed via Adequacy Model with method "L-BFGS-B" in R for GLBXII distribution, its sub-models and its competing models.Table 3 displays MLEs and their standard errors (in parentheses) for data set II. Table 4 displays goodness-of-fit values.

Concluding Remarks
We have developed GLBXII distribution on the basis of the GLPE.We have studied certain properties including structural properties, plots, sub-models, moments, factorial moments, moments of order statistics, L-moments, TL-moments, incomplete moments, inequality measures, residual life functions, reliability and uncertainty measures and compounding.The GLBXII distribution is characterized via different techniques.Maximum Likelihood estimates are computed.Goodness of fit shows that GLBXII distribution is better fit.Two applications of the GLBXII model to survival times of patients and eruptions data are illustrated to show significance and flexibility of GLBXII distribution.We have proved that GLBXII distribution is empirically better for lifetime applications.

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Bonferroni and Lorenz curves for a specified probability p are computed by distribution and  have sized biased moment Erlang distribution with pdf (

7 . 4 . 1 ,
It is easy to see that for certain functions, e.g. ( ) Proposition 7.4.1 provides a characterization of GLBXII distribution.Clearly there are other functions ( ) , Paranaiba et al. (2011), Usta (2013) and Paranaiba et al. (2013) studied BXII distribution.Okasha et al. (2015) studied BXII distribution and its various properties.Silver and Cordeiro (2015) developed a new compounding family by mixing BXII and power series distribution.Gomes et al. (2015) presented McDonald Burr XII distribution along with properties and application.Muhammad (2016) established generalized BXII-Poisson distribution to analyze strength of material.Afify et al. (2016) considered Weibull BXII distribution for analyzing the strengths of glass fibers.Thupeng (2016) modeled concentrations of daily extreme nitrogen dioxide with BXII distribution.Doğru and Arslan (2016) estimated the parameters of BXII distribution with optimal B-robust estimators.Yari and Tondpour (2017) derived a new Burr distribution to study the lifetime cancer data.Ghosh and Bourguignon (2017) presented properties of the extended BXII distribution.Mdlongwa et al. (2017) studied properties and applications of BXII modified Weibull distribution.
Mead and Afify (2017)also presented properties and applications of BXII distribution with five parameters.Bhatti et al. (2017) have studied the generalized log Pearson differential equation (GLPE) to develop probability distributions with various choices of the coefficients.Guerra et al. (2017) developed gamma burr XII distribution with flexible hazard function and studied different properties and application.Cadena (2017) studied different behaviors of extended Burr XII distribution.Kumar (2017) studied different properties of Burr type XII distribution.

2018 pp615-643 630 5.2 Estimation of Multicomponent Stress-Strength system Reliability for GLBXII Distribution Suppose
5. It means that a machine has at least "s" components working out of "k" component.The strengths of all components of system are 12 X , X ,....X k and stress Y is applied on 32X and X are independently identically distributed (i.i.d) and there is equal chance that 1 X is bigger than 2 X .Pak.j.stat.oper.res.Vol.XIV No.3