The Exponentiated Burr XII Poisson Distribution with Application to Lifetime Data

A five-parameter distribution, called the exponentiated Burr XII Poisson distribution, is defined and studied. The model has as special sub-models some important lifetime distributions discussed in the literature, such as the logistic, log-logistic, Weibull, Burr XII and exponentiated Burr XII distributions, among several others. We derive the ordinary and incomplete moments, generating and quantile functions, Bonferroni and Lorenz curves, mean deviations, reliability and two types of entropy. The order statistics and their moments are investigated. The method of maximum likelihood is proposed for estimating the model parameters. We obtain the observed information matrix. An application to a real data set demonstrates that the new distribution can provide a better fit than other classical lifetime models. We hope that this generalization may attract wider applications in reliability, biology and survival analysis.


Introduction
The statistics literature has numerous distributions for modeling lifetime data.But many if not most of these distributions lack motivation from a lifetime context.For example, there is not apparent physical motivation for the gamma distribution.It only has a more general mathematical form than the exponential distribution with one additional parameter, so it has nicer properties and provides better fits.The same arguments apply to the BXII distribution, among others.Zimmer et al. (1998) introduced the three parameter Burr XII (BXII) distribution with cumulative distribution function (cdf) and probability density function (pdf) (for x > 0) given by and respectively, where k > 0 and c > 0 are shape parameters and s > 0 is a scale parameter.If c > 1, the density function is unimodal with mode at x = s [(c − 1)/(ck + 1)] 1/c and is L-shaped if c = 1.If q < c k, the qth moment about zero is µ ′ q = s q k B(k − q c −1 , 1 + q c −1 ), where B(p, q) = Γ(p) Γ(q)/ Γ(p + q) and Γ(p) = ∫ ∞ 0 x p−1 e −x dx is the gamma function.
The BXII distribution, having as sub-models the logistic and Weibull distributions, is a very popular distribution for modeling lifetime data and phenomenon with monotone failure rates.When modeling monotone hazard rates, the Weibull model may be an initial choice because of its negatively and positively skewed density shapes.Nevertheless, it does not furnish a reasonable parametric fit for non-monotone failure rates such as the bathtub shaped and unimodal failure rates that are common in reliability and biological studies.
Several other authors including El-Bassiouny and Abdo (2010), Jayakumar and Mathew (2008), Brito et al. (2014) and Ramos et al. (2015) proposed and developed the structural properties of various generalized Burr XII distributions.
The cdf and the reliability function of the three-parameter BXII distribution can be expressed in closed-form, thus simplifying the computation of the percentiles and the likelihood function for censored data.This distribution has algebraic tails that are effective for modeling failures occurring with lesser frequency than with those models based on exponential tails.Hence, it represents an adequate distribution for modeling failure time data (Zimmer et al., 1998).Shao (2004) discussed maximum likelihood estimation of its parameters and Shao et al. (2004) studied models for extremes based on the BXII distribution with application to flood frequency analysis.According to Soliman (2005), this model generalizes a large number of distributions.Its versatility and flexibility turns it quite attractive as a tentative model for lifetime data.
For an arbitrary baseline cdf G(x), a random variable is said to have the exponentiated-G ("Exp-G" for short) distribution with parameter a > 0, say X ∼ Exp-G(a), if its pdf and cdf are H a (x) = G a (x) and h a (x) = aG a−1 (x) g(x), respectively.Thus, the cdf and pdf of the exponentiated Burr XII (Exp-BXII) distribution is given by and We provide four motivations for the proposed lifetime model called the exponentiated BXII Poisson (Exp-BXIIP) distribution.The first is based on failures of a system.Suppose that a system has N serial sub-systems functioning independently at a give time, where N is a truncated Poisson random variable with probability mass function (pmf) (5) for n = 1, 2, . . .Let X denote the time of failure of the first out of the N functioning systems defined by the independent random variables Y 1 ∼ Exp-BXII(α), . . ., Y N ∼ Exp-BXII(α) given by the cdf (3).Then, X = min(Y 1 , . . ., Y N ).So, the conditional cdf of X (for x > 0) given N is where s, k, c, α, λ > 0. Hence, the unconditional cdf of X is Then, where G(x) = G(x; s, k, c) is given by (1).We refer to the distribution (6) as the Exp-BXIIP distribution.Proving a new lifetime distribution is always precious for statisticians.The fact that the new model generalizes existing commonly used distributions is also a positive point.
The survival function associated with X becomes The probability density function (pdf) corresponding to ( 6) is given by Hereafter, a random variable X with density function ( 9) is denoted by X ∼ Exp-BXIIP (s, k, c, α, λ).Plots of the density function of X for selected parameter values are displayed in Figure 1.The Exp-BXIIP hazard rate function (hrf) is given by Plots of the hazard rate functions for selected parameter values are displayed in Figure 2.For a second motivation suppose that an ith system is made of α parallel components, so that the system will fail if all of the components fail.Assume that the failure times of the components for the ith system, say Z i,1 , Z i,2 , . . ., Z i,α , are independent and identically BXII random variables with parameters s, k, c.Let Y i denote the failure time of the ith system and that there is an unknown number N of independent systems.The cdf of the failure time X of the first system out of the N functioning system is given by (6).
For the third motivation, we assume that N is the unknown number of carcinogenic cells for an individual left active after the initial treatment has pmf ( 5) and that Y i is the time spent for the ith carcinogenic cell to produce a detectable cancer mass.Assuming that Y 1 , . . ., Y N is a sequence of iid Exp-BXII random variables independent of N, the time to relapse of cancer of a susceptible individual can be modeled by the Exp-BXIIP family of distributions.
Finally, the fourth motivation considers that the failure of a device occurs due to the presence of an unknown number N of initial defects of the same kind, which can be identifiable only after causing failure and are repaired perfectly.Define by Y i the time to the failure of the device due to the ith defect, for i ≥ 1.If we assume that the Y i 's are iid Exp-BXII random variables independent of N having pmf (5), then the time to the first failure is appropriately modeled by the Exp-BXIIP distribution.For reliability studies, the Exp-BXIIP models can arise in series and parallel systems with identical components, which appear in many industrial applications and biological organisms.These points indicate that the new family of distributions is well-motivated for industrial applications and biological studies.
In this paper, we study some mathematical properties of the Exp-BXIIP model and illustrate its potentiality.In Section 2, we demonstrate that the cdf and pdf of X can be expressed as a mixture of Exp-BXII densities.Explicit expressions for the ordinary and incomplete moments are derived in Section 3. Generating and quantile functions are derived in Section 4 and 5, respectively.In Section 6, mean deviations and reliability are derived.In Section 7, we investigate the order statistics and some of their structural properties.Rényi and Shannon entropies are derived in Section 8. Maximum likelihood estimation of the model parameters is performed and the observed information matrix is determined in Section 9.In Section 10, we provide an application of the Exp-BXIIP to a real data set.
Finally, Section 11 ends with some concluding remarks.

Useful Expansions
Using the Taylor series 6) can be expressed as and then where where h ( j+1) α (x; s, k, c) denotes the Exp-BXII fdp with parameters s, k, c and power parameter ( j + 1)α.Equation ( 13) reveals that the Exp-BXIIP density function is a mixture of Exp-BXII densities.

Properties
Some of the most important features and characteristics of a distribution can be studied through moments (e.g., tendency, dispersion, skewness and kurtosis).
Theorem 1 If X ∼ Exp-BXIIP(s, k, c, α, λ), we have the following approximations: 1.1 For α > 0 and λ > 0 real non-integers, we have the mixture representation where g(x; s, k(r + 1), c) denotes the BXII density function with scale parameter s and shape parameters c and k(r + 1), and the coefficients are given by Clearly, ∑ ∞ r=0 v r = 1.Equation ( 14) reveals that the Exp-BXIIP density function is an infinite linear combination of BXII density functions.So, some structural properties of the Exp-BXIIP distribution can be obtained from those of the BXII distribution.
1.2 For α > 0 and λ > 0 real non-integers, we obtain 1.3 If n < kc, the nth ordinary moment of the Exp-BXIIP distribution is given by Proof 1.1.
First, if z ∈ R, we have the power series Second, if |z| < 1 and b is a nonnegative integer, the power series holds Using ( 18), the Exp-BXIIP density function ( 9) can be expressed as Further, using (19), we obtain Finally, we have where v r is given by (15) and g(x; s, k(r + 1), c) was defined before.
Using Theorem 1.1 we obtain ( 16) by simple integration.
The nth moment of X comes from Theorem 1.1 where Y r+1 ∼ BXII(s, k(r + 1), c).Using a result in Zimmer et al. (1998), we obtain for n < kc The central moments (µ s ) and cumulants (κ s ) of X can be determined from (17) as respectively, where κ 1 = µ ′ 1 .For lifetime models, it is usually of interest to compute the nth incomplete moment of X defined by m n (y) = ∫ y 0 x n f (x)dx.The quantity m n (y) can be calculated from ( 14) as ) c ] −1 , we can write where dt is the incomplete beta function.

Moment Generating Function
An explicit expression for M(t) can be obtained from equation ( 14) as an infinite weighted sum where M r+1 (t) is the moment generating function (mgf) of Y r+1 and v r is defined by (15).We provide a simple representation for the mgf M BXII (t) of the BXII(s, k, c) distribution.We can write for t < 0 k+1) dy.

Now, we use the Meijer G-function defined by
where i = √ −1 is the complex unit and L denotes an integration path; see Section 9.3 in Gradshteyn and Ryzhik (2000) for a description of this path.The Meijer G-function contains as particular cases many integrals with elementary and special functions (Prudnikov et al., 1986).
We now assume that c = m/k, where m and k are positive integers.This condition is not restrictive since every positive real number can be approximated by a rational number.Using the integral (38) given in Appendix A, we conclude for t < 0 that Now, from equation ( 25), the mgf of the Exp-BXIIP(s, k, c, α, λ) distribution (for t < 0) follows as Equation ( 27) is the main result of this section.For the special cases c = 1 and c = 2, we can obtain simple expressions for M BXII (t) and, consequently, for M(t) using equations (1) (on page 16) and (2) (on page 20) of the book by Prudnikov et al. (1992).For c = 1 and t < 0, we have where Γ(v, x) = ∫ ∞ x t v−1 e −st dt is the complementary incomplete gamma function.For c = 2 and t < 0, we obtain is a generalized hypergeometric function and (a) k = a(a + 1) . . .(a + k − 1) denotes the ascending factorial.

Quantile Function
The Exp-BXIIP quantile function, say x = Q(u), can be obtained by inverting (6).We have The shortcomings of the classical kurtosis measure are well-known.For example, the moments of X in (9) are valid only for n < kc.There are many heavy-tailed distributions for which this quantile is infinite.So, it becomes uniformative precisely when it needs to be.Indeed, our motivation to use quantile-based measures stemmed from the non-existence of classical kurtosis for many generalized distributions.The Bowley skewness (see Kenney and Keeping, 1962) is based on quartiles whereas the Moors kurtosis (see Moors, 1998) is based on octiles where Q(•) denotes the Exp-BXIIP quantile function given by ( 28).Plots of the B and M functions for selected parameter values are displayed in Figure 3.

Other Measures
In this section, we calculate the following measures: means deviations, Bonferroni and Lorenz curves and the reliability of the Exp-BXIIP distribution.

Mean Deviations
Here, we determine the mean deviations and Bonferroni and Lorenz curves of X.The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median.These are known as the mean deviation about the mean and the mean deviation about the median -defined by respectively, where µ ′ 1 = E(X), F(µ ′ 1 ) is obtained from (6), the median M of X is calculated from the quantile function ( 28) by M = Q(1/2) and m 1 (q) = ∫ q 0 x f (x)dx is the incomplete mean of X given by ( 24) with n = 1.Setting u = y c , we can write from equation ( 14) where the integral can be calculated using Maple as Here, where csc(•) is the cosecant function and 2 F 1 is the hypergeometric function defined by Equation ( 29) is the main result of this section from which δ 1 (X) and δ 2 (X) are immediately determined.The mean deviations can be used to plot Lorenz and Bonferroni curves in fields like economics, reliability, demography, insurance and medicine.For a given probability π, they are defined by L(π) = m 1 (q)/µ ′ 1 and B(π) = m 1 (q)/(π µ ′ 1 ), respectively, where q = Q(π) comes directly from (28).

Reliability
In reliability, the stress-strength model describes the life of a component which has a random strength X 1 that is subjected to a random stress X 2 .The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever X 1 > X 2 .Hence, R = Pr(X 2 < X 1 ) is a measure of component reliability.It has many applications especially in engineering concepts, economics and physical science.We derive the reliability R when X 1 and X 2 have independent Exp-BXIIP(s, k 1 , c, α 1 , λ 1 ) and Exp-BXIIP(s, k 2 , c, α 2 , λ 2 ) distributions with identical scale parameter s and shape parameter c.The reliability is given by The cdf of X 2 and density of X 1 are obtained from Theorem 1 where Setting u = 1 + (x/s) c , we have , and then we obtain R.

Order Statistics
We now derive an explicit expression for the density of the ith order statistic X i:n , say f i:n (x), in a random sample of size n from the Exp-BXIIP distribution.It is well-known that for i = 1, . . ., n.Using the binomial expansion in the last equation, we readily obtain We use the identity for k, v positive integer where c v,0 = a v 0 and Then, we can write Substituting (33) into equation (37), we can rewrite L as where c l, j can be obtained from (32).
Using the power series expansion, we can write L as Setting u = (x/s) c , we obtain Finally, equation (36) reduces to ] .

Rényi and Shannon Entropy
The entropy of a random variable X with density function f (x) is a measure of variation of the uncertainty.For any real parameter ω > 0 and ω 1, the Rényi entropy of the Exp-BXIIP distribution is given by The details of the proof are given in Appendix B.
For the Shannon entropy, we have where v r is defined in Theorem 1 and E{log[g(X; s, k, c)]} can be computed from (2) at least numerically.The details of the proof are given in Appendix C.

Estimation
Let X i be a random variable following (9) with the vector θ = (s, k, c, α, λ) T of parameters.The data encountered in survival analysis and reliability studies are often censored.The censored log-likelihood l(θ) for the model parameters is The score functions for the parameters s, k, c α and λ are given by and The maximum likelihood estimate (MLE) θ of θ is obtained by solving the nonlinear likelihood equations U s (θ) = 0, U k (θ) = 0, U c (θ) = 0, U α (θ) = 0 and U λ (θ) = 0.These equations cannot be solved analytically and statistical software can be used to solve them numerically.We can use iterative techniques such as a Newton-Raphson type algorithm to obtain θ.The computations are performed using the software R version 3.0.0(package bbmle).
For interval estimation of (s, k, c, α, λ) and hypothesis tests on these parameters, we obtain the observed information matrix since its expectation requires numerical integration.The 5 × 5 observed information matrix J(θ) is

Conclusions
We define and study a new five-parameter lifetime model called the exponentiated Burr XII Poisson distribution, which extends some well-known lifetime distributions.Due to its flexibility in accommodating different forms of the hazard rate function, it is an important model for modeling lifetime data.We provide a mathematical treatment of the proposed distribution including a useful expansion for its density function.We derive explicit expressions for the moments, generating and quantile functions, mean deviations, reliability and entropies, which hold in generality for any parameter values.The model parameters are estimated by maximum likelihood.Additionally, the observed information matrix is determined.In one application to a real data set, we illustrate the potentiality of the new model.

Figure 1 .
Figure 1.Plots for the Exp-BXIIP density for some parameter values.

Figure 3 .
Figure 3. Plots of the B and M functions for some parameter values

Figure 4 .
Figure 4. Fitted densities to the histogram of the current data.

Table 2 .
Measures W * and A * More information is provided by a visual comparison of the fitted densities to the histogram of the data.The plots of the fitted Exp-BXIIP, BBXII, KwBXII and McBXII density functions are displayed in Figure 4.These plots indicate that the new distribution provides a good fit to these data and that it is also a very compettitive model to other lifetime distributions.