The exponentiated Lomax Poisson distribution

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(1)Advances and Applications in Statistics Volume 34, Number 2, 2013, Pages 107-135 Published Online: June 2013 Available online at http://pphmj.com/journals/adas.htm Published by Pushpa Publishing House, Allahabad, INDIA. THE EXPONENTIATED LOMAX POISSON DISTRIBUTION WITH AN APPLICATION TO LIFETIME DATA Manoel Wallace A. Ramos, Pedro Rafael D. Marinho, Ronaldo V. da Silva and Gauss M. Cordeiro Department of Statistics University Federal of Pernambuco 50740-540, Recife, PE, Brazil e-mail: [email protected] [email protected] [email protected] [email protected] Abstract Generalizing lifetime distributions is always precious for applied statisticians. By compounding the exponentiated Lomax and Poisson distributions, a new continuous distribution is introduced, called the exponentiated Lomax Poisson distribution. The new model extends the Lomax distribution and some other distributions and it is quite flexible to analyze positive data. Various structural properties of the new distribution are derived including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, Lorenz and Bonferroni curves, reliability, Rényi and © 2013 Pushpa Publishing House 2010 Mathematics Subject Classification: 60E05, 62-07, 62P30, 68U10. Keywords and phrases: exponentiated distribution, exponentiated Lomax Poisson distribution, Poisson distribution, maximum likelihood estimation. Communicated by K. K. Azad Received February 6, 2013.

(2) Manoel Wallace A. Ramos et al.. 108. Shannon entropies, order statistics and their moments. The estimation of the model parameters is performed by maximum likelihood. We also determine the observed information matrix. The potentiality of the new model is illustrated by means of a real data set. We hope that the new distribution will serve as an alternative model to other distributions available in the literature for modeling lifetime real data in many areas.. 1. Introduction The Lomax (Pareto II) distribution was originally proposed by Lomax [13]. It is an important model for lifetime analysis and it has been quite widely applied in some areas. For example, in the analysis of income data, biological sciences and originally in the business failure data. The Lomax distribution with parameters γ > 0 and β > 0, say Lomax( γ , β) , is given by. the cumulative distribution function (cdf). G ( x; γ, β ) = 1 − (1 + βx )− γ ,. x > 0,. (1). where γ and β are the shape and scale parameters, respectively. The corresponding probability density function (pdf) to (1) is. g ( x; γ, β) = γβ(1 + βx )−( γ +1) ,. x > 0.. (2). The Lomax distribution has been studied by several authors. Adding new parameters to existing distributions in order to obtain more flexibility has been investigated by several authors in the last twenty years or so. Abdul-Moniem and Abdel-Hameed [4] proposed the exponentiated Lomax distribution and studied some of its properties. This distribution with parameters α, γ and β, say EL(α, γ, β) , has cdf given by. H α ( x; γ, β ) = [1 − (1 + βx )− γ ]α , and corresponding density function hα ( x; γ , β ) = αγβ(1 + βx )−( γ +1) [1 − (1 + βx )− γ ]α −1.. (3).

(3) The Exponentiated Lomax Poisson Distribution …. 109. Balakrishnan and Ahsanullah [6] considered some moments of the Lomax distribution. Al-Awadhi and Ghitany [1] used it as a mixing distribution for the Poisson parameter and obtained the discrete Poisson-Lomax distribution. Abd-Ellah [3] investigated Bayesian prediction bounds for certain order statistics from Lomax samples and Abd-Elfattah et al. [2] studied the Bayesian and non-Bayesian estimation of the reliability. Nadarajah [18] presented sums, products and ratios for the bivariate Lomax distribution. Holland et al. [11] performed application of this distribution in biological sciences and even for modeling the distribution of the sizes of computer files on servers. Ghitany et al. [8] proposed a new parametric distribution using the Marshall-Olkin generator. Hassan and Al-Ghamdi [10] determined the optimal times of changing stress level for simple stress plans under a cumulative exposure model using the Lomax distribution. More recently, Lemonte and Cordeiro [14] proposed an extension of the Lomax distribution using the McDonald generator. Moghadam et al. [15] based on generalized order statistics investigated the Bayesian and the classical estimations of its model parameters, the reliability and hazard functions. Some authors studied the Lomax distribution in the multivariate case. This can be seen in Nayak [20], Roy and Gupta [26], Petropoulos and Kourouklis [21] and Nadarajah [19]. Most of the distributions lack physical motivation for modeling lifetime data. We now provide a physical interpretation for the proposed model. Consider a company formed by N systems working independently at any given time, and that each system consists of α parallel units, so the system will fail if all units fail. We assume that the random variables Z i , 1 , ..., Z i, α , representing the failure times of the parallel components of the ith system, are independent and have the same Lomax distribution with positive parameters γ and β. Let Yi denote the failure time of the ith system. Consider that N is a random variable having a truncated Poisson distribution defined by the probability mass function (pmf). Pr ( N = n ) =. λn. (eλ − 1) n !. ,. n = 1, 2, … .. (4).

(4) Manoel Wallace A. Ramos et al.. 110. Further, suppose that X represents the time to failure of the first out of the N functioning systems. In other words, X = min(Y1, ..., YN ). Then we obtain the cdf of X, say F ( x ) , from the conditional cdf of X as follows:. F ( x | N ) = 1 − Pr ( X > x | N ) = 1 − Pr N (Y1 > x ) = 1 − [1 − Pr α ( Z1, 1 ≤ x )]N = 1 − {1 − [1 − (1 + βx )− γ ]α }N . Then the unconditional cdf of X (for x > 0) is F ( x) =. ∞. n. ∑ [1 − {1 − [1 − (1 + βx )− γ ]α }n ] (eλ − 1)−1 λn! n =1. =. eλ [1 − exp{−λ[1 − (1 + βx )− γ ]α }]. (eλ − 1). ,. (5). where λ, α, γ and β are positive parameters. A random variable X having cdf (5) is called the exponentiated Lomax Poisson (ELP) distribution. Hereafter, we denote a random variable with cdf (5) by X ~ ELP(λ, α, γ , β ). Some distributions arise as special cases of the ELP model: • When λ → 0 + , the EL distribution is a limiting case of (5). • For λ → 0 + and α = 1, equation (5) reduces to the Lomax distribution. • When λ → 0 + and β = 1 in (5), we obtain the exponentiated Pareto (EP) distribution due to Gupta et al. [9]. • For α = 1, equation (5) reduces to the Poisson Lomax distribution (PL) (Al-Awadhi and Ghitany [1]). Some authors have presented compounding distributions as the one proposed in this paper. Rezaei et al. [24] performed the composition of the geometric and exponentiated exponential distribution to define the.

(5) The Exponentiated Lomax Poisson Distribution …. 111. exponentiated exponential geometric (EEG) model. Ristic and Nadarajah [25] proposed the exponentiated exponential Poisson (EEP) distribution by composing the Poisson and exponentiated exponential distribution. Bakouch et al. [5] introduced the exponentiated exponential binomial (EEB) distribution by compounding the exponentiated exponential and binomial distributions. In this paper, we study some mathematical properties of the ELP model and illustrate its potentiality. In Section 2, we define its density and hazard rate functions. In Section 3, we demonstrate that the cdf and pdf of X can be expressed as a mixture of EL densities. Explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, reliability, Rényi and Shannon entropies are derived in Section 4. In Section 5, we investigate the order statistics and some of their structural properties. Maximum likelihood estimation of the model parameters is performed and the observed information matrix is determined in Section 6. In Section 7, we provide an application of the ELP distribution to a real data set. Finally, some conclusions are addressed in Section 8. 2. Density and Hazard Functions. The pdf corresponding to (5) (for x > 0) is given by. f ( x; λ, α, γ , β ) =. λαγβ e λ [1 − (1 + β x )− γ ]α −1 exp{−λ[1 − (1 + β x )− γ ]α }. (eλ − 1) (1 + βx )γ +1. ,. (6). where λ > 0, α > 0, γ > 0 and β > 0. Some possible shapes of the density (6) for selected parameter values are displayed in Figure 1. It is evident that monotonically decreasing and unimodal shapes are possible. It is difficult to determine analytically the parameter regions corresponding to these shapes. However, some graphical analysis shows that monotonically decreasing shapes correspond to α ≤ 1 and that unimodal shapes correspond to α > 1..

(6) Manoel Wallace A. Ramos et al.. 112. Figure 1. Plots of the density function (6) for some parameter values.. The hazard rate function (hrf) (for x > 0) of the ELP distribution is given by h( x ) =. λαγβ(1 + βx )− γ −1[1 − (1 + βx )− γ ]α −1 exp{−λ[1 − (1 + βx )− γ ]α } exp{−λ[1 − (1 + βx )− γ ]α } − e − λ. , (7). where λ > 0, α > 0, γ > 0 and β > 0. In Figure 2, we provide some plots of this hrf..

(7) The Exponentiated Lomax Poisson Distribution …. 113. Figure 2. Plots of the hrf (7) for some parameter values. 3. Useful Expansion. Equations (5) and (6) can be expressed as linear combinations of EL distributions. Using the power series expansion 1− e. −z. ∞. =. ∑. k =0. (−1)k z k +1 , (k + 1)!. we can rewrite (5) as F ( x) =. eλ. ∞. ∑. (eλ − 1) k = 0. (−1)k λk +1 {[1 − (1 + βx )− γ ]α }k +1 (k + 1)!. (8).

(8) 114. Manoel Wallace A. Ramos et al.. which leads to an infinite linear combination F (x) =. ∞. ∑ ωk H (k +1) α ( x; γ, β),. (9). k =0. where ωk =. (−1)k eλ λk +1 (eλ − 1) (k + 1)!. and H ( k +1) α ( x; γ, β) denotes the EL cdf with parameters (k + 1) α, γ and β. Differentiating (9) gives. f (x) =. ∞. ∑ ωk h(k +1) α ( x; γ, β),. (10). k =0. where h( k +1) α ( x; γ, β) is the EL density function with scale parameters γ and β and shape parameter (k + 1) α. Clearly,. ∞. ∑k = 0 ωk. = 1.. 4. Properties of the ELP Distribution. In this section, we study some structural properties of the ELP distribution. 4.1. Moments. Many of the interesting characteristics and features of a distribution can be obtained using ordinary moments. The nth moment of X can be easily obtained from equation (10). We have. E( X n ) =. ∞. ∑ ωk E[Y(nk +1) ],. k =0. where Y( k +1) denotes the EL random variable with density function. h( k +1) α ( x; γ, β ) for k = 0, 1, 2, … . Abdul-Moniem and Abdel-Hameed [4] demonstrated that the nth moment of Y(1) can be expressed (for n < γ ) as.

(9) The Exponentiated Lomax Poisson Distribution …. 115. n ⎛n⎞ α n−l (−1)l ⎜ ⎟ B ⎡⎢1 − , α⎤ , E[Y(1n) ] = n γ ⎣ ⎦⎥ l ⎝ ⎠ β l =0. ∑. where B(a, b ) =. 1. ∫0 w. αe λ. a −1. (1 − w)b −1 dw is the beta function. Thus, we obtain. ∞. ∑. E( X ) = n λ β (e − 1) k = 0 n. (−1)k λk +1 k!. n. ⎛n⎞. ∑ (−1)l ⎜⎝ l ⎟⎠ B ⎡⎢⎣1 − n γ− l , (k + 1) α⎤⎥⎦ l =0. ∞. =. ∑ sk B ⎡⎢⎣1 − n γ− l , (k + 1) α⎤⎥⎦ ,. (11). k =0. where n (−1)k + l λk +1 ⎛ n ⎞ αe λ sk = n λ ⎜ ⎟. k! ⎝l⎠ β (e − 1) l = 0. ∑. For empirical purposes, the shape of many distributions can be usefully described by the incomplete moments. These types of moments play an important role for measuring inequality, for example, income quantiles and Lorenz and Bonferroni curves. These curves depend on the first incomplete moment of the distribution. The nth incomplete moment of X follows from equation (10) as mn ( z ) = E ( X n | X < z ) =. ∞. ∑ ωk ∫0 xnh(k +1) α ( x; γ, β) dx z. k =0. =. αγβeλ. (e. λ. ∞. ∑ − 1). k =0. (−1)k λk +1 k!. z n. ∫0 x. (1 + βx )− γ −1[1 − (1 + βx )− γ ]( k +1) α −1 dx.. Let u = (1 + βx )− γ . Thus, the last equation becomes αe λ. ∞. ∑. mn ( z ) = − n λ β (e − 1) k = 0. −γ (−1)k + n λk +1 (1+ βz ) ⎛⎜. × (1 − u )( k +1) α −1 du.. k!. ∫1. ⎜⎜1 − u ⎝. −1 ⎞ n γ ⎟. ⎟⎟ ⎠.

(10) Manoel Wallace A. Ramos et al.. 116. If z < 1 and b > 0 is a nonnegative integer, then the power series holds. (1 − z )b −1 =. ∞. ∑. p =0. (−1) p Γ(b ) p z . Γ(b − p ) p!. (12). n. −1 ⎞ ⎛ ⎜ ( k +1) α −1 Using (12) for (1 − u ) and the binomial expansion for ⎜1 − u γ ⎟⎟ , ⎜ ⎟ ⎝ ⎠. we obtain ∞. λ. n. αγe mn ( z ) = − n λ β (e − 1) k , p = 0 r = 0 ×. (1+ βz )− γ. ∫1 ∞. =. ∑∑ r γ. p−. u. ⎛n⎞ ⎝r⎠ [( p + 1) γ − r ]Γ[(k + 1) α − p ] p! k!. (−1)r + n + p + k λk +1⎜ ⎟ Γ[(k + 1) α]. du. n. ∑ ∑ vk , r , p [1 − (1 + βx )−( p +1) γ + r ],. (13). k , p =0 r =0. where. ⎛n⎞ ⎝r⎠ . vk , r , p = n λ β [( p + 1) γ − r ] (e − 1) Γ[(k + 1) α − p ] p! k!. (−1)r + n + p + k γαλk +1eλ ⎜ ⎟ Γ[(k + 1) α]. 4.2. Generating function. We derive an explicit expression for the moment generating function (mgf) M (t ; α, γ, β) of the EL distribution. We have from equation (3), M (t ; α, γ , β ) = αγβ. ∞ tx. ∫0. e (1 + β x )− γ −1[1 − (1 + β x )− γ ]α −1 dx.. Using (12), it follows that M (t ; α, γ , β ) = αγβ. ∞. ∑. p =0. (−1) p Γ(α ) Γ(α − p ) p!. ∫. ∞ tx e (1 + 0. β x )− ( p +1)γ −1 dx..

(11) The Exponentiated Lomax Poisson Distribution …. 117. Using the Mathematica software (Wolfram [28]) to calculate the above integral (for t < 0 ) , we obtain M (t ; α, γ , β ) =. where Γ(a, x ) =. αγΓ(α ) et β. ∞. ∫x. ∞. (−1) p (−t )( p +1) γ. ∑ β( p +1) γΓ(α − p ) p! Γ[−( p + 1) γ, −t β],. (14). p =0. wa −1e − w dw is the complementary incomplete gamma. function. From equations (10) and (14), we can express the ELP generating function (for t < 0) as M (t ; λ, α, γ, β ) =. ∞. ∞. ∑ ∑ qk , pΓ[−( p + 1) γ, −t β],. (15). k =0 p =0. where qk , p =. (−1) p (k + 1) αγωk (−t )( p +1) γ Γ[(k + 1) α] . et ββ( p +1) γ Γ[(k + 1) α − p ] p!. Equations (14) and (15) are the main results of this section. 4.3. Quantile function. The ELP quantile function, say x = Q(u ) , can be obtained by inverting (5). We have x = Q(u ) = F −1 (u ) −1 ⎡ ⎤ 1 γ ⎛ ⎞ −1 ⎢ −1 λ λ = β ⎢⎜⎜ −{−λ log[−u (e − 1) + e ] + 1}α + 1⎟⎟ − 1⎥⎥ . ⎠ ⎢⎣⎝ ⎥⎦. (16). The shortcomings of the classical kurtosis measure are well-known. There are many heavy-tailed distributions for which this quantity is infinite. So, it becomes uninformative 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..

(12) Manoel Wallace A. Ramos et al.. 118. The Bowley’s skewness is based on quartiles (Kenney and Keeping [12]): Q(3 4 ) − 2Q(1 2 ) + Q(1 4) B= Q(3 4) − Q(1 4) and the Moors’ kurtosis (Moors [16]) is based on octiles. M =. Q(7 8) − Q(5 8) − Q(3 8) + Q(1 8) , Q (6 8 ) − Q ( 2 8 ). where Q(⋅) is given by (16). Plots of the skewness and kurtosis for some choices of α, γ and β as function of λ and for some choices of λ, γ and β as function of α are displayed in Figure 3. These plots indicate that there is a great flexibility of the skewness and kurtosis curves of the new distribution.. Figure 3. Plots of the ELP skewness and kurtosis as functions of λ for selected values of α, γ and β and as functions of α for selected values of λ, γ and β..

(13) The Exponentiated Lomax Poisson Distribution …. 119. 4.4. Mean deviations. 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 δ1 ( X ) = 2μ1′ F (μ1′ ) − 2m1(μ1′ ) and δ 2 ( X ) = μ1′ − 2m1( M ) , respectively, where μ1′ = E ( X ) comes from (11) with n = 1 and M is the median. From (16), it follows that −1 ⎡ ⎤ 1 γ ⎛ ⎞ −1 ⎢ −1 λ λ M = β ⎢⎜⎜ − {−λ log[−1 2 (e − 1) + e ] + 1}α + 1⎟⎟ − 1⎥⎥ , ⎠ ⎢⎣⎝ ⎥⎦. F (μ1′ ) is easily calculated from the cdf (5) and m1( z ) =. z. ∫ 0 x f ( x ) dx. (17). is the. first incomplete moment of X obtained from (13) with n = 1, m1( z ) =. ∞. ∑ {bp [1 − (1 + βz )−( p +1) γ ] + c p [1 − (1 + βz )−( p +1) γ +1]},. (18). p =0. where ∞. (−1) p +1 λαeλ (−1)k λk Γ[(k + 1) α] bp = ( p + 1) β(eλ − 1) p! k = 0 Γ[(k + 1) α − p ] k!. ∑. and ∞. (−1) p λαγeλ (−1)k λk Γ[(k + 1) α] cp = . β[( p + 1) γ − 1] (e λ − 1) p! k = 0 Γ[(k + 1) α − p ] k!. ∑. Based on (18), δ1 ( X ) and δ 2 ( X ) are immediately calculated. An application of the mean deviations is to the Lorenz and Bonferroni curves that are useful in fields like economics, reliability, demography, insurance and medicine. For a given probability π, they are defined by L(π) = m1(q ) μ1′ and B(π).

(14) Manoel Wallace A. Ramos et al.. 120. = m1(q ) (πμ1′ ) , respectively, where q = Q(π) comes from (16). In economics, if π = F (q ) is the proportion of units whose income is lower than or equal to q, then L(π) gives the proportion of total income volume accumulated by the. set of units with an income lower than or equal to q. In a similar manner, the Bonferroni curve B(π) gives the ratio between the mean income of this group and the mean income of the population. The curves L(π) and B(π) for the ELP distribution as functions of π are readily calculated from (18) and plotted for some parameter values in Figure 4.. Figure 4. Plots of L(π) and B(π) for α = 5, γ = 2 and β = 3. 4.5. Reliability. In the context of 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 < X1 ) is a measure of component reliability. It has many applications especially in engineering concepts. Now, we obtain the reliability R when X 1 and X 2 have independent ELP(λ1, α1, γ, β) and ELP(λ 2 , α 2 , γ, β) distributions. The reliability is defined by.

(15) The Exponentiated Lomax Poisson Distribution … ∞. ∫0. R=. 121. f1( x ) F2 ( x ) ds.. The pdf of X 1 and cdf of X 2 are obtained from equations (10) and (9) as f1 ( x ) =. ∞. ∑ ωk (λ1 ) h(k +1) α1 ( x; γ, β). k =0. and ∞. ∑ ωq (λ 2 ) H (q +1) α2 ( x; γ, β),. F2 ( x ) =. q =0. where. ωk (λ1 ) =. (−1)k eλ1 λk1 +1 , (eλ1 − 1) (k + 1)!. and ωq (λ 2 ) is obtained analogously. Hence, ∞. ∑. R=. ωk (λ1 ) ωq (λ 2 ) (k + 1) α1γβ. k, q =0. ×. ∞. ∫ 0 (1 + βx ). [1 − (1 + βx )− γ ]( k +1) α1 + ( q +1) α 2 −1 dx.. − γ −1. Setting u = (1 + βx )− γ , we have ∞. R=. ∑. ωk (λ1 ) ωq (λ 2 ) (k + 1) α1. k, q =0. 1. ( k +1) α1 + ( q +1) α 2 −1 du.. ∫0 (1 − u ). Hence, the reliability of the ELP distribution is given by ∞. R=. ∑. k , q =0. (k + 1) α1 ω ( λ ) ω ( λ ). (k + 1) α1 + (q + 1) α 2 k 1 q 2. (19).

(16) Manoel Wallace A. Ramos et al.. 122 4.6. Entropies. The entropy of a random variable X with density function f ( x ) is a measure of variation of the uncertainty. Two popular entropy measures are due to Shannon and Rényi (Shannon [27] and Rényi [23]). A large value of the entropy indicates the greater uncertainty in the data. The Rényi entropy is defined by I R (δ ) =. ∞ δ 1 log f ( x ) dx 1− δ 0. ∫. for δ > 0 and δ ≠ 1. If (δ − 1) γ + δ > 0 and (α − 1) δ + 1 > 0, based on the ELP density function, then we obtain ∞. ∫0. f δ ( x ) dx. δ ⎞ ∞ (−1)k λk δ k ⎜ ⎟ = (1 + βx )( − γ −1) δ [1 − (1 + βx )− γ ](α −1) δ + kα dx k! 0 ⎜⎝ e λ − 1 ⎟⎠ k =0. ∫. ∞ ⎛ λαγβ e λ. ∑. δ. ∞ ⎛ λαγβ eλ ⎞ (−1)k λk δ k ⎡ δ − 1 −1 ⎜ ⎟ ( ) B =⎜ λ γβ + δ, (α − 1) δ + αk + 1⎤ . ⎟ ⎢ ⎥⎦ k ! γ ⎣ ⎝ e −1 ⎠ k =0. ∑. So, the Rényi entropy for the ELP distribution becomes ⎧⎪⎛ λαγβ e λ ⎞δ 1 ⎟ ( γβ)−1 log ⎨⎜⎜ λ I R (δ ) = ⎟ 1− δ ⎩⎪⎝ e − 1 ⎠ ∞. ×. ∑. k =0. (−1)k λk δk k!. ⎫⎪ δ −1 B⎡ + δ, (α − 1) δ + αk + 1⎤ ⎬. ⎢⎣ γ ⎥⎦ ⎭⎪. (20). The Shannon entropy is given by. E{−log[ f ( x )]} = − log(λαγβ eλ ) + log(eλ − 1) + ( γ + 1) E[log(1 + βX )] + (1 − α ) E{log[1 − (1 + βx )− γ ]} + λE{[1 − (1 + βx )− γ ]α }. (21).

(17) The Exponentiated Lomax Poisson Distribution … Using the power series for the logarithm function, we can express (21) as E{−log[ f ( x )]} = − log(λαγβ e λ ) + log(e λ − 1) +. + (α − 1). ∞. ( γ + 1) γ. ∞. ∑ k E{[1 − (1 + βX )− γ ]k } 1. k =1. ∑ k E[(1 + βX )− kγ ] + λE{[1 − (1 + βX )− γ ]α }. 1. k =1. Setting y = (1 + βx )− γ , we have α λαeλ 1 k E[(1 + βX )− kγ ] = λ y (1 − y )α −1 e − λ (1− y ) dy e −1 0. ∫. λαeλ. ∞. ∑. (−1)r λr. ∞. (−1)r λr. = λ e − 1 r =0 λαeλ. ∑. = λ e − 1 r =0. r!. r!. 1. ∫0 y. k. (1 − y )( r +1) α −1 dy. B[k + 1, ( r + 1) α].. Setting z = 1 − (1 + βx )− γ , we obtain. λαeλ 1 k + α −1 − λz α E{[1 − (1 + βX )− γ ]k } = λ z e dz e −1 0. ∫. k. λα eλ ⎡ ⎛ α + k ⎞ ⎛ α + k , λ ⎞⎤ . = λ Γ⎜ ⎟ − Γ⎜ ⎟⎥ ⎢ α ⎠ ⎝ α ⎠⎦ e −1⎣ ⎝ Analogously, we can write eλ − λ − 1 λαeλ 1 2α −1 − λz α . z e dz = E{[1 − (1 + βX )− γ ]α } = λ λ(eλ − 1) e −1 0. ∫. 123.

(18) Manoel Wallace A. Ramos et al.. 124. Then we can express the Shannon entropy of X as E{−log[ f ( X )]} −k. ⎛ eλ − 1 ⎞ ( γ + 1) e λ ∞ λ α ⎡ ⎛ α + k ⎞ ⎛ α + k , λ ⎞⎤ ⎟+ = log⎜⎜ Γ⎜ ⎟⎥ ⎟ − Γ⎜ λ⎟ λ ⎢ k α ⎣ ⎠⎦ ⎝ α ⎠ ⎝ ⎝ λαγβ e ⎠ γ (e − 1) k =1. ∑. +. λαeλ (α − 1) eλ − 1. ∞. ∞. ∑∑. (−1)r λr. k =1 r = 0. kr!. B[k + 1, (r + 1) α] +. eλ − λ − 1 eλ − 1. .. 5. Order Statistics. Order statistics make their appearance in many areas of statistical theory and practice. The density f i:n ( x ) of the ith order statistic, for i = 1, ..., n, from independent identically distributed random variables X 1 , ..., X n from the ELP distribution is given by f i :n ( x ) =. f (x) F ( x )i −1[1 − F ( x )]n − i . B(i, n − i + 1). Substituting (5) and (6) in this equation, we can write f i :n ( x ) =. f ( x) (1 − e − λ )− i +1[1 − exp{−λ[1 − (1 + βx )− γ ]α }]i −1 B(i, n − i + 1) × {1 − (1 − e −λ )−1[1 − exp{−λ[1 − (1 + βx )− γ ]α }]}n − i .. Using the binomial expansion, f i:n ( x ) can be expressed as f ( x) f i :n ( x ) = B(i, n − i + 1). n −i. ⎛n − i⎞ j − λ − j − i +1 ⎟ (−1) (1 − e ) j ⎠. ∑ ⎜⎝ j =0. × [1 − exp{−λ[1 − (1 + βx )− γ ]α }] j + i −1. Applying the binomial expansion to the last term, we obtain f ( x) fi:n ( x ) = B(i, n − i + 1). n − i j + i −1. ⎛ n − i ⎞ ⎛ j + i − 1⎞ j+r − λ − j −i +1 ⎜ ⎟⎜ ⎟ (−1) (1 − e ) j ⎠⎝ r ⎝ ⎠ j =0 r =0. ∑∑. × exp[−λr{1 − (1 + βx )− γ }α ]..

(19) The Exponentiated Lomax Poisson Distribution …. 125. Then f i:n ( x ) reduces to n! f i :n ( x ) = (i − 1)! (n − i )!. ×. n − i j + i −1. ⎛ n − i ⎞ ⎛ j + i − 1⎞ j+r ⎟⎜ ⎟ ( −1) ⎜ j ⎠⎝ r ⎝ ⎠ j =0 r =0. ∑∑. [1 − e −( r +1) λ ] f ( x; (r + 1) λ, α, γ , β). (r + 1) (1 − e − λ ) j + i. Finally, we obtain f i :n ( x ) =. n − i j + i −1. ∑∑. tr , j f ( x; (r + 1) λ, α, γ , β) ,. (22). j =0 r =0. where. [1 − e −( r +1) λ ] (−1) j + r n! tr , j = (i − 1)! (n − i )! (r + 1) (1 − e − λ ) j + i. ⎛ n − i ⎞ ⎛ j + i − 1⎞ ⎜ ⎟⎜ ⎟. r ⎝ j ⎠⎝ ⎠. So, the pdf of X i:n can be represented as a double finite mixture of ELP density functions. Moments of order statistics play an important role in quality control testing and reliability, where a practitioner needs to predict the failure of future items based on the times of a few early failures. These predictors are often based on moments of order statistics. Thus, from equation (22), the sth ordinary moment and generating function of X i:n are E ( X is:n ). n − i j + i −1. =. ∑∑. tr , j E (Wrs ) ,. j =0 r =0. E[exp(tX i:n )] =. n − i j + i −1. ∑ ∑ tr , j E[exp(tWr )], j =0 r =0. respectively, where Wr ~ ELP((r + 1) λ, α, γ, β). Here, the quantities E (Wrs ) and E[exp(tWr )] are immediately obtained from (11) and (15) as.

(20) Manoel Wallace A. Ramos et al.. 126 E (Wrs ) =. ∞. s. ∑∑ ak , l B ⎡⎢⎣1 − s −γ l , (k + 1) α⎤⎥⎦ ,. s < γ,. k =0 l =0. where ak , l =. (−1)k + l αe( r +1) λ [(r + 1) λ ]k +1 ⎛ s ⎞ ⎜ ⎟ ⎝l ⎠ β s [e( r +1) λ − 1] k!. and ∞. ∑. E[exp(tWr )] =. qk∗, p Γ[−( p + 1) γ, −t β],. k , p =0. where. (−1) p + k (k + 1) αγ[(r + 1) λ ]k +1(−t )( p +1) γ Γ[(k + 1) α] . qk∗, p = ( p +1) γ t β − ( r +1) λ ( r +1) λ β e [e − 1] Γ[(k + 1) α − p ] (k + 1)! p! 6. Maximum Likelihood Estimation. In this section, we derive the maximum likelihood estimators (MLEs) of the unknown parameters λ, α, γ and β of (6). We consider a random sample X 1 , ..., X n of size n from the ELP(λ, α, γ , β ) distribution, with observed values x1 , ..., xn . Let θ = (λ, α, γ, β)T be the vector of the model parameters. The log-likelihood function for θ reduces to l (θ ) = n log(λαγβ ) − n log(1 − eλ ) + (α − 1). n. ∑ log[1 − (1 + βxi )− γ ] i =1. − ( γ + 1). n. n. i =1. i =1. ∑ log(1 + βxi ) − λ ∑ [1 − (1 + βxi )− γ ]α .. The score functions for the parameters λ, α, γ and β are given by U λ (θ ) =. n ne − λ [1 − (1 + βxi )− γ ]α , − − λ 1 − e− λ i =1 n. ∑.

(21) The Exponentiated Lomax Poisson Distribution … n + α. U α (θ ) =. n. ∑. log[1 − (1 + βxi )− γ ] − λ. i =1. 127. n. ∑ [1 − (1 + βxi )− γ ]α i =1. × log[1 − (1 + βxi )− γ ], n + (α − 1) γ. U γ (θ ) =. n. ∑ i =1. (1 + βxi )− γ log(1 + βxi ) − 1 − (1 + βxi )− γ. n. ∑ log(1 + βxi ) i =1. n. − λα. ∑ [1 − (1 + βxi )− γ ]α −1(1 + βxi )− γ log(1 + βxi ) i =1. and U β (θ ) =. n + (α − 1) γ β. n. ∑ i =1. xi (1 + βxi )− γ −1 xi + (− γ − 1) −γ 1 + βxi 1 − (1 + βxi ) i =1 n. ∑. n. − λαγ. ∑ [1 − (1 + βxi )− γ ]α −1(1 + βxi )− γ −1 xi . i =1. The MLE θˆ of θ is obtained by solving the nonlinear likelihood equations U λ (θ ) = 0, U α (θ ) = 0, U γ (θ ) = 0 and U β (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 the estimate θˆ . For interval estimation of (λ, α, γ, β) and hypothesis tests on these parameters, we obtain the observed information matrix since its expectation requires numerical integration. The 4 × 4 observed information matrix J (θ ) is ⎡U λλ ⎢ ⋅ J (θ ) = − ⎢ ⎢ ⋅ ⎢ ⎢⎣ ⋅. U λα. U λγ. U αα. U αγ. ⋅. U γγ. ⋅. ⋅. whose elements are given in Appendix A.. U λβ ⎤ U αβ ⎥⎥ , U γβ ⎥ ⎥ U ββ ⎥⎦.

(22) Manoel Wallace A. Ramos et al.. 128. 7. Application. Here, we present an application of the ELP distribution to a real data set. We shall compare the fits of the Lomax, EL, McDonald Lomax (McLomax), Beta Lomax (BLomax), Kumaraswamy Lomax (KLomax) and Gamma Lomax (GLomax) distributions. The density functions of the McLomax, BLomax, KLomax and GLomax distributions are given by f McLomax ( x ) =. f BLomax ( x ) =. cγβ(1 + β x )− γ −1 −1. B(ac , b ). [1 − (1 + βx )− γ ]a −1{1 − [1 − (1 + βx )− γ ]c }b −1,. γβ(1 + β x )− γb −1 [1 − (1 + βx )− γ ]a −1, B( a, b ). f KLomax ( x ) = abγβ(1 + βx )− γ −1[1 − (1 + βx )− γ ]a −1{1 − [1 − (1 + βx )− γ ]a }b −1, f GLomax ( x ) =. 1 a γ β(1 + β x )− γ −1[log(1 + β x )]a −1, Γ( a ). respectively, where all the parameters in these densities are positive. We consider an uncensored data set corresponding to failure times for a particular windshield model including 88 observations that are classified as failed times of windshields (Murthy et al. [17]): 0.040, 1.866, 2.385, 3.443, 0.301, 1.876, 2.481, 3.467, 0.309, 1.899, 2.610, 3.478, 0.557, 1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779, 1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.82, 3, 4.035, 1.281, 2.085, 2.890, 4.121, 1.303, 2.089, 2.902, 4.167, 1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255, 1.505, 2.154, 2.964, 4.278, 1.506, 2.190, 3.000, 4.305, 1.568, 2.194, 3.103, 4.376, 1.615, 2.223, 3.114, 4.449, 1.619, 2.224, 3.117, 4.485, 1.652, 2.229, 3.166, 4.570, 1.652, 2.300, 3.344, 4.602, 1.757, 2.324, 3.376, 4.663. Table 1 lists the MLEs of the model parameters and the values for these distributions of the following statistics. AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion), BIC (Bayesian Information.

(23) The Exponentiated Lomax Poisson Distribution …. 129. Criterion), and the statistics of the Cramér-von Mises (W ∗ ) and AndersonDarling ( A∗ ) described in details by Chen and Balakrishnan [7] to verify which distribution gives a better fit to these data. In general, the smaller the values of the statistics W ∗ and A∗ , the better the fit to the data. Let H ( x; θ ) be the cdf, where the form of H is known but θ (a k-dimensional parameter vector, say) is unknown. To obtain the statistics W ∗ and A∗ , we can proceed as follows: (i) Compute vi = H ( xi ; θˆ ) , where the xi ’s are in ascending order, (ii) Compute yi = Φ −1 (vi ) , where Φ(⋅) is the standard normal cdf and Φ −1 (⋅) its inverse, (iii) Compute ui = Φ {( yi − y ) s y }, where y =. (1 n ) ∑in=1 yi and s 2y = (n − 1)−1 ∑in=1 ( yi − y )2 ; (iv) Calculate n. W2 =. ∑ i =1. 2 ⎧u − ( 2i − 1) ⎫ + 1 ⎨ i ⎬ 2n ⎭ 12n ⎩. and. A2 = − n −. 1 n. n. ∑ {(2i − 1) log(ui ) + (2n + 1 − 2i ) log(1 − ui )}; i =1. (v) Modify W 2 into W ∗ = W 2 (1 + 0.5 n ) and A2 into. A∗ = A2 (1 + 0.75 n + 2.25 n 2 ). The required numerical evaluations are carried out using the R software (R Development Core Team [22]). The figures in this table indicate that the ELP model corresponds to the smallest values of the statistics W ∗ and A∗ among all fitted models, that is, the new model yields a better fit to the failed times of windshields data than the other models. More information is provided by a visual comparison in Figure 5 which displays the histogram of the data and the fitted density functions of the ELP, McLomax, BLomax,.

(24) Manoel Wallace A. Ramos et al.. 130. KLomax and GLomax models. The Lomax and Elomax models do not fit the data well. Clearly, the ELP distribution provides a closer fit to the histogram and then it is the best model to explain these data. Table 1. MLEs (standard errors in parentheses) and the statistics W ∗ and A∗ Distributions McLomax (a, b, c, γ, β). A∗. Estimates 2.792. 13.802. 3.907. 2.230. W∗. AIC. CAIC. BIC. 0.113 3,256 0,473 280,048 280,807 292,261. (0.473) (6.481) (1,667) (0.991) (0.064) ELP (λ, α, γ, β). 49,961. 2,666. 17,628. 0,005. 0,710 0,071 273,832 274,332 283,602. (0.002) (9.618) (2.431) (0.562) BLomax (a, b, γ, β). 3.645. 5.991. 26.179. 0.008. 1.435 0.172 285.870 286.370 295.640. (0.562) (2.431) (9.618) (0.002) KLomax (a, b, γ, β). 3.100. 50.445. (0.603) (14.249) GLomax (a, γ, β). 3.570. 90.741. 0.499. 0.324. 1.279 0.149 284.123 284.623 293.894. (0.098) (0.101) 0.016. 1.442 0.173 283,965 284,261 291,292. (0,538) (30,757) (0,005) EL (α, γ, β). 4,325. 0,329. 3,184. 3,184 0,457 313,950 314,246 321,278. (1,046) (0,066) (0,421) Lomax (γ, β). 28,329. 0,014. 1,467 0,177 336,398 336,544 341,283. (14,843) (0,007). Figure 5. Fitted distributions..

(25) The Exponentiated Lomax Poisson Distribution …. 131. 8. Concluding Remarks. We generalize the Lomax distribution by defining the exponentiated Lomax Poisson (ELP) distribution. We provide a mathematical treatment of the new distribution including expansions for the density and cumulative functions and explicit expressions for the ordinary and incomplete moments, generating and quantile functions, order statistics and their moments, Rényi and Shannon entropies, mean deviations and Lorentz and Bonferroni curves. We estimate the model parameters using maximum likelihood and determine the observed information matrix. An application to a real data set illustrates the potentiality of the proposed model. The new model provides to the current data a better fit than other useful lifetime models available in the literature. We hope that the proposed model may attract wider applications in survival analysis for modeling positive real data sets, since the formulae derived are manageable using modern computer facilities. Appendix A - Observed Information Matrix. The elements of the observed information matrix for the parameters. (λ, α, γ, β) are: U λλ = −. n λ2. +. ne −λ 1 − e−λ. +. ne −2λ. (1 − e − λ )2. ,. n. U λα = −. [1 − (1 + βxi )− γ ]α log[1 − (1 + βxi )− γ ], ∑ i =1 n. U λγ = −. α(1 + β xi )− γ log(1 + β xi ) [1 − (1 + β xi )− γ ]α −1 , ∑ i =1 n. U λβ = −. αγxi (1 + β xi )− γ −1[1 − (1 + βxi )− γ ]α −1 , ∑ i =1.

(26) Manoel Wallace A. Ramos et al.. 132 n. n. U αα = −. α. 2. −λ. [1 − (1 + βxi )− γ ]γ {log[1 − (1 + βxi )− γ ]}2 , ∑ i =1. n (1 + βxi )− γ log(1 + βxi ) (1 + βxi )− γ log(1 + βxi ) −λ − γ − α +1 1 − (1 + β xi )− γ i =1 i =1 [1 − (1 + β xi ) ] n. U αγ =. ∑. ∑. × {α log(1 − (1 + βxi )− γ ) + 1}, n. U αβ = γ. ∑ i =1. xi (1 + β xi )− γ −1[1 − (1 + β xi )− γ ]−1 − λγ. n. xi (1 + β xi )− γ −1 ∑ i =1. × [1 − (1 + βxi )− γ ]α −1{α log[1 − (1 + βxi )− γ ] + 1}, U γγ = −. +. n γ2. − (α − 1). ⎧⎪ (1 + β xi )− γ [log(1 + βxi )]2 ⎨ 1 − (1 + βxi )− γ ⎩ i =1 ⎪ n. ∑. (1 + βxi )−2 γ [log(1 + βxi )]2 ⎫⎪ ⎬ ⎪⎭ [1 − (1 + βxi )− γ ]2 ⎧⎪ α 2 (1 + β xi )− 2 γ [log(1 + β xi )]2 α(1 + β xi )− γ [log(1 + β xi )]2 − ⎨ − γ −α + 2 ⎪ − + β [ [1 − (1 + βxi )− γ ]− α +1 1 ( 1 x ) ] ⎩ i i =1 n. −λ. −. ∑. α(1 + β xi )−2 γ [log(1 + β xi )]2 ⎫⎪ ⎬, [1 − (1 + βxi )− γ ]− α + 2 ⎪⎭. U γβ = − (α − 1). ⎧⎪ γxi (1 + β xi )− γ −1 log(1 + β xi ) xi (1 + β xi )− γ −1 − ⎨ 1 − (1 + βxi )− γ 1 − (1 + βxi )− γ ⎩ i =1 ⎪ n. ∑. xi γx (1 + β xi )− 2 γ −1 log(1 + β xi ) ⎫⎪ + i ⎬− −γ 2 ⎪⎭ i =1 1 + β xi [1 − (1 + βxi ) ] n. ∑. ⎧⎪ γα 2 xi (1 + β xi )− 2 γ −1 log(1 + βxi ) ⎨ ⎪ [1 − (1 + βxi )− γ ]− α + 2 i =1 ⎩ n. −λ. ∑.

(27) The Exponentiated Lomax Poisson Distribution … −. +. U ββ = −. 133. γαxi (1 + β xi )γ −1 log(1 + β xi ). [1 − (1 + βxi )− γ ]− α +1. αxi (1 + β xi )γ −1. − [1 − (1 + βxi )− γ ]− α +1. n β2. − (α − 1). γαxi (1 + β xi )−2 γ −1 log(1 + β xi ) ⎫⎪ ⎬, [1 − (1 + βxi )− γ ]− α + 2 ⎭⎪. ⎧⎪ γ 2 xi2 (1 + β xi )− γ − 2 γxi2 (1 + β xi )− γ − 2 + ⎨ −γ ⎪ − + β 1 1 x 1 − (1 + β xi )− γ ( ) i =1 ⎩ i n. ∑. xi2 γ 2 xi2 (1 + β xi )− 2( γ +1) ⎫⎪ + γ + 1 ( ) ⎬ 2 [1 − (1 + βxi )− γ ]2 ⎪⎭ i =1 (1 + β xi ) n. +. ∑. ⎧⎪ α 2 γ 2 xi2 (1 + β xi )− 2( γ +1) αγ 2 xi2 (1 + βxi )− γ − 2 − ⎨ − γ −α + 2 ⎪ [1 − (1 + βxi )− γ ]− α +1 i =1 ⎩ [1 − (1 + β xi ) ] n. −λ. −. ∑. αγ 2 xi2 (1 + β xi )−2( γ +1) ⎫⎪ − ⎬. [1 − (1 + βxi )− γ ]− α +1 [1 − (1 + βxi )− γ ]− α + 2 ⎪⎭ αγxi2 (1 + β xi )− γ − 2. References [1] S. A. Al-Awadhi and M. E. Ghitany, Statistical properties of Poisson-Lomax distribution and its application to repeated accidents data, J. Appl. Statist. Sci. 10 (2001), 365-372. [2] A. M. Abd-Elfattah, F. M. Alaboud and A. H. Alharby, On sample size estimation for Lomax distribution, Aust. J. Basic Appl. Sci. 1 (2007), 373-378. [3] A. H. Abd-Ellah, Bayesian one sample prediction bounds for the Lomax distribution, Indian J. Pure Appl. Math. 30 (2003), 101-109. [4] I. B. Abdul-Moniem and H. F. Abdel-Hameed, On exponentiated Lomax distribution, Internat. J. Math. Arch. 3 (2012), 2144-2150. [5] H. S. Bakouch, M. M. Ristic, A. Asgharzadeh, L. Esmaily and B. M. Al-Zahrani, An exponentiated exponential binomial distribution with application, Statist. Probab. Lett. 82 (2012), 1067-1081. [6] N. Balakrishnan and M. Ahsanullah, Relations for single and product moments of record values from Lomax distribution, Sankhya B 56 (1994), 140-146..

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