Exponentiated Extended Weibull-Power Series Class of Distributions
aa r X i v : . [ s t a t . O T ] M a r Exponentiated Extended Weibull-Power SeriesClass of Distributions
S. Tahmasebi , A. A. Jafari , ∗ Department of Statistics, Persian Gulf University, Bushehr, Iran Department of Statistics, Yazd University, Yazd, Iran
Abstract
In this paper, we introduce a new class of distributions by compounding the exponen-tiated extended Weibull family and power series family. This distribution contains sev-eral lifetime models such as the complementary extended Weibull-power series, generalizedexponential-power series, generalized linear failure rate-power series, exponentiated Weibull-power series, generalized modified Weibull-power series, generalized Gompertz-power seriesand exponentiated extended Weibull distributions as special cases. We obtain several prop-erties of this new class of distributions such as Shannon entropy, mean residual life, hazardrate function, quantiles and moments. The maximum likelihood estimation procedure viaa EM-algorithm is presented.
Keywords:
EM-algorithm, Exponentiated family, Maximum likelihood estimation, Power se-ries distributions.
The extended Weibull (EW) family contains various well-known distributions such as exponen-tial, Pareto, Gompertz, Weibull, linear failure rate (Barlow, 1968), modified Weibull (Lai et al.,2003), additive Weibull (Xie and Lai, 1995; Almalki and Yuan, 2013) and Chen (Chen, 2000)distributions. For more details see Nadarajah and Kotz (2005) and Pham and Lai (2007).Using the given method by Gupta and Kundu (1999), the EW family can be generalized.We call it exponentiated extended Weibull (EEW) distribution. The cumulative distributionfunction (cdf) of this distribution is G ( x ; α, β, Θ ) = [1 − e − αH ( x ; Θ ) ] β , α > , β > , x ≥ , (1.1) ∗ Corresponding:[email protected] nd its probability density function (pdf) isg( x ; α, β, Θ ) = αβh ( x ; Θ ) e − αH ( x ; Θ ) [1 − e − αH ( x ; Θ ) ] β − , (1.2)where Θ is a vector of parameters, and H ( x ; Θ ) is a non-negative, continuous, monotoneincreasing, differentiable function of x such that H ( x ; Θ ) → x → + and H ( x ; Θ ) → ∞ as x → ∞ . It is denoted by EEW( α, β, Θ ).The EEW distribuyion is a flexible family and contains many exponentiated distributionssuch as generalized exponential (Gupta and Kundu, 1999), exponentiated Weibull (Mudholkar and Srivastava,1993), generalized Rayleigh (Surles and Padgett, 2001; Kundu and Raqab, 2005), generalizedmodified Weibull (Carrasco et al., 2008), generalized linear failure rate (Sarhan and Kundu,2009), and generalized Gompertz (El-Gohary et al., 2013) distributions.In recent years, many distributions to model lifetime data have been introduced. The ba-sic idea of introducing these models is that a lifetime of a system with N (discrete randomvariable) components and the positive continuous random variable, say X i (the lifetime of ithomponent), can be denoted by the non-negative random variable Y = min( X , . . . , X N ) or Y = max( X , . . . , X N ), based on whether the components are series or parallel.In this paper, we compound the EEW family and power series distributions, and introducea new class of distribution. This class of distributions can be applied to reliability prob-lems and its some properties are investigated in this paper. We call it exponentiated extendedWeibull-power series (EEWPS) class of distributions. In similar way, some distributions are pro-posed in literature: The exponential-power series (EP) distribution by Chahkandi and Ganjali(2009), Weibull-power series (WPS) distributions by Morais and Barreto-Souza (2011), gen-eralized exponential-power series (GEP) distribution by Mahmoudi and Jafari (2012), com-plementary exponential power series by Flores et al. (2013), extended Weibull-power series(EWPS) distribution by Silva et al. (2013), double bounded Kumaraswamy-power series byBidram and Nekoukhou (2013), Burr-power series by Silva and Cordeiro (2013), generalizedlinear failure rate-power series (GLFRP) distribution by Alamatsaz and Shams (2014), Birnbaum-Saunders-power series distribution by Bourguignon et al. (2014), linear failure rate-power se-ries by Mahmoudi and Jafari (2014), and complementary extended Weibull-power series byCordeiro and Silva (2014). Similar procedures are used by Roman et al. (2012), Lu and Shi(2011), Nadarajah et al. (2014a) and Louzada et al. (2014). For compounding continuous dis-tributions with discrete distributions, Nadarajah et al. (2013) introduced the package Com-pounding in R software (R Development Core Team, 2014).e provide three motivations for the EEWPS class of distributions, which can be ap-plied in some interesting situations as follows: (i) This new class of distributions due to thestochastic representation Y = max( X , . . . , X N ), can arises in parallel systems with identicalcomponents, where each component has the EEW distribution lifetime. This model appears inmany industrial applications and biological organisms which the lifetime of the event is onlythe maximum ordered lifetime value among all causes. (ii) The EEWPS class of distributionsgives a reasonable parametric fit to some modeling phenomenon with non-monotone hazardrates such as the bathtub-shaped, unimodal and increasing-decreasing-increasing hazard rates,which are common in reliability and biological studies. (iii) The time to the last failure can beappropriately modeled by the EEWPS class of distributions.The remainder of this paper is organized as follows: The pdf and failure rate function of thenew class of distributions are given in Section 2. The special cases of the EEWPS distributionare considered in Section 3. Some properties such as quantiles, moments, order statistics,Shannon entropy and mean residual life are given in Section 4. Estimation of parameters bymaximum likelihood are discussed in Section 5. Application to a real data set is presented inSection 6. A discrete random variable, N is a member of power series distributions (truncated at zero) ifits probability mass function (pmf) is given by p n = P ( N = n ) = a n λ n C ( λ ) , n = 1 , , . . . , (2.1)where a n ≥ C ( λ ) = ∞ P n =1 a n λ n , and λ ∈ (0 , s ) is chosen in a way such that C ( λ ) is finiteand its first, second and third derivatives are defined and shown by C ′ ( . ), C ′′ ( . ) and C ′′′ ( . ),respectively. The term “power series distribution” is generally credited to Noack (1950). Thisfamily of distributions includes many of the most common distributions, including the binomial,Poisson, geometric, negative binomial, logarithmic distributions. For more details about powerseries distributions, see Johnson et al. (2005), page 75. Theorem 2.1.
Let N be a random variable denoting the number of failure causes which it is amember of power series distributions with pmf in (2.1) . Also, For given N , let X , X , ..., X N beindependent identically distributed random variables from EEW distribution with pdf in (1.2) .Then X ( N ) = max ≤ i ≤ N { X i } has EEWPS class of distributions is denoted by EEWPS( α, β, λ, ) and has the following pdf: f ( x ) = αβλh ( x ; Θ ) e − αH ( x ; Θ ) (1 − e − αH ( x ; Θ ) ) β − C ′ (cid:0) λ (1 − e − αH ( x ; Θ ) ) β (cid:1) C ( λ ) , x > . (2.2) Proof.
The conditional cdf of X ( N ) | N = n has EEW( α, nβ, Θ ). Hence, P ( X ( N ) ≤ x, N = n ) = a n λ n C ( λ ) [1 − e − αH ( x ; Θ ) ] nβ , (2.3)and the marginal cdf of X ( N ) is F ( x ) = C ( λ (1 − e − αH ( x ; Θ ) ) β ) C ( λ ) , x > . (2.4)The derivative of F with respect to x is (2.2). Therefore, X ( N ) has EEWPS distribution. Proposition 1.
The pdf of EEWPS class can be expressed as infinite linear combination ofdensity of order distribution, i.e. it can be written as f ( x ) = ∞ X n =1 p n g ( n ) ( x ; α, nβ, Θ ) , (2.5) where g ( n ) ( x ; α, nβ, Θ ) is the pdf of EEW distribution with parameters α , nβ and Θ .Proof. Consider t = 1 − e − αH ( x ; Θ ) . So f ( x ) = αβλh ( x ; Θ ) e − αH ( x ; Θ ) t β − C ′ (cid:0) λt β (cid:1) C ( λ )= αβλh ( x ; Θ ) e − αH ( x ; Θ ) t β − ∞ P n =1 na n ( λt β ) n − C ( λ )= ∞ X n =1 a n λ n C ( λ ) nαβh ( x ; Θ ) e − αH ( x ; Θ ) t nβ − = ∞ X n =1 p n g ( n ) ( x ; α, nβ, Θ ) . Proposition 2.
The limiting distribution of
EEWPS( β, λ, Θ ) when λ → + is lim λ → + F ( x ) = [1 − e − αH ( x ; Θ ) ] cβ , which is a EEW distribution with parameters α , cβ and Θ , where c = min { n ∈ N : a n > } .roof. Consider t = 1 − e − αH ( x ; Θ ) . Solim λ → + F ( x ) = lim λ → + C ( λt β ) C ( λ ) = lim λ → + ∞ P n =1 a n λ n t nβ ∞ P n =1 a n λ n = lim λ → + a c t cβ + ∞ P n = c +1 a n λ n − c t nβ a c + ∞ P n = c +1 a n λ n − c = t cβ . Proposition 3.
The hazard rate function of the EEWPS class of distributions is given by r ( x ) = αλβh ( x ; Θ )(1 − t ) t β − C ′ (cid:0) λt β (cid:1) C ( λ ) − C ( λt β ) , (2.6) where t = 1 − e − αH ( x ; Θ ) .Proof. Using (2.2), (2.4) and definition of hazard rate function as r ( x ) = f ( x ) / (1 − F ( x ), theproof is obvious. In this Section, we consider some special cases of the EEWPS distribution. If β = 1, then the pdf in (2.2) becomes to f ( x ) = αλh ( x ; Θ ) e − αH ( x ; Θ ) C ′ (cid:0) λ (1 − e − αH ( x ; Θ ) ) (cid:1) C ( λ ) , x > , (3.1)which is the pdf of complementary extended Weibull power series (CEWPS) class of distribu-tions introduced by Cordeiro and Silva (2014). If H ( x ; Θ ) = x , then the pdf in (2.2) becomes to f ( x ) = αβλe − αx (1 − e − αx ) β − C ′ (cid:0) λ (1 − e − αx ) β (cid:1) C ( λ ) , x > . (3.2)which is the pdf of generalized exponential-power series (GEPS) class of distributions intro-duced by Mahmoudi and Jafari (2012). The GEPS class of distributions contains complemen-tary exponentiated exponential-geometric distribution introduced by Louzada et al. (2013),omplementary exponential-geometric distribution introduced by Louzada et al. (2011), Poisson-exponential distribution introduced by Cancho et al. (2011) and Louzada-Neto et al. (2011),complementary exponential -power series class of distributions introduced by Flores et al.(2013), generalized exponential distribution introduced by Gupta and Kundu (1999) and gen-eralized exponential-geometric distribution introduced by Bidram et al. (2013) . If H ( x ; Θ ) = axα + bx α , then the pdf in (2.2) becomes to f ( x ) = βλ ( a + bx ) e − ax − bx (1 − e − ax − bx ) β − C ′ (cid:18) λ (1 − e − ax − bx ) β (cid:19) C ( λ ) , x > . (3.3)which is the pdf of generalized linear failure rate-power series (GLFRPS) class of distributionsintroduced by Alamatsaz and Shams (2014). It is a modification of generalized linear failurerate distribution introduced by Sarhan and Kundu (2009) and generalized linear failure rate-geometric distribution introduced by Nadarajah et al. (2014b). If b = 0, it becomes to GEPSclass of distributions. Also, If β = 1, it becomes to linear failure rate-power series introducedby Mahmoudi and Jafari (2014). If H ( x ; Θ ) = x γ , then the pdf in (2.2) becomes to f ( x ) = αβλγx γ − e − αx γ (1 − e − αx γ ) β − C ′ (cid:0) λ (1 − e − αx γ ) β (cid:1) C ( λ ) , x > . (3.4)which is the pdf of exponentiated Weibull-power series (EWPS) class of distributions introducedby Mahmoudi and Shiran (2012). It is a modification of exponentiated Weibull distributionintroduced by Mudholkar and Srivastava (1993).It is contain the complementary Weibull ge-ometric distribution introduced by Tojeiro et al. (2014). Also, the Marshall-Olkin extendedWeibull distribution introduced by Cordeiro and Lemonte (2013) is a special case of EWPS. If H ( x ; Θ ) = x γ exp( τ x ), then the pdf in (2.2) becomes to f ( x ) = αβλx γ − ( γ + τ x ) e τx − αx γ exp( τx ) C ′ (cid:0) λ (1 − e − αx γ exp( τx ) ) β (cid:1) (1 − e − αx γ exp( τx ) ) − β C ( λ ) , x > , (3.5)nd we call generalized modified Weibull-power series (GMWPS) class of distributions. It iscontained the generalized modified Weibull distribution introduced by Carrasco et al. (2008).If τ = 0, then GMWPS class of distributions becomes to EWPS class of distributions. If H ( x ; Θ ) = γ ( e γx − f ( x ) = αβλe γx e − αγ ( e γx − (1 − e − αγ ( e γx − ) β − C ′ (cid:16) λ (1 − e − αγ ( e γx − ) β (cid:17) C ( λ ) , x > . (3.6)and we call generalized Gompertz-power series class of distributions. It is contained the gen-eralized Gompertz distribution introduced by El-Gohary et al. (2013). In this section, some properties of EEWPS class of distributions such as quantiles, moments,order statistics, Shannon entropy and mean residual life are derived. Using (2.5), we can obtain F ( x ) = ∞ X n =1 p n G ( n ) ( x ; α, nβ, Θ ) = ∞ X n =1 p n t nβ , (4.1)where t = 1 − e − αH ( x ; Θ ) . Based on the mathematical quantities of the baseline pdf g ( n ) ( x ; α, nβ, Θ ), we can obtain some statistical quantities such as ordinary and incomplete moments, gen-erating function and mean deviations of this family of distributions. Let X = G − (cid:18) C − ( C ( λ ) U ) λ (cid:19) , where U has a uniform distribution on (0 , G − ( y ) = H − [ − α ln(1 − y β )] and C − ( . ) is theinverse function of C ( . ). Then X has the EEWPS( α, β, λ, Θ ) distribution. This result helps insimulating data from the EEWPS distribution with generating uniform distribution data. Theorem 4.1.
Consider X ∼ EEWPS( α, β, λ, Θ ) . Then the moment generating function ofEEWPS is M X ( t ) = ∞ X n =1 ∞ X j =0 p n (cid:18) nβj + 1 (cid:19) ( − j M Y ( t ) , (4.2) where Y has EEW( α ( j + 1) , , Θ ) .roof. The Laplace transform of the EEWPS class can be expressed as L ( s ) = E ( e − sX ) = ∞ X n =1 P ( N = n ) L n ( s ) , where L n ( s ) is the Laplace transform of EEW distribution with parameters α , nβ and Θ givenas L n ( s ) = Z + ∞ e − sx nαβh ( x ; Θ ) e − αH ( x ; Θ ) [1 − e − αH ( x ; Θ ) ] nβ − dx = nαβ Z + ∞ e − sx h ( x ; Θ ) ∞ X j =0 (cid:18) nβ − j (cid:19) ( − j e − ( j +1) αH ( x ; Θ ) dx = ∞ X j =0 nβ (cid:18) nβ − j (cid:19) ( − j Z + ∞ α ( j + 1) j + 1 h ( y ; Θ ) e − ( j +1) αH ( y ; Θ ) − sy dy = ∞ X j =0 (cid:18) nβj + 1 (cid:19) ( − j L ( s ) , where L ( s ) is the Laplace transform of the EEW( α ( j + 1) , , Θ ). Therefore, the momentgenerating function of EEWPS is M X ( t ) = ∞ X n =1 p n L n ( − t )= ∞ X n =1 ∞ X j =0 p n (cid:18) nβj + 1 (cid:19) ( − j L ( − t )= ∞ X n =1 ∞ X j =0 p n (cid:18) nβj + 1 (cid:19) ( − j M Y ( t ) . Theorem 4.2.
The noncentral moment functions of EEWPS is µ r = ∞ X n =1 a n λ n C ( λ ) ∞ X j =0 (cid:18) nβj + 1 (cid:19) ( − j µ ′ r = ∞ X n =1 ∞ X j =0 p n (cid:18) nβj + 1 (cid:19) ( − j µ ′ r , (4.3) where µ ′ r = E [ Y r ] and Y has EEW( α ( j + 1) , , Θ ) .Proof. We can use M X ( t ) to obtain µ r . But from the direct calculation, proof is obvious.With considering H ( x ) = x γ and C ( λ ) = λ (1 − λ ) − , we calculated the first four momentswith different values of parameters for the EEWPS distribution using (4.3). Also, we computedthese values from the direct definition by numerical integration. We found that the results aresame. The values are given in Tables 1.able 1: The four moments of EEWPS model. α β λ γ µ µ µ µ Let X , X , . . . , X m be a random sample of size m from EEWPS( α, β, λ, Θ ), then the pdf ofthe i th order statistic, say X i : m , is given by f i : m ( x ) = m !( i − m − i )! f ( x ) (cid:20) C ( λt β ) C ( λ ) (cid:21) i − (cid:20) − C ( λt β ) C ( λ ) (cid:21) m − i = m !( i − m − i )! f ( x ) m − i X j =0 (cid:18) m − ij (cid:19) ( − j (cid:20) C ( λt β ) C ( λ ) (cid:21) j + i − = m !( i − m − i )! ∞ X n =1 m − i X j =0 p n g ( n ) ( x ; α, nβ, Θ ) (cid:18) m − ij (cid:19) ( − j (cid:20) C ( λt β ) C ( λ ) (cid:21) j + i − = m !( i − m − i )! ∞ X n =1 m − i X j =0 w j p n g ( n ) ( x ; α, nβ, Θ ) (cid:20) C ( λt β ) C ( λ ) (cid:21) j + i − , here f is the pdf of EEWP class of distributions, t = 1 − e − αH ( x ; Θ ) and w j = (cid:0) m − ij (cid:1) ( − j .Also, the cdf of X i : m is given by F i : m ( x ) = m X k = i m − k X j =0 ( − j (cid:18) m − kj (cid:19)(cid:18) mk (cid:19) (cid:20) C ( λt β ) C ( λ ) (cid:21) j + k . An analytical expression for r th moment of order statistics X i : m is obtained as E [ X ri : m ] = m !( i − m − i )! ∞ X n =1 m − i X j =0 w j p n E [ Z r ( F ( Z )) j + i − ] , where Z has a EEW distribution with parameters α , nβ and Θ . The maximum entropy method is a powerful technique in the field of probability and statistics.It is introduced by Jaynes (1957) and closely related to the Shannon’s entropy. Also, it isapplied in a wide variety of fields and used for the characterization of pdf’s; see, for example,Kapur (1994) Soofi (2000) and Zografos and Balakrishnan (2009). Shore and Johnson (1980)treated the maximum entropy method axiomatically.Considers a class of pdf’s F = { f ( x ; α, β, λ, Θ ) : E f ( T i ( X )) = β i , i = 0 , , ...., m } , (4.4)where T ( X ) , ..., T m ( X ) are absolutely integrable functions with respect to f , and T ( X ) = 1.Also, consider the shannon’s entropy of none-negative continuous random variable X with pdf f defined by Shannon (1948) as H sh ( f ) = E [ − log f ( X )] = − Z + ∞ f ( x ) log( f ( x )) dx. (4.5)The maximum entropy distribution is the pdf of the class F , denoted by f ME determined as f ME ( x ; λ, β, Θ ) = arg max f ∈ F H sh ( f ) . Now, suitable constraints are derived in order to provide a maximum entropy character-ization for the class (4.4) based on Jaynes (1957). For this purpose, the next result plays animportant role.
Proposition 4.
Let X has EEWPS( α, β, λ, Θ ) with the pdf given by (2.2) . Then,i. E h log( C ′ ( λ (1 − e − αH ( X ; Θ ) ) β )) i = λC ( λ ) E h C ′ ( λ (1 − e − αH ( Y ; Θ ) ) β ) log( C ′ ( λ (1 − e − αH ( Y ; Θ ) ) β )) i , ii. E [log( h ( X ; Θ ))] = λC ( λ ) E h C ′ ( λ (1 − e − H ( Y ; Θ ) ) β ) log( h ( Y ; Θ )) i , iii. E h log(1 − e − αH ( X ; Θ ) ) i = λC ( λ ) E h C ′ ( λ (1 − e − αH ( Y ; Θ ) ) β ) log(1 − e − αH ( Y ; Θ ) ) i , where Y follows the EEW distribution with the pdf in (1.2) . An explicit expression of Shannon entropy for EEWPS distribution is obtained as H sh ( f ) = − log( αβλ ) − λC ( λ ) E [ C ′ ( λ (1 − e − H ( Y ; Θ ) ) β ) log( C ′ ( λ (1 − e − H ( Y ; Θ ) ) β ))]+ log[ C ( λ )] − ( β − λC ( λ ) E [ C ′ ( λ (1 − e − H ( Y ; Θ ) ) β ) log(1 − e − H ( Y ; Θ ) )] − λC ( λ ) E [ C ′ ( λ (1 − e − H ( Y ; Θ ) ) β ) log( h ( Y ; Θ ))] . (4.6)Also, the mean residual life function of X is given by m ( t ) = E [ X − t | X > t ] = R + ∞ t ( x − t ) f ( x ) dx − F ( t ) = C ( λ ) ∞ P n =1 p n R + ∞ t zg ( n ) ( z ; α, nβ, Θ ) dzC ( λ ) − C ( λG ( x )) − t = C ( λ ) ∞ P n =1 p n E [ ZI ( Z>t ) ] C ( λ ) − C ( λG ( x )) − t, (4.7)where Z has a EEW distribution with parameters α , nβ and Θ . In the context of reliability, the stress - strength model describes the life of a component whichhas a random strength X subjected to a random stress Y . The component fails at the instantthat the stress applied to it exceeds the strength, and the component will function satisfactorilywhenever X > Y . Hence, R = P ( X > Y ) is a measure of component reliability. It has manyapplications especially in engineering concept. Here, we obtain the form for the reliability R when X and Y are independent random variables having the same EEWPS distribution. Thequantity R can be expressed as R = Z ∞ f ( x ; α, β, λ, Θ ) F ( x ; α, β, λ, Θ ) dx = Z ∞ λg ( x ) C ′ ( λG ( x )) C ( λG ( x )) C ( λ ) dx = ∞ X n =1 p n Z ∞ g ( n ) ( x ; α, nβ, Θ ) C ( λG ( x )) C ( λ ) dx. (4.8) Estimation
In this section, we first study the maximum likelihood estimations (MLE’s) of the parameters.Then, we propose an Expectation-Maximization (EM) algorithm to estimate the parameters.
Let x , . . . , x n be observed value from the EEWPS distribution with parameters ξ = ( α, β, λ, Θ ) T . The log-likelihood function is given by l n = l n ( ξ ; x ) = n [log( α ) + log( β ) + log( λ ) − log( C ( λ ))] + n X i =1 log[ h ( x i ; Θ )] − α n X i =1 H ( x i ; Θ ) + ( β − n X i =1 log t i + n X i =1 log( C ′ ( λt βi )) , where x = ( x , . . . , x n ) and t i = 1 − e − αH ( x i ; Θ ) . The components of the score function U ( ξ ; x ) =( ∂l n ∂α , ∂l n ∂β , ∂l n ∂λ , ∂l n ∂ Θ ) T are ∂l n ∂α = nα − n X i =1 H ( x i ; Θ ) , (5.1) ∂l n ∂β = nβ + n X i =1 log( t i ) + n X i =1 λt βi log( t i ) C ′′ ( λt βi ) C ′ ( λt βi ) , (5.2) ∂l n ∂λ = nλ − nC ′ ( λ ) C ( λ ) + n X i =1 t βi C ′′ ( λt βi ) C ′ ( λt βi ) , (5.3) ∂l n ∂ Θ k = n X i =1 ∂h ( x i ; Θ ) ∂ Θ k . h ( x i ; Θ ) − α n X i =1 ∂H ( x i ; Θ ) ∂ Θ k +( β − n X i =1 ∂t i ∂ Θ k t i + βλ n X i =1 [ ∂t i ∂ Θ k ] t β − i C ′′ ( λt βi ) C ′ ( λt βi ) , (5.4)where Θ k is the k th element of the vector Θ .The MLE of ξ , say ˆ ξ , is obtained by solving the nonlinear system U ( ξ ; x ) = . We cannotget an explicit form for this nonlinear system of equations and they can be calculated by usinga numerical method, like the Newton method or the bisection method. Only, for given Θ , from(5.1) we have α = n P ni =1 H ( x i ; Θ ) . Therefore, (5.4) becomes n X i =1 ∂h ( x i ; Θ ) ∂ Θ k . h ( x i ; Θ ) − n P ni =1 H ( x i ; Θ ) n X i =1 ∂H ( x i ; Θ ) ∂ Θ k ( β − n X i =1 ∂t i ∂ Θ k t i + βλ n X i =1 [ ∂t i ∂ Θ k ] t β − i C ′′ ( λt βi ) C ′ ( λt βi ) . (5.5) Theorem 5.1.
The pdf, f ( x | Θ ) , of EEWPS distribution satisfies on the regularity condistions,i.e.i. the support of f ( x | Θ ) does not depend on Θ ,ii. f ( x | Θ ) is twice continuously differentiable with respect to Θ ,iii. the differentiation and integration are interchangeable in the sense that ∂∂ Θ Z ∞−∞ f ( x | Θ ) dx = Z ∞−∞ ∂∂ Θ f ( x | Θ ) dx, ∂ ∂ Θ ∂ Θ T Z ∞−∞ f ( x | Θ ) dx = Z ∞−∞ ∂ ∂ Θ ∂ Θ T f ( x | Θ ) dx. Proof.
The proof is obvious and for more details, see Casella and Berger (2001) Section 10.The asymptotic confidence intervals of these parameters will be derived based on Fisherinformation matrix. It is well-known that under regularity conditions, the asymptotic distribu-tion of √ n (ˆ ξ − ξ ) is multivariate normal with mean and variance-covariance matrix J − n ( ξ ),where J n ( ξ ) = lim n →∞ I n ( ξ ), and I n ( ξ ) is the observed information matrix as I n ( ξ ) = − U αα U αβ U αλ | U Tα Θ U αβ U ββ U βλ | U Tβ Θ U αβ U λβ U λλ | U Tλ Θ − − − − − U α Θ U β Θ U λ Θ | U ΘΘ , whose elements are obtained by derivative the equations (5.1)-(5.4) with respect to parameters. The traditional methods to obtain the MLE’s are numerical methods for solving the equations(5.1)-(5.4), and sensitive to the initial values. Therefore, we develop an EM algorithm forobtaining the MLE’s of the parameters of EEWPS class of distributions. It is a very powerfultool in handling the incomplete data problem (Dempster et al., 1977). It is an iterative method,and there are two steps in each iteration: Expectation step or the E-step and the Maximizationstep or the M-step. The EM algorithm is especially useful if the complete data set is easy toanalyze.Using (2.3), we define a hypothetical complete-data distribution with a joint pdf in theform g ( x, z ; ξ ) = a z λ z C ( λ ) zαβh ( x ; Θ )(1 − t ) t zβ − , x > , z ∈ N , here t = 1 − e − αH ( x ; Θ ) . The E-step of an EM cycle requires the expectation of ( Z | X ; ξ ( r ) )where ξ ( r ) = ( α ( r ) , β ( r ) , λ ( r ) , Θ ( r ) ) is the current estimate (in the r th iteration) of ξ . Theexpected value of Z | X = x is E ( Z | X = x ) = 1 + λt α C ′′ ( λt β ) C ′ ( λt β ) . (5.6)The M-step of EM cycle is completed by using the MLE over Θ , with the missing z ’sreplaced by their conditional expectations given above. Therefore, the log-likelihood for thecomplete-data y = ( x , . . . , x n , z , ..., z n ) is l ∗ ( y ; ξ ) ∝ n X i =1 z i log( λ ) + n log( αβ ) + n X i =1 log h ( x i ; Θ ) − α n X i =1 H ( x i ; Θ )+ n X i =1 ( z i β −
1) log(1 − e − αH ( x i ; Θ ) ) − n log( C ( λ )) . (5.7)On differentiation of (5.7) with respect to parameters α , β , λ and Θ k , we obtain the componentsof the score function as ∂l ∗ n ∂α = nα − n X i =1 H ( x i ; Θ ) + n X i =1 ( z i β − H ( x i ; Θ ) e − αH ( x i ; Θ ) − e − αH ( x i ; Θ ) ,∂l ∗ n ∂β = nβ + n X i =1 z i log(1 − e − αH ( x i ; Θ ) ) ,∂l ∗ n ∂λ = n X i =1 z i λ − n C ′ ( λ ) C ( λ ) ,∂l ∗ n ∂ Θ k = n X i =1 ∂h ( x i ; Θ ) ∂ Θ k . h ( x i ; Θ ) − α n X i =1 ∂H ( x i ; Θ ) ∂ Θ k + n X i =1 ( z i β − ∂H ( x i ; Θ ) ∂ Θ k − e − αH ( x i ; Θ ) . Therefore, we obtain the iterative procedure of the EM-algorithm asˆ β ( j +1) = − n n P i =1 ˆ z ( j ) i log[1 − e − ˆ α ( j ) H ( x i ; ˆ Θ ( j ) ) ] , ˆ λ ( j +1) = C (ˆ λ ( j +1) ) nC ′ (ˆ λ ( j +1) ) n X i =1 ˆ z ( j ) i ,n ˆ α ( j +1) − n X i =1 H ( x i ; ˆ Θ ( j ) ) + n X i =1 ( ˆ z i ( j ) ˆ β ( j ) − H ( x i ; ˆ Θ ( j ) ) e − ˆ α ( j +1) H ( x i ; ˆ Θ ( j ) ) − e − ˆ α ( j +1) H ( x i ; ˆ Θ ( j ) ) = 0 , n X i =1 ∂h ( x i ; ˆ Θ ( j +1) ) ∂ Θ k . h ( x i ; ˆ Θ ( j +1) ) − ˆ α ( j ) n X i =1 ∂H ( x i ; ˆ Θ ( j +1) ) ∂ Θ k + n X i =1 ∂H ( x i ; ˆ Θ ( j +1) ) ∂ Θ k . ˆ z i ( j ) ˆ β ( j ) − − e − ˆ α ( j ) H ( x i ; ˆ Θ ( j +1) ) = 0 , here ˆ λ ( j +1) , ˆ α ( j +1) and ˆΘ ( j +1) k are found numerically. Here, we haveˆ z ( j ) i = 1 + λ ∗ ( j ) C ′′ ( λ ∗ ( j ) ) C ′ ( λ ∗ ( j ) ) , i = 1 , , ..., n, where λ ∗ ( j ) = ˆ λ ( j ) [1 − e − ˆ α ( j ) H ( x i ; ˆΘ k ( j ) ) ] ˆ β ( j ) .We can use the results of Louis (1982) to obtain the standard errors of the estimators fromthe EM-algorithm. Consider ℓ c ( Θ ; x ) = E ( I c ( Θ ; y ) | x ), where I c ( Θ ; y ) = − [ ∂U ( y ; Θ ) ∂ Θ ] is the( k + 3) × ( k + 3) observed information matrix. If ℓ m ( Θ ; x ) = V ar [ U ( y ; Θ ) | x ], then, we obtainthe observed information as J ( ˆ Θ ; x ) = ℓ c ( ˆ Θ ; x ) − ℓ m ( ˆ Θ ; x ) . (5.8)The standard errors of the MLE’s based on the EM-algorithm are the square root of thediagonal elements of the J ( ˆ Θ ; x ). The computation of these matrices are too long and tedious.Therefore, we did not present the details. Reader can see Mahmoudi and Jafari (2012) how tocalculate these values. In this section, we analyze the real data set given by Murthy et al. (2004) to demonstrate theperformance of EEWPS class of distributions in practice. This data set consists of the failuretimes of 20 mechanical components, and is also studied by Silva et al. (2013):0.067, 0.068, 0.076, 0.081, 0.084, 0.085, 0.085, 0.086, 0.089, 0.0980.098, 0.114, 0.114, 0.115, 0.121, 0.125, 0.131, 0.149, 0.160, 0.485Since the EEWPS distribution can be used for modeling of failure times, we consider thisdistribution for fitting these data. But, this distribution is a large class of distributions. Here,we consider five sub-models of EEWPS distribution. Some of them are suggested in literature.i. The exponentiated Weibull geometric (EWG) distribution, i.e. the EEWPS distributionwith H ( x, Θ ) = x γ and C ( λ ) = λ (1 − λ ) − .ii. The complementary Weibull geometric (CWG) distribution, i.e. the EEWPS distribu-tion with H ( x, Θ ) = x γ , C ( λ ) = λ (1 − λ ) − and β = 1. This distribution is considered byCordeiro and Silva (2014).iii. The generalized exponential geometric (GEG) distribution, i.e. the EEWPS distributionwith H ( x, Θ ) = x and C ( λ ) = λ (1 − λ ) − . This distribution is considered by Mahmoudi and Jafari(2012).v. The exponentiated Chen logarithmic (ECL) distribution, i.e. the EEWPS distribution with H ( x, Θ ) = exp( x γ ) and C ( λ ) = − log(1 − λ ).iv. The complementary Chen logarithmic (CCL) distribution, i.e. the EEWPS distributionwith H ( x, Θ ) = exp( x γ ), C ( λ ) = − log(1 − λ ) and β = 1. This distribution is considered byCordeiro and Silva (2014).The MLE’s of the parameters for the distributions are obtained by the EM algorithmgiven in Section 5. Also, the standard errors of MLE’s are computed and given in paracenteses.To test the goodness-of-fit of the distributions, we calculated the maximized log-likelihood(log( L )), the Kolmogorov-Smirnov (K-S) statistic with its respective p-value, the AIC (AkaikeInformation Criterion), AICC (AIC with correction), BIC (Bayesian Information Criterion),CM (Cramer-von Mises statistic) and AD (Anderson-Darling statistic) for the five submodels ofdistribution. The R software (R Development Core Team, 2014) is used for the computations.The results are given in Table 2, and from K-S, it can be concluded that all five modelsare appropriate for this data set. But, the EWG and ECL distributions are better than otherdistributions. In fact, we have a better fit when there is the parameter β (exponentiatedparameter) in model. The plots of the densities (together with the data histogram) and cdf’sgiven in Figure 1 confirm this conclusion.Table 2: Parameter estimates (standard errors), K-S statistic, p -value, AIC, AICC, BIC, CMand AD for the data set. Distribution EWG CWG GEG ECL CCLˆ α γ λ β . e — 13.825 7 . e —(s.e.) (1 . e ) — (8.471) 2 . e —log ( L ) 37.978 26.422 32.976 37.794 25.759K-S 0.124 0.264 0.160 0.121 0.262p-value 0.917 0.122 0.683 0.931 0.127AIC -67.957 -46.845 -59.952 -67.588 -45.518AICC -65.29 -45.345 -58.452 -64.922 -44.018BIC -63.974 -43.858 -56.965 -63.606 -42.531CM 0.048 0.436 0.153 0.051 0.463AD 0.402 2.537 1.136 0.423 2.663 istogram x D en s i t y EWGCWGGEGECLCCL 0.1 0.2 0.3 0.4 0.5 . . . . . . Empirical Distribution x c d f EWGCWGGEGECLCCL
Figure 1: The histogram of the data set with the estimated pdf’s (left), the empirical cdf ofthe data set, and estimated cdf’s (right) for fitted of five submodels.
Acknowledgments
The authors are thankful to the referees for helpful comments and suggestions.
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