The Generalized Weighted Lindley Distribution: Properties, Estimation and Applications P. L. RAMOS,∗ F. LOUZADA† 6 1 0 2 Institute of Mathematical Science and Computing l u Universidade de Sao Paulo, Sao Carlos-SP, Brazil J 8 1 January/2016 ] T S . h t Abstract a m In this paper, we proposed a new lifetime distribution namely generalized weighted [ Lindley (GLW) distribution. The GLW distribution is a useful generalization of the 2 weightedLindleydistribution, whichaccommodatesincreasing, decreasing, decreasing- v increasing-decreasing, bathtub, or unimodal hazard functions, making the GWL dis- 0 1 tribution a flexible model for reliability data. A significant account of mathematical 4 properties of the new distribution are presented. Different estimation procedures are 7 also given such as, maximum likelihood estimators, method of moments, ordinary and 0 . weighted least-squares, percentile, maximum product of spacings and minimum dis- 1 0 tance estimators. The different estimators are compared by an extensive numerical 6 simulations. Finally, we analyze two data sets for illustrative purposes, proving that 1 the GWL outperform several usual three parameters lifetime distributions. : v Keywords: Generalized weighted Lindley distribution, Maximum Likelihood Estima- i X tion, Maximum product of spacings. r a 1 Introduction In recent years, several new extensions of the exponential distribution has been introduced in the literature for describing real problems. Ghitany et al. (2008) investigated different properties of the Lindley distribution and outlined that in many cases the Lindley distri- bution is a better model than one based on the exponential distribution. Since then, many ∗Email: [email protected] †Email: [email protected] 1 generalizations of the Lindley distribution have been introduced, such as generalized Lind- ley (Zalerzadeh and Dolati, 2009), extended Lindley (Bakouch et al., 2012), exponential Poisson Lindley (Barreto-Souza and Bakouch, 2013), Power Lindley (Ghitany et al., 2013) distribution, among others. Ghitany et al. (2011) introduced a new class of weighted Lindley (WL) distribution adding more flexibility to the Lindley distribution. Let T be a random variable with a WL distribution the probability density function (p.d.f) is given by λφ+1 f(t|λ,φ) = tφ−1(1+t)e−λt, (1) (λ+φ)Γ(φ) for all t > 0 , φ > 0 and λ > 0 and Γ(φ) = (cid:82)∞e−xxφ−1dx is known as gamma function. 0 One of its peculiarities is that the hazard function can has an increasing (φ ≥ 1) or bathtub (0 < φ < 1) shape. Different properties of this model and estimation methods were studied by Mazucheli et al. (2013), Ali (2013), Wang (2015), Al-Mutairi et al. (2015) among others. In this paper, a new lifetime distribution family is proposed, which is an direct general- ization of the weighted Lindley distribution. The p.d.f is given by αλαφ f(t|φ,λ,α) = tαφ−1(λ+(λt)α)e−(λt)α, (2) (λ+φ)Γ(φ) for all t > 0, φ > 0,λ > 0 and α > 0. Important probability distributions can be obtained from the GWL distribution as the weighted Lindley distribution (α = 1) , Power Lindley dis- tribution(φ = 1)andtheLindleydistribution(φ = 1andα = 1). Duethisrelationship, such model could also be named as weighted power Lindley or generalized power Lindley distribu- tion. We present a proof that this model has different forms of the hazard function, such as: increasing, decreasing, bathtub, unimodal or decreasing-increasing-decreasing shape, making the GWL distribution a flexible model for reliability data. Moreover, a significant account of mathematical properties of the new distribution are provided. The inferential procedures of the parameters of the GLW distribution are presented con- sidering different estimation methods such as: maximum likelihood estimators (MLE), meth- ods of moments (ME), ordinary least-squares estimation (OLSE), weighted least-squares es- timation (WLSE), maximum product of spacings (MPS), Cramer-von Mises type minimum distance (CME), Anderson-Darling (ADE) and Right-tail Anderson-Darling (RADE). We compare the performances of the such different methods using extensive numerical simula- tions. Finally, we analyze two data sets for illustrative purposes, proving that the GWL outperform several usual three parameters lifetime distributions such as: the generalized Gamma distribution (Stacy, 1962), the generalized Weibull (GW) distribution (Mudholkar et al., 1996), the generalized exponential-Poisson (GEP) distribution (Barreto-Souza and Cribatari-Neto, 2009) and the exponentiated Weibull (EP) distribution (Mudholkar et al., 1995). The paper is organized as follows. In Section 2, we provide a significant account of mathematicalpropertiesofthenewdistribution. InSection3,wediscusstheeightestimation 2 methods considered in this paper. In Section 4 a simulation study is presented in order to identify the most efficient procedure. In Section 5 the methodology is illustrated in two real data sets. Some final comments are presented in Section 6. 2 Generalized Weighted Lindley distribution The Generalized weighted Lindley distribution (2) can be expressed as a two-component mixture f(t|φ,λ,α) = pf (t|φ,λ,α)+(1−p)f (t|φ,λ,α) 1 2 where p = λ/(λ+φ) and T ∼ GG(φ+j−1,λ,α), for j = 1,2, i.e, f (t|λ,φ) has Generalized j j Gamma distribution, given by α f (t|φ,λ,α) = λα(φ+j−1)tα(φ+j−1)−1e−(λt)α. (3) j Γ(φ+j −1) The behaviours of the p.d.f. (2) when t → 0 and t → ∞ are, respectively, given by  ∞, if αφ < 1    αλ2 f(0) = , if αφ = 1 , f(∞) = 0. (λ+φ)Γ(φ)    0, if αφ > 1 Figure 1 gives examples of the shapes of the density function for different values of φ,λ and α. 5 5 1. f =0.2, l =1.5, a =2.1 1. f =0.2, l =2.5, a =0.2 f =0.5, l =1.5, a =1.5 f =0.5, l =2.5, a =0.6 f =0.7, l =1.5, a =1.0 f =1.0, l =2.5, a =0.8 f =0.7, l =1.5, a =1.5 f =1.5, l =1.5, a =0.5 f =3.5, l =20, a =0.7 f =2.0, l =1.5, a =1.2 0 0 1. 1. f(t) f(t) 5 5 0. 0. 0 0 0. 0. 0 1 2 3 4 0 1 2 3 4 t t Figure 1: Density function shapes for GWL distribution considering different values of φ,λ and α. 3 The cumulative distribution function from the GWL distribution is given by γ[φ,(λt)α](λ+φ)−(λt)αφe−(λt)α F(t|φ,λ,α) = . (4) (λ+φ)Γ(φ) where γ[y,x] = (cid:82)xwy−1e−wdw is the lower incomplete gamma function. 0 2.1 Moments Many important features and properties of a distribution can be obtained through its mo- ments, such as mean, variance, kurtosis and skewness. In this section, we present some important moments, such as the moment generating function, r-th moment, r-th central moment among others. Theorem 2.1. For the random variable T with GWL distribution, the moment generating function is given by (cid:88)∞ tr (cid:0)r +φ+λ(cid:1)Γ(r +φ) M (t) = α α . (5) X λrr! (λ+φ)Γ(φ) r=0 Proof. Note that, the moment generating function from GG distribution (3) is given by (cid:88)∞ tr Γ(r +φ+j −1) M (t) = α . X,j r! λrΓ(φ+j −1) r=0 Therefore, as the GWL (2) distribution can be expressed as a two-component mixture, we have (cid:90) ∞ (cid:2) (cid:3) M (t) = E etX = etxf(x|φ,λ,α)dx = pM (t)+(1−p)M (t) X X,1 X,2 0 λ (cid:88)∞ tr Γ(r +φ) φ (cid:88)∞ tr Γ(r +φ+1) = α + α (λ+φ) r! λrΓ(φ) (λ+φ) r! λrΓ(φ+1) r=0 r=0 1 (cid:88)∞ tr λΓ(r +φ) 1 (cid:88)∞ tr (cid:0)r +φ(cid:1)Γ(r +φ) = α + α α (λ+φ) r! λrΓ(φ) (λ+φ) r! λrΓ(φ) r=0 r=0 (cid:88)∞ tr (cid:0)r +φ+λ(cid:1)Γ(r +φ) = α α . λrr! (λ+φ)Γ(φ) r=0 Corollary 2.2. For the random variable T with GWL distribution, the r-th moment is given by (cid:0) (cid:1) r +φ+λ Γ(r +φ) µ = E[Tr] = α α . (6) r (λ+φ)λrΓ(φ) 4 Proof. From the literature µ = M(r)(0) = dnMX(0) and the result follows. r X dtn Corollary 2.3. For the random variable T with GWL distribution, the r-th central moment is given by r (cid:18) (cid:19) (cid:88) r M = E[T −µ]r = (−µ)r−iE[Ti] r i i=0 (7) (cid:88)r (cid:18)r(cid:19)(cid:32) (cid:0)1 +φ+λ(cid:1)Γ(cid:0)1 +φ(cid:1)(cid:33)r−i (cid:0)i +φ+λ(cid:1)Γ(i +φ) = − α α α α i λ(λ+φ)Γ(φ) (λ+φ)λiΓ(φ) i=0 Corollary 2.4. A random variable T with GWL distribution, has the mean and variance given by (cid:0) (cid:1) (cid:0) (cid:1) 1 +φ+λ Γ 1 +φ µ = α α , (8) λ(λ+φ)Γ(φ) λ(λ+φ)(cid:0)2 +φ+λ(cid:1)Γ(cid:0)2 +φ(cid:1)−(cid:0)1 +φ+λ(cid:1)2Γ(cid:0)1 +φ(cid:1)2 σ2 = α α α α . (9) λ2(λ+φ)2Γ(φ)2 Proof. From (6) and considering r = 1 follows µ = µ. The second result follows from (7) 1 considering r = 2 and with some algebra follow the results. Different type of moments can be easily achieved for GWL distribution, one in particular, that has play a important role in information theory is given by (ψ(φ)−αlogλ+(λ+φ)−1) E[log(T)] = . (10) α 2.2 Survival Properties In this section, we present the survival, the hazard and mean residual life function for the GWL distribution. The survival function of T ∼ GWL(φ,λ,α) with the probability of an observation does not fail until a specified time t is Γ[φ,(λt)α](λ+φ)+(λt)αφe−(λt)α S(t|φ,λ,α) = (11) (λ+φ)Γ(φ) where Γ(x,y) = (cid:82)xwy−1e−xdw is called upper incomplete gamma. The hazard function 0 quantify the instantaneous risk of failure at a given time t and is given by f(t|φ,λ,α) αλαφtαφ−1(λ+(λt)α)e−(λt)α h(t|φ,λ,α) = = . (12) S(t|φ,λ,α) Γ[φ,(λt)α](λ+φ)+(λt)αφe−(λt)α 5 The behaviours of the hazard function (12) when t → 0 and t → ∞, respectively, are given by  ∞, if αφ < 1 0, if αφ < 1    αλ2  h(0) = , if αφ = 1 and h(∞) = λ, if αφ = 1 (λ+φ)Γ(φ)    ∞, if αφ > 1. 0, if αφ > 1 Theorem 2.5. The hazard rate function h(t) of the generalized weighted Lindley distribution is increasing, decreasing, bathtub, unimodal or decreasing-increasing-decreasing shaped. Proof. Is not straightforward to apply the theorem proposed by Glaser (1980) in the GLW distribution. Moreover, since the hazard rate function (12) is complex, we consider the following cases: 1. Let α = 1, then GWL distribution reduces to the WL distribution. In this case, Ghitany et al. (2008) proved that the hazard function is bathtub shaped (increasing) if 0 < φ < 1 (φ > 0), for all λ > 0. 2. Let φ = 1, then GWL distribution reduces to the PL distribution. In this case, considering β = λα, Ghitany et al. (2013) proved that the hazard function is • increasing if {0 < α ≥ 1,β > 0}; • decreasing if (cid:8)0 < α ≤ 1,β > 0(cid:9) or (cid:8)1 < α < 1,β ≥ (2α−1)2(4α(1−α))−1(cid:9); 2 2 • decreasing-increasing-decreasingif(cid:8)1 < α < 1,0 < β < (2α−1)2(4α(1−α))−1(cid:9). 2 3. Let α = 2 and λ = 1, from Glasers theorem (1980), we conclude straightforwardly that the hazard rate function is decreasing shaped (unimodal) if 0 < φ < 1 (φ > 1). These properties make the GWL distribution a flexible model for reliability data. Figure 2 gives examples from the shapes of the hazard function for different values of φ,λ and α. Themeanresiduallife(MRL)hasbeenusedwidelyinsurvivalanalysisandrepresentsthe expected additional lifetime given that a component has survived until time t, the following result presents the MRL function of the GWL distribution Proposition 2.6. The mean residual life function r(t|φ,λ,α) of the GWL distribution is given by (cid:0)φ+ 1 +λ(cid:1)Γ(cid:0)φ+ 1,(λt)α(cid:1)−λt(λ+φ)Γ(φ,(λt)α) r(t|φ,λ,α) = α α . (13) λ[(λ+φ)Γ(φ,(λt)α)+(λt)αφe−(λt)α] 6 10 6 f =0.2, l =1.5, a =2.1 f =0.2, l =2.5, a =0.2 f =0.5, l =1.5, a =1.5 f =0.5, l =2.5, a =0.6 8 f =0.7, l =1.5, a =1.0 5 f =1.0, l =2.5, a =0.8 f =0.7, l =1.5, a =1.5 f =1.5, l =1.5, a =0.5 f =3.5, l =20, a =0.7 f =2.0, l =1.5, a =1.2 4 6 h(t) h(t) 3 4 2 2 1 0 0 0 1 2 3 4 0 1 2 3 4 t t Figure 2: Hazard function shapes for GWL distribution and considering different values of φ,λ and α Proof. Note that 1 (cid:90) ∞ r(t|φ,λ,α) = yf(y|λ,φ)dy −t S(t) t 1 (cid:20) (cid:90) ∞ (cid:90) ∞ (cid:21) = p yf (y|λ,φ)dy +(1−p) yf (y|λ,φ)dy −t 1 2 S(t) t x (cid:0)φ+ 1 +λ(cid:1)Γ(cid:0)φ+ 1,(λt)α(cid:1)−λt(λ+φ)Γ(φ,(λt)α) = α α . λ[(λ+φ)Γ(φ,(λt)α)+(λt)αφe−(λt)α] The behaviors of the mean residual life function when t → 0 and t → ∞, respectively, are given by  ∞, if α < 1   1 1 r(0) = and r(∞) , if α = 1 . λ((λ+φ)Γ(φ)) λ   0, if α > 1 2.3 Entropy In information theory, entropy has played a central role as a measure of the uncertainty associated with a random variable. Proposed by Shannon (1948), Shannon’s entropy is one of the most important metrics in information theory. Shannon’s entropy for the GWL distribution can be obtained by solving the following equation (cid:90) ∞ (cid:18)αλαφtαφ−1(λ+(λt)α)e−(λt)α(cid:19) H (φ,λ,α) = − log f(t|φ,λ,α)dt. (14) S (λ+φ)Γ(φ) 0 7 Proposition 2.7. A random variable T with GWL distribution, has Shannon’s Entropy given by φ(1+φ+λ) H (φ,λ,α) = log(λ+φ)+logΓ(φ)−logα−logλ− S (λ+φ) (15) ψ(φ)(αφ−1) (αφ−1) η(φ,λ) − − − . α α(λ+φ) (λ+φ)Γ(φ) where (cid:90) ∞ (cid:90) 1 η(φ,λ) = (λ+y)log(λ+y)yφ−1e−ydy = (λ−logu)log(λ−logu)(−logu)φ−1du. 0 0 Proof. From the equation (14) we have H (φ,λ,α) = −logα−αφlogλ+log(λ+φ)+log(Γ(φ))+λαE[Tα] S (16) −(αφ−1)E[logT]−E[log(λ+(λT)α)] Note that (cid:90) ∞ αλαφtαφ−1(λ+(λt)α)e−(λt)α E[log(λ+(λT)α)] = log(λ+(λT)α dt, (λ+φ)Γ(φ) 0 using the change of variable y = (λt)α and after some algebra 1 (cid:90) ∞ E[log(λ+(λT)α)] = (λ+y)log(λ+y)yφ−1e−ydy (λ+φ)Γ(φ) 0 η(φ,λ) = . (λ+φ)Γ(φ) Through equations (6) and (10), we can easily find the solution of E[Tα] and E[logT] and the result follows. Other popular entropy measure is proposed by Renyi (1961). Some recent applications of the Reenyi entropy can be seen in Popescu & Aiordachioaie (2013). If T has the probability density function (1) then Renyi entropy is defined by 1 (cid:90) ∞ log fρ(x)dx. (17) 1−ρ 0 Proposition 2.8. A random variable T with GWL distribution, has the Renyi entropy given by (ρ−1)(logα+logλ)−ρ(log(λ+φ)+logΓ(φ))−log(δ(ρ,φ,λ,α)) H (ρ) = (18) R 1−ρ where δ(ρ,φ,λ,α) = (cid:82)∞yρφ−ρα+1−α(λ+y)ρe−ρydy. 0 8 Proof. The Renyi entropy is given by 1 (cid:18) αρλρ (cid:90) ∞ (cid:16) 1(cid:17) (cid:19) H (ρ) = log (λt)αρ φ−α (λ+(λt)α)ρe−ρ(λt)αdt R 1−ρ (λ+φ)ρΓ(φ)ρ 0 1 (cid:18) αρλρ (cid:90) ∞ ρφ−ρ+1−α (cid:19) = log y α (λ+y)ρe−ρydy 1−ρ (λ+φ)ρΓ(φ)ρ 0 1 (cid:18) αρλρ (cid:19) = log δ(ρ,φ,λ,α) 1−ρ (λ+φ)ρΓ(φ)ρ and with some algebra the proof is completed. 2.4 Lorenz curves The Lorenz curve (see Bonferroni, 1930) are well-known measures used in reliability, income inequality, life testing and renewal theory. The Lorenz curve for a non-negative T random variable is given through the consecutive plot of (cid:82)txf(x)dx 1 (cid:90) t L(F(t)) = 0 = xf(x)dx. (cid:82)∞xf(x)dx µ 0 0 Proposition 2.9. The Lorenz curve of the GWL distribution is γ(cid:0)φ+1+ 1,(λF−1(p))α(cid:1)+λγ(cid:0)φ+ 1,(λF−1(p))α(cid:1) L(p) = α α (cid:0) (cid:1) (cid:2) (cid:3) 1 +φ+λ Γ 1 +φ α α or (cid:0)1 +φ+λ(cid:1)γ(cid:0)φ+ 1,(λF−1(p))α(cid:1)−(λF−1(p))αφ−1e−(λF−1(p))α L(p) = α α (cid:0) (cid:1) (cid:2) (cid:3) 1 +φ+λ Γ 1 +φ α α where F−1(p) = t . p 3 Methods of estimation In this section we describe eight different estimation methods to obtain the estimates of the parameters φ,λ and α of the GWL distribution. 3.1 Maximum Likelihood Estimation Among the statistical inference methods, the maximum likelihood method is widely used due its better asymptotic properties. Under the maximum likelihood method, the estimators are obtained from maximizing the likelihood function (see for example, Casella & Berger, 2002). Let T ,...,T be a random sample such that T ∼ GWL(φ,λ,α). In this case, the likelihood 1 n function from (2) is given by, 9 (cid:40) (cid:41) (cid:40) (cid:41) αnλnαφ (cid:89)n (cid:89)n (cid:88)n L(φ,λ,α;t) = tαφ−1 (λ+(λt )α)exp −λα tα . (19) (λ+φ)Γ(φ)n i i i i=1 i=1 i=1 The log-likelihood function l(φ,λ,α;t) = logL(φ,λ,α;t) is given by, n (cid:88) l(φ,λ,α;t) = nlogα+nαφlogλ−nlog(λ+φ)−nlogΓ(φ)+(αφ−1) log(t ) i i=1 (20) n n (cid:88) (cid:88) + log(λ+(λt )α)−λα tα. i i i=1 i=1 From the expressions ∂ l(φ,λ,α;t) = 0, ∂ l(φ,λ,α;t) = 0, ∂ l(φ,λ,α;t) = 0, we get the ∂φ ∂λ ∂α likelihood equations, n (cid:88) n ˆ ˆ nαˆlog(λ)+αˆ log(t ) = +nψ(φ) (21) i ˆ ˆ λ+φ i=1 nαˆφˆ (cid:88)n 1+αˆλˆαˆ−1tαˆ (cid:88)n n + i = αˆλˆαˆ−1 tαˆ + (22) λˆ λˆ +(t )αˆ i λˆ +φˆ i=1 i i=1 n (cid:88)n (cid:88)n (λˆt )αˆlog(λˆt ) (cid:88)n +nφˆlog(λˆ)+φˆ log(t )+ i i = λˆαˆ t αˆlog(λˆt ), (23) αˆ i λˆ +(ˆˆλt )αˆ i i i=1 i=1 i i=1 where ψ(k) = ∂ logΓ(k) = Γ(cid:48)(k). The solutions of such non-linear system provide the ∂k Γ(k) maximum likelihood estimators. Numerical methods such as Newton-Rapshon are required to find the solution of the nonlinear system. Note that from (21) and (23) and after some algebra we have (cid:18) (cid:19) 1 n ˆ αˆ = +nψ(φ) (24) (cid:16)nlog(λˆ)+(cid:80)n log(t )(cid:17) λˆ +φˆ i=1 i (cid:16) n(cid:17) λˆαˆ(cid:80)n t αˆlog(λˆt )−(cid:80)n (λˆti)αˆlog(λˆti) − φˆ= i=1 i i i=1 λˆ+(ˆˆλti)αˆ αˆ , (25) (cid:16) (cid:17) nlog(λˆ)+(cid:80)n log(t ) i=1 i The obtained MLE’s (maximum likelihood estimators) of φ,λ and α are biased consid- ering small sample sizes. For large sample sizes the obtained estimators are not biased and they are asymptotically efficient. The MLE estimates are asymptotically normal distributed with a joint multivariate normal distribution given by, (φˆ,λˆ,αˆ) ∼ N [(φ,λ,α),I−1(φ,λ,α)] for n → ∞, (26) 3 10