A new extension of power Lindley distribution for analyzing bimodal data

Transcription

1 Chlean Journal of Statstcs Vol. 8, No. 1, Aprl 2017, Research Paper A new extenson of power Lndley dstrbuton for analyzng bmodal data Morad Alzadeh 1, S.M.T.K MrMostafaee 2,, and Indranl Ghosh 3 1 Department of Statstcs, Persan Gulf Unversty, , Bushehr, Iran, 2 Department of Statstcs, Unversty of Mazandaran, , Mazandaran, Iran, 3 Department of Mathematcs and Statstcs, Unversty of North Carolna Wlmngton, USA (Receved: October 6, 2016 Accepted n fnal form: Aprl 6, 2017) Abstract In ths artcle, we ntroduce a new three-parameter odd log-logstc power Lndley dstrbuton and dscuss some of ts propertes. These nclude the shapes of the densty and hazard rate functons, mxture representaton, the moments, the quantle functon, and order statstcs. Maxmum lkelhood estmaton of the parameters and ther estmated asymptotc standard errors are derved. Three algorthms are proposed for generatng random data from the proposed dstrbuton. A smulaton study s carred out to examne the bas and mean square error of the maxmum lkelhood estmators of the parameters. An applcaton of the model to two real data sets s presented fnally and compared wth the ft attaned by some other well-known two and three-parameter dstrbutons for llustratve purposes. It s observed that the proposed model has some advantages n analyzng lfetme data as compared to other popular models n the sense that t exhbts varyng shapes and shows more flexblty than many currently avalable dstrbutons. 1. Introducton The Lndley dstrbuton specfed by the probablty densty functon (pdf) f(x) = θ2 (1 + x) exp( θx), x > 0, θ > 0. θ + 1 was ntroduced by Lndley (1958) n the context of Bayesan statstcs. Ghtany et al. (2008) nvestgated propertes of the Lndley dstrbuton wth applcaton and outlned that the Lndley dstrbuton s a better model than one based on the exponental dstrbuton, n other words many mathematcal propertes of the Lndley dstrbuton are more flexble than those of the exponental dstrbuton. Ghtany et al. (2008) showed that the Lndley dstrbuton can be wrtten as a mxture of an exponental dstrbuton and a gamma dstrbuton wth shape parameter 2. Gleaton and Lynch (2004, 2006) ntroduced a new famly of dstrbutons whch s called the Generalzed log-logstc famly of dstrbutons. The cumulatve dstrbuton functon Correspondng author. Emal: m.mrmostafaee@umz.ac.r ISSN: (prnt)/issn: (onlne) c Chlean Statstcal Socety Socedad Chlena de Estadístca

2 68 Alzadeh, MrMostafaee and Ghosh (cdf) of ths famly s gven by F (x; α, ξ) = G(x; ξ) α G(x; ξ) α + G(x; ξ) α, (1) where α > 0 s the shape parameter, G(x; ξ) s the cdf of the baselne dstrbuton, G(x; ξ) = 1 G(x; ξ) s the survval functon and ξ s the set of parameters of the baselne dstrbuton G( ). In addton, the pdf of the famly s f(x; α, ξ) = α g(x; ξ) G(x; ξ)α 1 G(x; ξ) α 1 G(x; ξ) α + G(x; ξ) α] 2. Ths famly was called later the odd log-logstc famly of dstrbutons. If the baselne dstrbuton possesses a closed form cdf, the generated new dstrbuton wll also possess a closed form cdf. One can easly show that log log F (x;α,ξ) F (x;α,ξ) G(x;ξ) G(x;ξ) ] ] = α. Therefore α s the quotent of the log-odds rato for the generated and baselne dstrbutons. Recently, Ghtany et al. (2013) proposed a generalzaton of the Lndley dstrbuton, the power Lndley dstrbuton, wth cdf G(x) = 1 (1 + λ 1 + λ xβ )e λxβ, x > 0, β, λ > 0. Now, by lettng G(x; ξ) n (1) to be the cdf of the power Lndley dstrbuton, where ξ = (β, λ) s the set of parameters, we can obtan a new extenson of the power Lndley dstrbuton, called the odd log-logstc power Lndley (henceforth, OLL-PL) dstrbuton. The cdf, pdf and hazard rate functon of ths dstrbuton are gven by F (x; α, β, λ) = ] α 1 (1 + λ 1+λ xβ )e λxβ ] α, (2) 1 (1 + λ 1+λ xβ )e λxβ + (1 + λ 1+λ xβ ) α e λα xβ f(x; α, β, λ) = x > 0, ] α 1 αβ λ 2 x β 1 (1 + x β ) e 1 α λxβ (1 + λ 1+λ xβ )e λxβ (1 + λ 1+λ xβ ) α 1 { ] α } 2, (1 + λ) 1 (1 + λ 1+λ xβ )e λxβ + (1 + λ 1+λ xβ ) α e λα xβ (3) and h(x; α, β, λ) = (1 + λ) (1 + λ 1+λ xβ ) αβ λ 2 x β 1 (1 + x β ) 1 (1 + λ 1+λ xβ )e λxβ ] α 1 { 1 (1 + λ 1+λ xβ )e λxβ ] α + (1 + λ 1+λ xβ ) α e λα xβ }, (4)

3 Chlean Journal of Statstcs 69 respectvely, where α, β, λ > 0. We wrte X OLL-PL(α, β, λ) f the pdf of X can be wrtten as (3). The new dstrbuton s very flexble n the sense that t can be skewed and symmetrc dependng upon the specfc choces of the parameters. Furthermore, the assocated cdf s n closed form. Consequently, ths dstrbuton can be appled to modellng censored data too. Ths s a major motvaton to carry out ths work. Furthermore, n relablty engneerng and lfetme analyss, we often assume that the falure tmes of the components wthn each system follow the exponental lfetmes; see, for example, Adamds and Loukas (1998) among others and the references theren. Ths assumpton may seem unreasonable because, for the exponental dstrbuton, the hazard rate s a constant, whereas many real-lfe systems do not have constant hazard rates, and the components of a system are often more rgd than the system tself, such as bones n a human body, balls of a steel ppe, etc. Accordngly, t becomes reasonable to consder the components of a system to follow a dstrbuton wth a non-constant hazard functon that has flexble hazard functon shapes. From Fgure 1, we see that ths model can be bmodal (α = 0.5, β = 3, λ = 2) whch s rare n classc lfetme dstrbutons. We can also observe that when α and λ are fxed, then for small values of β, the densty s often decreasng. However, for larger values of β, the densty may not be decreasng any longer and t can be unmodal when α = 1.5 and 3 and unmodal, bmodal or decreasng-ncreasng-decreasng when α = 0.1 and 0.5. In addton, t seems that λ affects the heght of the densty plots. From Fgure 2, we can see that the hazard functon s often decreasng for small values of β. For larger values of β, the hazard functon can be ncreasng (for example for the case (α = 0.5, β = 3, λ = 2)) and bath-tub shaped (for example for the case (α = 0.1, β = 3, λ = 2)). In addton, the hazard functon can be upsde-down bath-tub shaped too (for example for the case (α = 3, β = 0.5, λ = 2)). In addton, as we wll see n subsecton 2.4, the addtonal parameter α plays the role of controllng the tal weghts of the new dstrbuton. An nterpretaton of the OLL-PL dstrbuton can be gven as follows: Let X be a lfetme random varable havng power Lndley dstrbuton. The odds rato that an ndvdual (or component) followng the lfetme X wll de (fal) at tme x s y = G(x; β, λ)/ḡ(x; β, λ). Here, one can consder ths odds of death as a random varable, say Y. Now, f we model the randomness of the odds of death usng the log-logstc dstrbuton wth scale parameter 1 and shape parameter α, (F Y (y) = y α /1 + y α ] for y > 0), then we can wrte P r(y y) = F Y ( G(x; β, λ)/ Ḡ(x; β, λ) ), whch s gven by (2), see Cooray (2006) for more detals regardng ths nterpretaton. Specal Cases: For α = 1, we obtan the power Lndley dstrbuton. For β = 1, we obtan the odd log-logstc Lndley dstrbuton (Ozel et al., 2016). For α = β = 1, we obtan the Lndley dstrbuton. We hope that ths new dstrbuton can be appled to descrbng lfetme data more properly than the exstng dstrbutons. The major motvaton of ntroducng the OLL-PL dstrbuton can be summarzed as follows. () The OLL-PL dstrbuton contans several lfetme dstrbutons as specal cases, such as the power Lndley (PL) dstrbuton due to Ghtany et al. (2013) for α = 1. () It s shown n Secton 2 that the OLL-PL dstrbuton can be vewed as a mxture of exponentated power Lndley (EPL) dstrbutons ntroduced by Warahena-Lyanage and Parara (2014) and Ashour and Eltehwy (2015). () The OLL- PL dstrbuton s a flexble model whch can be wdely used for modelng lfetme data. (v) The OLL-PL dstrbuton exhbts monotone as well as non-monotone hazard rates

4 70 Alzadeh, MrMostafaee and Ghosh densty α=0.1 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β=3 densty α=0.5 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β= x x densty α=1.5 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β=3 densty α=3 λ=0.5 ; β=0.9 λ=0.5 ; β=1.5 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β= x x Fgure 1. Pdfs of the OLL-PL model for selected α, β and λ. but does not exhbt a constant hazard rate, whch makes ths dstrbuton to be superor to other lfetme dstrbutons, whch exhbt only monotoncally ncreasng/decreasng, or constant hazard rates. (v) The OLL-PL dstrbuton outperforms several of the well-known lfetme dstrbutons wth respect to some real data examples. The rest of the artcle s organzed as follows: In Secton 2, we dscuss some structural propertes of the OLL-PL dstrbuton. Secton 3 deals wth the classcal method of estmaton (usng maxmum lkelhood) of the model parameters of the OLL-PL dstrbuton, and a small smulaton study s conducted to verfy the effcacy of the sad estmaton procedure. In Secton 4, two real data sets are consdered as an example to llustrate the applcablty of OLL-PL dstrbuton. In Secton 5, we provde some concludng remarks. 2. Structural propertes In ths secton, we dscuss some structural propertes of the OLL-PL dstrbuton.

5 Chlean Journal of Statstcs 71 hazard α=0.1 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β=3 hazard α=0.5 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β= x x hazard α=1.5 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=1 ; β=3 λ=1.5 ; β=0.5 λ=2 ; β=1 λ=2 ; β=3 hazard α=3 λ=0.5 ; β=0.5 λ=0.5 ; β=1 λ=0.5 ; β=3 λ=2 ; β=0.5 λ=2 ; β=1 λ=2 ; β= x x Fgure 2. Hazard rate functons of the OLL-PL model for selected α, β and λ. 2.1 Mxture representatons for the pdf and cdf The EPL dstrbuton, ntroduced by Warahena-Lyanage and Parara (2014) and Ashour and Eltehwy (2015), has the pdf f EP L (x; α, β, λ) = αλ2 λ + 1 (1 + xβ )x β 1 e λx β 1 (1 + λ ] α λ xβ )e λx β, x > 0, α, β, λ > 0.(5) We wrte X EPL(α, β, λ) f the pdf of X can be expressed as (5). In addton, the cdf of the EPL model s F EP L (x; α, β, λ) = 1 (1 + λ ] α 1 + λ xβ )e λxβ, x > 0. Now, we show that the OLL-PL dstrbuton can be vewed as a mxture of EPL dstrbutons. Usng the generalzed bnomal expanson, the numerator of (2) can be wrtten

6 72 Alzadeh, MrMostafaee and Ghosh as 1 (1 + λ ] α 1 + λ xβ )e λxβ = k=0 a k 1 (1 + λ ] k 1 + λ xβ )e λxβ, where a k = ( )( ) α =k ( 1)+k and the denomnator of (2) can be wrtten as k 1 (1 + λ ] α 1 + λ xβ )e λxβ + (1 + λ 1 + λ xβ ) α e λα xβ = k=0 b k 1 (1 + λ ] k 1 + λ xβ )e λxβ, ( ) α where b k = a k +( 1) k. Therefore the cdf of the OLL-PL dstrbuton can be expressed k as F (x) = where c 0 = a0 b 0 k=0 a k k=0 b k ] k 1 (1 + λ 1+λ xβ )e λxβ ] k = 1 (1 + λ 1+λ xβ )e λxβ = 0 and for k 1 we have c k = b 1 0 a k b 1 0 k=0 c k 1 (1 + λ ] k 1 + λ xβ )e λxβ, ] k b r c k r. r=1 Or equvalently, we can wrte the cdf of OLL-PL as F (x) = c k F EP L (x; k, β, λ) = k=1 c k+1 F EP L (x; k + 1, β, λ), (6) where F EP L (x; k + 1, β, λ) denotes the cdf of the EPL dstrbuton wth parameters k + 1, β and λ. We note that k=0 c k+1 = 1. By dfferentatng equaton (6), the pdf of the OLL-PL dstrbuton can be expanded as f(x) = k=0 c k+1 f EP L (x; k + 1, β, λ), k=0 where f EP L (x; k + 1, β, λ) denotes the pdf of the EPL dstrbuton wth parameters k + 1, β and λ. We defne and compute A(a 1, a 2, a 3 ; β, λ) = where a 1 > 1, a 2 > 0 and a 3 R Moments x a1 (1 + x β ) e a2 xβ 1 (1 + λ ] a3 1 + λ xβ )e λxβ dx,

7 Chlean Journal of Statstcs 73 Usng the generalzed bnomal expanson, one can obtan A(a 1, a 2, a 3 ; β, λ) = l 1 l 2+1 l 1=0 l 2=0 l 3=0 ( )( a3 l1 Next, the m-th moment of the OLL-PL dstrbuton wll be E X m ] = β λ2 1 + λ l 1 l 2 )( ) l2 + 1 ( 1) l 1 λ l2 Γ( a1+1 β + l 3 ) l 3 β (λ + 1) l1 (λ l 1 + a 2 ) a 1 +1 (k + 1) c k+1 A(m + β 1, λ, k; β, λ). k=0 β +l3. For nteger values of m, let µ m = E(X m ) and µ = µ 1 = E(X), then one can also fnd the m-th central moment of the OLL-PL dstrbuton through the followng well-known equaton µ m = E(X µ) m = m r=0 ( ) m µ r r( µ) m r. (7) Usng (7), the varance, skewness and kurtoss measures can be obtaned from the followng relatons: Var(X) = E(X 2 ) E(X)] 2, Skewness(X) = E(X3 ) 3E(X)E(X 2 ) + 2E(X)] 3, V ar(x)] 3 2 Kurtoss(X) = E(X4 ) 4E(X)E(X 3 ) + 6E(X 2 )E(X)] 2 3E(X)] 4 V ar(x)] 2. Fgure 3 shows the behavor of the skewness and kurtoss of the OLL-PL dstrbuton wth respect to α and λ when β = Skewness 0.4 Kurtoss Lambda Alpha Lambda Alpha Next, we defne and compute B(a 1, a 2, a 3, y; β, λ) = Fgure 3. Skewness (left) and kurtoss (rght) plots when β = 2. y 0 x a1 (1 + x β ) e a2 xβ 1 (1 + λ ] a3 1 + λ xβ )e λxβ dx,

8 74 Alzadeh, MrMostafaee and Ghosh where a 1 > 1, a 2 > 0 and a 3 R. Usng generalzed bnomal expanson, one can obtan B(a 1, a 2, a 3, y; β, λ) = l 1 l 2+1 l 1=0 l 2=0 l 3=0 ( )( a3 l1 l 1 l 2 )( ) l2 + 1 ( 1) l 1 λ l2 γ( a1+1 β + l 3, (λ l 1 + a 2 ) y β ) l 3 β (λ + 1) l1 (λ l 1 + a 2 ) a 1 +1 where γ(λ, z) = z 0 tλ 1 e t dt denotes the ncomplete gamma functon. Now, the r-th ncomplete moment of the OLL-PL dstrbuton s found to be m r (y) = y 0 x r f(x)dx = λ2 β 1 + λ (k + 1) c k+1 B(r + β 1, λ, k, y; β, λ). k=0 β +l3, 2.3 Stochastc orderng Let X and Y be two random varables. X s sad to be stochastcally less than or equal to Y, denoted by X Y f P (X > x) P (Y > x) for all x n the support set of X. st Theorem 2.1 Suppose X OLL-PL(α 1, β 1, λ) and Y OLL-PL(α 2, β 2, λ). If α 1 < α 2, β 1 < β 2, then X Y. st Proof Straghtforward and hence omtted. 2.4 Quantle Functon Quantle functons are n wdespread use n statstcs and often fnd representatons n terms of lookup tables for key percentles. Let X OLL-PL(α, β, λ). The quantle functon of X, say Q(p), s defned by F (Q(p)) = p and s the root of the followng equaton (1 + λ + λ Q(p) β ) e λ Q(p)β = (1 + λ) (1 p) 1 α, (8) + (1 p) 1 α for 0 < p < 1. Insertng Z(p) = 1 λ λ Q(p) β, nto (8), we have Hence, the soluton of Z(p) s p 1 α Z(p) e Z(p) = (1 + λ) (1 p) 1 α e 1 λ. p 1 α + (1 p) 1 α ] (1 + λ) (1 p) 1 α e 1 λ Z(p) = W 1, p 1 α + (1 p) 1 α where W 1.] s the negatve branch of the Lambert W functon (Corless et al. 1996). Thus, we obtan the quantle functon as { ]} Q(p) = 1 1 λ 1 λ W (1 + λ) (1 p) 1 α e 1 λ 1 β 1. (9) p 1 α + (1 p) 1 α

9 Chlean Journal of Statstcs 75 The partcular case of (9) for α = β = 1 has been derved recently by Jodrá (2010). Here, we also propose three dfferent algorthms for generatng random data from the OLL-PL dstrbuton. The frst algorthm s based on generatng random data from the Lndley dstrbuton usng the exponental-gamma mxture (see Ghtany et al., 2008). Algorthm 1 (Mxture Form of the Lndley Dstrbuton) (1) Generate U Unform(0, 1), = 1,, n; (2) Generate V Exponental(λ), = 1,, n; (3) Generate W Gamma(2, λ), = 1,, n; (4) If U 1 α α U 1 +(1 U ) α 1 λ 1+λ set X = V 1 β, otherwse, set X = W 1 β, = 1,, n. The second algorthm s based on generatng random data usng the Webull-generalzed gamma (GG) mxture (see Ghtany et al., 2013). Algorthm 2 (Mxture Form of the power Lndley Dstrbuton) (1) Generate U Unform(0, 1), = 1,, n; (2) Generate Y W ebull(β, λ), = 1,, n; (3) Generate Z GG(2, β, λ), = 1,, n; (4) If U 1 α α U 1 +(1 U ) α 1 λ 1+λ set X = Y, otherwse, set X = Z, = 1,, n. The thrd algorthm s based on generatng random data from the nverse cdf n (2) of the OLL-PL dstrbuton, see (9). Algorthm 3 (Inverse cdf) (1) Generate U Unform(0, 1), = 1,, n; (2) Set { X = 1 1 λ 1 λ W (1 + λ) (1 U ) 1 α e 1 λ 1 + (1 U ) 1 α U 1 α ]} 1 β, = 1,, n. Algorthm 1 s the smplest data generaton algorthm and therefore s preferable. Data generaton from the classcal dstrbutons lke unform, exponental and gamma dstrbutons s ncluded normally n many statstcal software. We have used Algorthm 1 n our smulaton study, see Subsecton 3.1. Algorthm 3 nvolves the Lambert W functon and therefore s somehow complcated, see Corless et al. (1996) for more detals regardng the Lambert W functon. 2.5 Asymptotc propertes Let X OLL-PL(α, β, λ), then the asymptotcs of equatons (2), (3) and (4) as x 0 are gven by F (x) (λ x β ) α as x 0, f(x) α β λ α x αβ 1 as x 0, h(x) α β λ α x αβ 1 as x 0. The asymptotcs of equatons (2), (3) and (4) as x are gven by

10 76 Alzadeh, MrMostafaee and Ghosh λ 1 F (x) ( 1 + λ )α x α β e αλ xβ as x, λ f(x) α λ β( 1 + λ )α x β(α+1) 1 e αλ xβ as x, h(x) αβλ x β 1 as x. These equatons show the effect of parameters on the tals of the OLL-PL dstrbuton. 2.6 Extreme values Let X 1,..., X n be a random sample from (3) and X = (X X n )/n denote the sample mean, then by the usual central lmt theorem, the dstrbuton of n( X E(X))/ V ar(x) approaches the standard normal dstrbuton as n. Sometmes one would be nterested n the asymptotcs of the extreme values M n = max(x 1,..., X n ) and m n = mn(x 1,..., X n ). For (2), t can be seen that and F (t x) lm t 0 F (t) = xαβ, 1 F (t x) lm t 1 F (t) = e αλ xβ. Thus, t follows from Theorem n Leadbetter et al. (1983) that there must be normalzng constants a n > 0, b n, c n > 0 and d n such that and P r a n (M n b n ) x] e e λα xβ, P r a n (m n b n ) x] 1 e xαβ, as n. Usng Corollary of Leadbetter et al. (1983), we can obtan the form of normalzng constants a n, b n, c n and d n. 2.7 Order statstcs Order statstcs make ther appearance n many areas of statstcal theory and practce. Suppose that X 1,..., X n are a random sample from an OLL-PL dstrbuton. Let X :n denote the -th order statstc. The pdf of X :n can be expressed as (see Arnold et al., 1992) f :n (x) = K f(x) F 1 (x) {1 F (x)} n = K n ( ) n ( 1) j f(x) F (x) j+ 1, j j=0 where K = n! ( 1)! (n )!.

11 Chlean Journal of Statstcs 77 We use the result of Gradshteyn and Ryzhk (2000) for a power seres rased to a postve nteger n (for n 1) ( ) n a u = =0 d n, u, =0 where the coeffcents d n, (for = 1, 2,...) are determned from the recurrence equaton (wth d n,0 = a n 0 ) d n, = ( a 0 ) 1 m(n + 1) ] a m d n, m. m=1 We can demonstrate that the densty functon of the -th order statstc of an OLL-PL dstrbuton can be expressed as f :n (x) = n r,k=0 j=0 m r,k,j f EP L(x; r + k + + j, β, λ), (10) where f EP L (x; α, β, λ) denotes the densty functon of EPL dstrbuton wth parameters α, β and λ and the coeffcents m r,k,j m r,k,j (, n) s are gven by m r,k,j = n! (r + 1) c r+1 ( 1) j a j+ 1,k ( 1)! (n j)! j! (r + k + + j), n whch the coeffcents c r s are defned n Subsecton 2.1 and the quanttes a j+ 1,k can be determned such that a j+ 1,0 = cj+ 1 1 and for k 1 a j+ 1,k = (k c 1) 1 k q (j + ) k] c q+1 a j+ 1,k q. q=1 Equaton (10) s the man result of ths secton. It reveals that the pdf of the OLL-PL order statstc s a lnear combnaton of EPL dstrbutons. So, several mathematcal quanttes of these order statstcs lke ordnary and ncomplete moments, factoral moments, moment generatng functon, mean devatons and others can be derved usng ths result. 3. Estmaton In ths secton, we dscuss maxmum lkelhood estmaton (MLE) and nference for the OLL-PL dstrbuton. Let x 1,..., x n be a random sample from OLL-PL model where α, β and λ are the unknown parameters. The log-lkelhood for the parameters of the OLL-PL dstrbuton gven the data set x 1,..., x n reduces to ( ) α β λ 2 l n = n log + (β 1) 1 + λ + (α 1) log(x ) + =1 log t (1 t )] 2 =1 log(1 + x β ) λ =1 =1 log t α + (1 t ) α ], (11) =1 x β

12 78 Alzadeh, MrMostafaee and Ghosh ( ) where t = λ 1+λ xβ e λxβ. The assocated nonlnear log-lkelhood system ln h where l n α = n α + log t (1 t )] 2 =1 l n β = n β + log(x ) + =1 +(α 1) l n λ = 2n λ 2α =1 =1 x β log(x ) 1 + x β λ =1 t + (1 α) =1 n 1 + λ x β + (α 1) =1 =1 t α 1 (1 t ) α 1 t α + (1 t ) α, = 0, (for h = α, β, λ), follows as t α log(t ) + (1 t ) α log(1 t ) t α + (1 t ) α, x β log(x ) =1 1 t 2α =1 =1 t + (1 α) = λ2 1 + λ xβ (1 + xβ xβ )e λ log(x ), t α 1 =1 = xβ e λx β (1 + λ) 2 + x β ( λ ) λ xβ e λx β. (1 t ) α 1 t α + (1 t ) α, 1 t For estmatng the model parameters, numercal teratve technques should be used to solve these equatons. We can nvestgate the global maxma of the log-lkelhood by settng dfferent startng values for the parameters. The nformaton matrx wll be requred for nterval estmaton. Let θ = (α, β, λ) T, then the asymptotc dstrbuton of n( θ θ) s N 3 (0, K(θ) 1 ), under standard regularty condtons (see Lehmann and Casella, 1998, pp ), where K(θ) s the expected nformaton matrx. The asymptotc behavor remans vald f K(θ) s superseded by the observed nformaton matrx multpled by 1 n, say 1 n I(θ), approxmated by θ,.e. 1 n I( θ). We have where I αα = 2 l n α 2 = n α 2 2 I αβ = I βα = 2 l n α β = 2α =1 t α 1 =1 =1 I(θ) = I αα I αβ I αλ I βα I ββ I βλ, I λα I λβ I λλ t α (1 t ) α log(t ) log( t 1 t ) + t α (1 t ) α log(1 t ) log( 1 t t ) t α + (1 t ) α ] 2, t =1 1 t 2 (1 t ) α 1 log( t 1 t ) t α + (1 t ) α ] 2, =1 t α 1 (1 t ) α 1 t α + (1 t ) α

13 Chlean Journal of Statstcs 79 I αλ = I λα = 2 l n α λ = 2α + 2α =1 =1 =1 I ββ = 2 l n β 2 = n β (1 α) =1 2α(α 1) t =1 1 t 2 =1 t α 1 log(t ) (1 t ) α 1 log(1 t ) t α + (1 t ) α t α 1 (1 t ) α 1 t α + (1 t ) α t α log(t ) + (1 t ) α log(1 t )] t α 1 (1 t ) α 1] t α + (1 t ) α ] 2, =1 x β ( ) 2 log(x ) λ 1 + x β t (ββ) (1 t ) + =1 ] 2 (1 t ) 2 2α ] 2 t α 2 x β log(x )] 2 + (α 1) =1 =1 t (ββ) + (1 t ) α 2 t α + (1 t ) α + 2α 2 I βλ = I λβ = 2 l n β λ = x β log(x ) + (α 1) + (1 α) =1 2α(α 1) I λλ = 2 l n λ 2 + (1 α) t (βλ) =1 = 2n λ 2 + =1 2α(α 1) n whch =1 (1 t ) + (1 t ) 2 2α t α 2 =1 =1 t (βλ) + (1 t ) α 2 t α + (1 t ) α + 2α 2 n (1 + λ) 2 + (α 1) t (λλ) (1 t ) + =1 ] 2 =1 (1 t ) 2 2α ] 2 t α 2 t (ββ) = λ2 1 + λ xβ e λxβ t (λλ) t =1 t α 1 =1 t (βλ) t 2 t (λλ) + (1 t ) α 2 t α + (1 t ) α + 2α 2 =1 (1 t ) α 1 t α + (1 t ) α { t t α 1 =1 t 2 t α 1 t (ββ) t t 2 (1 t ) α 1 } 2 t α + (1 t ) α, (1 t ) α 1 t α + (1 t ) α { t α 1 (1 t ) α 1 } 2 t α + (1 t ) α, t α 1 =1 ] 2 (1 t ) α 1 t α + (1 t ) α { (1 t ) α 1 } 2 t α + (1 t ) α, t α 1 log(x )] (2 λ)x β λ x2β t (βλ) = λ xβ (1 + λ) 2 e λxβ log(x )(1 + x β )λxβ (1 + λ) λ 2], t (λλ) = 2xβ e λx β (1 + λ) 3 + 2x2β e λxβ (1 + λ) 2 x 2β (1 + λ 1 + λ xβ )e λxβ ],. ] 2

14 80 Alzadeh, MrMostafaee and Ghosh 3.1 A smulaton study In order to assess the performance of the maxmum lkelhood method, a small smulaton study s performed usng the statstcal software R. However, one can also perform n SAS by PROC NLMIXED procedure. The number of Monte Carlo replcatons was 30,000. For maxmzng the log-lkelhood functon, one can use the MaxBFGS subroutne wth analytcal dervatves. The evaluaton of the estmates was performed based on the followng quanttes for each sample sze: the emprcal mean squared errors (MSEs) are calculated usng the R package from the Monte Carlo replcatons. The maxmum lkelhood (ML) estmates are determned for each smulated data, say, ( ˆα, ˆβ, ˆλ ) for = 1, 2,, 30, 000 and the bases and MSEs are computed by and bas h (n) = (ĥ h), =1 MSE h (n) = (ĥ h) 2, for h = α, β, λ. We consder the sample szes at n = 100, 300 and 500 and consder dfferent values for the parameters. The emprcal results are gven n Table 1, and ndcate that the estmates are qute stable and, more mportantly, are close to the true values for these sample szes. Furthermore, as the sample sze ncreases, the MSEs decreases as expected. =1 Table 1. The bases and MSEs of the estmates under the maxmum lkelhood method. Sample Sze Actual Value Bas MSE n α β λ ˆα ˆβ ˆλ ˆα ˆβ ˆλ

15 Chlean Journal of Statstcs Applcaton In ths secton, we llustrate the power of OLL-PL dstrbuton usng two real data sets. The frst data set, denoted as D1, refers to remsson tmes n months of a random sample of 128 bladder cancer patents reported n Lee and Wang (2003). The second data set, referred as D2, ncludes the breakng tmes (n hours) for 76 Kevlar 49/epoxy strands that were subject to a stress of ks and a temperature of 110 C. The data were taken from Go mez et al. (2014) and they were prevously reported by Glaser (1983). For the purpose of comparson, we ftted the followng models as well as the OLL-PL dstrbuton to the above two data sets: () the Lndley dstrbuton, () the power Lndley (PL) dstrbuton, () the OLL-Lndley dstrbuton, (v) the EPL dstrbuton, (v) the beta Lndley (BL) dstrbuton whose pdf s gven by (see Merovc and Sharma, 2014) fbl (x) = 1 FL (x)]α 1 1 FL (x)]β 1 fl (x), B(α, β) x > 0, α, β, λ > 0, where fl ( ) and FL ( ) are the pdf and cdf of the Lndley dstrbuton wth parameter λ, respectvely and B(a, b) s the complete beta functon. and (v) the generalzed Lndley (GL) dstrbuton, Nadarajah et al. (2011), a sub-model of beta Lndley dstrbuton whch wll be obtaned by settng β = Hstogram for D Hstogram for D Densty OLL PL EPL GL BL Densty OLL PL OLL L PL Lndley OLL PL EPL GL BL Densty 0.3 OLL PL OLL L PL Lndley 0.1 Densty 60 Hstogram for D2 0.4 Hstogram for D Fgure 4. Hstograms and the ftted pdfs for D1 and D2 (left plots for sub-models and rght plots for the others). We appled formal goodness-of-ft tests to determnng whch model fts the data best.

16 82 Alzadeh, MrMostafaee and Ghosh To ths end, we consdered the Cramér-von Mses (W ) and Anderson-Darlng (A ) test statstcs, see Chen and Balakrshnan (1995) for detals regardng these statstcs. Generally speakng, the smaller the values of A and W, the better the ft to the data. We have also consdered the Kolmogorov-Smrnov (K-S) statstc and ts correspondng p-value and the mnmum value of the mnus log-lkelhood functon (-Log(L)) for the sake of comparson. The ML estmates of the parameters (standard errors n parentheses) as well as the goodness-of-ft test statstcs for the two real data sets are presented n Table 2. Table 2. Parameter ML estmates (standard errors n the parentheses) and the goodness-of-ft test statstcs. Results for D1 Model α β λ -log(l) W A K-S p-value OLL-PL(α, β, λ) ( ) ( ) ( ) Lndley(λ) ( ) PL(β, λ) ( ) ( ) OLL-L(α, λ) ( ) ( ) EPL(α, β, λ) ( ) ( ) ( ) GL(α, λ) ( ) ( ) BL(α, β, λ) ( ) ( ) ( ) Results for D2 Model α β λ -log(l) W A K-S p-value OLL-PL(α, β, λ) ( ) ( ) ( ) Lndley(λ) ( ) PL(β, λ) ( ) ( ) OLL-L(α, λ) ( ) ( ) EPL(α, β, λ) (0.666) (0.1937) (0.3596) GL(α, λ) ( ) ( ) BL(α, β, λ) (0.355) (0.835) (1.056) As we can see from Table 2, the smallest values of -Log(L), W, A and K-S statstcs and the largest p-values belong to the OLL-PL dstrbuton for both data sets. Therefore the OLL-PL dstrbuton outperforms the other compettve consdered dstrbuton n the sense of these crtera. For the frst data set, the EPL dstrbuton provdes the second best ft and for the second data set, the OLL-L dstrbuton provdes the second best ft. These conclusons can also be drawn vsually from Fgures 4-6. Fgures 5 and 6 reveal that the plotted ponts for the OLL-PL dstrbuton best capture the dagonal lne n the probablty plots. Hence, the OLL-PL dstrbuton could, n prncple, be an approprate model for fttng these data sets.

17 Chlean Journal of Statstcs 83 The OLL PL dstrbuton The Lndley dstrbuton The OLL L dstrbuton The PL dstrbuton The EPL dstrbuton The GL dstrbuton The BL dstrbuton Fgure 5. Probablty plots for D1. 5. Concludng Remarks In ths paper, we ntroduce a new three-parameter dstrbuton, so-called the odd loglogstc power Lndley (OLL-PL) dstrbuton, that extends the generalzed log logstc famly of dstrbutons by mxng wth the Lndley dstrbuton proposed by Lndley (1958). Several of ts structural propertes are dscussed n detal. These nclude shape of the probablty densty functon, hazard rate functon and ts shape, quantle functon, lmtng dstrbutons of order statstcs, and the general r-th moments. Moreover, the maxmum lkelhood estmaton procedure s dscussed for estmatng the parameters. As can be seen from the shapes of the probablty densty and hazard functons, the new dstrbuton provdes more flexblty than other dstrbutons that are commonly used for fttng lfetme data. A smulaton study s also provded. Fnally, two real-data examples are analyzed to show the applcablty of the new dstrbuton n practcal stuatons. All the computatons are performed usng Maple 17 and R (R Core Team, 2016 and Marnho et al., 2013). It s worthwhle to menton that other attractve propertes of the new dstrbuton are not consdered n ths paper, such as the relablty parameter, cumulants, cumulatve resdual entropy, dstrbuton of the sum, product, dfference & rato of OLL-PL random varables, and bvarate & multvarate generalzatons of the OLL-PL dstrbuton, etc.

18 84 Alzadeh, MrMostafaee and Ghosh The OLL PL dstrbuton The Lndley dstrbuton The OLL L dstrbuton The PL dstrbuton The EPL dstrbuton The GL dstrbuton The BL dstrbuton Fgure 6. Probablty plots for D2. In an ongong project, we plan to address some of these propertes lsted above. Bayesan estmates of the parameters are currently under nvestgaton and wll be reported elsewhere. Addtonally, there are some open questons whch we wll try to address n future research: Is t possble to have characterzatons of such famles of dstrbutons va order statstcs? For the bvarate extenson of the OLL-PL dstrbuton and subsequently the multvarate extenson, can we consder copula-based constructon approach? If yes, then, what can we say regardng the dependence structure of the assocated copula model(s)? Is t reasonable to consder the dscrete analog of the contnuous OLL-PL dstrbuton? Acknowledgements We would lke to thank the revewer for hs/her mportant suggestons and crtcsms whch greatly mproved the paper.

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20 86 Alzadeh, MrMostafaee and Ghosh R Core Team R: A language and envronment for statstcal computng. R Foundaton for Statstcal Computng, Venna, Austra. URL: Warahena-Lyanage, G., and Parara, M A generalzed power Lndley dstrbuton wth applcatons. Asan Journal of Mathematcs and Applcatons, ID ama0169.