A new flexible Weibull distribution

Transcription

1 Communicaions for Saisical Applicaions and Mehods 2016, Vol. 23, No. 5, hp://dx.doi.org/ /csam Prin ISSN / Online ISSN A new flexible Weibull disribuion Sangun Park a, Jihwan Park 1, a, Youngsik Choi a a Deparmen of Applied Saisics, Yonsei Universiy, Korea Absrac Many of sudies have suggesed he modificaions on Weibull disribuion o model he non-monoone hazards. In his paper, we combine wo cumulaive hazard funcions and propose a new modified Weibull disribuion funcion. The newly suggesed disribuion will be named as a new flexible Weibull disribuion. Corresponding hazard funcion of he proposed disribuion shows flexible (monoone or non-monoone) shapes. We sudy he characerisics of he proposed disribuion ha includes ageing behavior, momen, and order saisic. We also discuss an esimaion mehod for is parameers. The performance of he proposed disribuion is compared wih exising modified Weibull disribuions using various ypes of hazard funcions. We also use real daa example o illusrae he efficiency of he proposed disribuion. Keywords: Weibull disribuion, modified Weibull disribuion, bahub shape, hazard funcion, maximum likelihood esimae, reliabiliy 1. Inroducion Weibull disribuion has been widely used in various fields such as reliabiliy engineering due o is flexibiliy in fiing failure imes, where is survival funcion is given as F() = exp ( (θ) λ), > 0, wih parameers λ, θ > 0. The corresponding hazard funcion (failure rae funcion) can hen be wrien as h() = λθ λ λ 1. However, Weibull disribuion is inappropriae o model he non-monoone hazard rae such as bahubshaped hazard rae because Weibull disribuion can produce only monoonic hazard raes. Hence, many of modificaions of Weibull disribuion have been suggesed, which can fi he non-monoone hazard rae. I may be a naural approach o combine wo differen survival funcions (wih increasing and decreasing hazards) and generae a disribuion funcion as F() = α F 1 () + (1 α) F 2 (), where 0 < α < 1, which is well-known as a mixure of disribuions, or F() = F α 1 () F β 2 (), (1.1) 1 Corresponding auhor: Deparmen of Applied Saisics, Yonsei Universiy, 50, Yonsei-ro, Seodaemun-gu, Seoul 03722, Korea. jh.park@yonsei.ac.kr Published 30 Sepember 2016 / journal homepage: hp://csam.or.kr c 2016 The Korean Saisical Sociey, and Korean Inernaional Saisical Sociey. All righs reserved.

2 400 Sangun Park, Jihwan Park, Youngsik Choi wih parameers α, β > 0. The cumulaive disribuion funcion can be wrien in erms of he cumulaive hazard funcion H(x) as F() = 1 e H(), where he cumulaive hazard funcion saisfies he following properies as 1. H() is nonnegaive and increasing, 2. lim 0 H() = 0 and lim H() =. Hence, we can also combine wo cumulaive hazard funcions o generae a disribuion funcion as which is evenually equivalen o (1.1), or H() = αh 1 () + βh 2 (), (1.2) H() = H α 1 () Hβ 2 (), (1.3) wih parameers α, β > 0. We can inroduce furher an inercep parameer in (1.3) o add more flexibiliy. The modified Weibull disribuions suggesed by Xie and Lai (1995), Almalki and Yuan (2013), and Lemone e al. (2014) belong o he class (1.2), and he modified Weibull disribuions suggesed by Xie e al. (2002), Lai e al. (2003), Nadarajah and Koz (2005), Bebbingon e al. (2007) and Aryal and Elbaal (2015) belong o he class (1.3), and Park and Park (2016) discussed is generalizaion. Only he difference in pas works is he choice of cumulaive hazard funcions. If we combine differen ypes of hazard funcions, where one is increasing and he oher is decreasing, he hazard funcions of he above disribuion funcions are expeced o be flexible. However, he popular choices in pas works include exp() and log(), which are no appropriae as cumulaive hazard funcions. In his paper, we choose exp() 1 and log( + 1) as wo cumulaive hazard funcions and produce a new flexible Weibull disribuion (NFW) by following he aforemenioned approach (1.3) as log H NFW () = µ + α log (exp () 1) + β log (log ( + 1)). We provide he properies including characerizaion of hazard funcion and ageing behavior. The proposed disribuion is also compared wih some curren modified Weibull disribuions for various ypes of disribuion funcions and real daa example. 2. New flexible Weibull disribuion and is properies We propose a new modified Weibull disribuion as log H NFW (; µ, α, β) = µ + α log (exp () 1) + β log (log ( + 1)), which we call new flexible Weibull disribuion. Since boh exp () 1 and log ( + 1) saisfy he properies of he cumulaive hazard funcion, he resuling new flexible Weibull disribuion becomes heoreically rigorous and more flexible compared o oher modified Weibull disribuions. The hazard funcion corresponding o α log (exp () 1) shows

3 A new flexible Weibull disribuion 401 he bahub-shape (α < 1) or he increasing shape (α 1), whereas he hazard funcion corresponding o β log (log ( + 1)) may be decreasing (β 1) or upside-down bahub-shaped (β > 1). The cumulaive hazard funcion of he new flexible Weibull disribuion is given by H NFW (; µ, α, β) = exp (µ){exp () 1} α {log ( + 1)} β, and he corresponding hazard funcion has he following form h NFW (; µ, α, β) = exp (µ){exp () 1} α 1 {log ( + 1)} β 1 [ β ] α exp () log ( + 1) + {exp () 1}. + 1 The cumulaive disribuion funcion of he new flexible Weibull disribuion can be wrien as F NFW (; µ, α, β) = 1 exp [ exp (µ){exp () 1} α {log ( + 1)} β], and is probabiliy densiy funcion can be obained as f NFW (; µ, α, β) = exp (µ){exp () 1} α 1 {log ( + 1)} β 1 [ β ] α exp () log ( + 1) + {exp () 1} + 1 exp [ exp (µ){exp () 1} α {log ( + 1)} β]. Figures 1 and 2 shapes of hazard funcion h NFW () relaing o he change of α and β, wih µ fixed o 0. As shown in boh figures, he hazard funcions of he new flexible Weibull disribuion can cover increasing, decreasing, bahub-shaped, modified bahub-shaped and upside-down bahub-shaped failure raes. We also sudy he limiing behavior of he hazard funcion of he hazard funcion. Noe ha lim h NFW (; µ, α, β) = 0 when α is close o 0 and h NFW () goes o oherwise. I is sraighforward ha lim 0 h NFW (; µ, α, β) = 0 when a leas one of he parameers α or β is greaer or equal o 1. However when neiher α nor β is greaer or equal o 1, he limiing behavior varies as follows, depending on he parameers. Figure 2 illusraes he differen behaviors of hazard funcion as goes o 0, by an example of changing β when α is fixed. 1. If α + β is less han 1, 2. If α + β is equal o 1, 3. If α + β is greaer han 1, lim h NFW (; µ, α, β) =. 0 lim h NFW (; µ, α, β) = exp (µ). 0 lim h NFW (; µ, α, β) = 0. 0 Limiing behavior plays an imporan role on he shape of he hazard funcion. The resuling hazard funcion is only decreasing if a limiing behavior is fixed, for example, o be h(0) = and

4 402 Sangun Park, Jihwan Park, Youngsik Choi Hazard : Alpha=1e 06 Hazard : Alpha=0.2 Hazard : Alpha=1 hazard β = 0.5 β = 1 β = 1.5 β = β = 0.5 β = 0.8 β = 0.9 β = 1.5 β = 0.5 β = 1 β = 1.5 β = Hazard : Bea=0.1 Hazard : Bea=1 Hazard : Bea=3 hazard α = 0.1 α = 0.5 α = 1 α = α = 0.01 α = 0.1 α = 0.2 α = 0.5 α = 0.01 α = 0.1 α = 1 α = 1.5 Figure 1: Hazard funcion of new flexible Weibull disribuion wih µ = 0. Hazard : Alpha = 0.2 h() β=0.5 (α+β<1) β=0.8 (α+β=1) β=0.9 (α+β>1) Figure 2: Change of limiing behavior of hazard funcion when α = 0.2.

5 A new flexible Weibull disribuion 403 Densiy 1s Momen 2nd Momen β = 0.1 β = 1 β = 2 β = bea bea Figure 3: Change of densiy and momen when α = 1e 06. Densiy 1s Momen 2nd Momen β = 0.1 β = 1 β = 2 β = Figure 4: Change of densiy and momen when α = 0.2. bea bea Densiy 1s Momen 2nd Momen β = 0.1 β = 1 β = 2 β = Figure 5: Change of densiy and momen when α = 1. bea bea h( ) = 0. Mos previous sudies have fixed limiing behaviors, which resul in limied ypes of capable hazard funcions. The new flexible Weibull disribuion, however, is able o cover various ypes of hazard funcions by an appropriae choice of parameers. We can wrie he r h momen of he new flexible Weibull disribuion as E[T r ] = = 0 0 r f ()d = 0 r r 1 S ()d r r 1 exp [ exp (µ){exp () 1} α {log ( + 1)} β] d. We use Gauss-Kronrod quadraure for numerical inegraion since he above inegral canno be compued in closed-form. Figures 3 5 shows he changing densiy and momens as β varies when α is fixed.

6 404 Sangun Park, Jihwan Park, Youngsik Choi Densiy : Alpha=0.5, Bea= r = 1 r = 3 r = 5 r = 7 r = Figure 6: Probabiliy densiy funcion of r h order saisic. We also sudy he order saisic of he proposed disribuion. We denoe he probabiliy densiy funcion of r h order saisic T (r) as f r:n (). The probabiliy densiy funcion f r:n () can be wrien as 1 f r:n () = B(r, n r + 1) Fr 1 ()[1 F()] n r f () 1 r 1 ( ) r 1 = ( 1) k exp{ H()(n + k + 1 r)}h(). B(r, n r + 1) k k=0 Since he probabiliy densiy funcion f () can be wrien in erms of hazard funcion h() and cumulaive hazard funcion H() as we can derive f r:n () as f () = h(x)e H(), 1 r 1 ( ) r 1 ( 1) k f r:n () = B(r, n r + 1) k (n + k + 1 r) f (; µ, α, β), k=0 where µ = µ + log(n + k + 1 r). Figure 6 shows he probabiliy densiy funcion of r h order saisic when α and β are boh fixed o 0.5 and n fixed o Parameer esimaion In order o esimae he unknown parameers of he disribuion, we can consider he Weibull-ype probabiliy plo employed in Bebbingon e al. (2007), Lai e al. (2003) and Park and Park (2016) by leing he heoreical cumulaive hazard funcion be as close o he empirical cumulaive hazard funcion as log Ĥ() = µ + α log (exp () 1) + β log (log ( + 1)), where Ĥ() is he nonparameric esimae of he cumulaive hazard funcion a.

7 A new flexible Weibull disribuion 405 Hence, we can obain he (weighed) leas square esimaion by esimaing H() wih he Nelson- Aalen esimaor or Kaplan-Meier esimaor. For simpliciy, one may consider he ordinary leas square esimaion; however, we consider he maximum likelihood esimaion as follows. The likelihood funcion of he new flexible Weibull disribuion given 1,..., n has he following form as L NFW (µ, α, β) = = n f NFW ( i ; µ, α, β) n exp (µ){exp ( i ) 1} α 1 {log ( i + 1)} β 1 [ ] β α exp ( i ) log ( i + 1) + {exp ( i ) 1} i + 1 exp [ exp (µ){exp ( i ) 1} α {log ( i + 1)} β]. Then he score funcions for parameers µ, α and β can be obained as follows. log L NFW µ log L NFW α log L NFW β = n = = exp (µ){exp ( i ) 1} α {log ( i + 1)} β, log {exp ( i ) 1} + exp (µ) exp ( i ) log ( i + 1)( i + 1) α exp ( i ) log ( i + 1)( i + 1) + β{exp ( i ) 1} {exp ( i ) 1} α {log ( i + 1)} β log {exp ( i ) 1}, log {log ( i + 1)} + exp (µ) {exp ( i ) 1} α exp ( i ) log ( i + 1)( i + 1) + β{exp ( i ) 1} {exp ( i ) 1} α {log ( i + 1)} β log {log ( i + 1)}. In order o obain maximum likelihood esimaes of µ, α and β, we solve he above hree equaions using he quasi-newon mehod wih iniial values se o he ordinary leas square esimaes. 4. Simulaion sudies In order o evaluae he performance of he new flexible Weibull disribuion, we consider he following five differen ypes of disribuions. The hazard funcions are given in Figure 7. Example 1. Consan hazard funcion: Exponenial disribuion wih a rae parameer λ = 1/3. Example 2. Decreasing hazard funcion: Addiive Weibull disribuion suggesed by Lemone e al. (2014) wih parameers a = 0.1, b = 1, c = 0.2, d = 0.8. Example 3. Increasing hazard funcion: Addiive Weibull disribuion suggesed by Lemone e al. (2014) wih parameers a = 0.1, b = 1.5, c = 0.2, d = 1.

8 406 Sangun Park, Jihwan Park, Youngsik Choi Hazard : Example Hazard : Example Hazard : Example hazard Hazard : Example Hazard : Example hazard Figure 7: Hazard funcions for simulaed examples. Example 4. Bahub-shaped hazard funcion: Addiive Weibull disribuion suggesed by Lemone e al. (2014) wih parameers a = 0.1, b = 1.5, c = 0.2, d = 0.5. Example 5. Upside-down bahub-shaped hazard funcion: Lognormal disribuion wih mean µ = 0 and variance σ = 1. We compare he new flexible Weibull disribuion wih he sandard Weibull disribuion (woparameer), modified Weibull disribuion (hree- parameer), flexible Weibull disribuion (hreeparameer), very flexible Weibull disribuion (hree-parameer), and wo-parameer lifeime disribuion suggesed by Chen (2000) as 1. Weibull 2. Modified Weibull by Lai e al. (2003) 3. Flexible Weibull by Bebbingon e al. (2007) log H() = α + β log(). log H() = µ + α + β log(). log H() = α β.

9 A new flexible Weibull disribuion 407 Table 1: Maximum likelihood fi of Examples 1 5 Model Example 1 Example 2 log L AIC K-S log L AIC K-S Weibull Modified Weibull Flexible Weibull Very flexible Weibull Chen New flexible Weibull Model Example 3 Example 4 log L AIC K-S log L AIC K-S Weibull Modified Weibull Flexible Weibull Very flexible Weibull Chen New flexible Weibull Model Example 5 log L AIC K-S Weibull Modified Weibull Flexible Weibull Very flexible Weibull Chen New flexible Weibull AIC = Akaike Informaion Crierion; K-S = Kolmogorov-Smirnov. 4. Very flexible Weibull by Park and Park (2016) log H() = µ + α + β log log( + 1). 5. Two-parameer lifeime disribuion by Chen (2000) log H() = µ + log ( exp ( β) 1 ). As we can see, he above four modified Weibull disribuions conain a leas one inappropriae cumulaive hazard funcion. In evaluaing he performance, we generaed a random sample of size 30 from each disribuion, and calculaed he log-likelihood value, Akaike Informaion Crierion (AIC), and Kolmogorov- Smirnov saisic (K-S). We repeaed 100,000 Mone Carlo simulaions and calculaed he averages which are abulaed in Table 1. The numerical resuls in Table 1 indicae ha he new flexible Weibull disribuion shows he robus performances over he five differen disribuions. For he firs hree examples where he Weibull disribuion can fi well, he Weibull disribuion shows he lowes AIC values bu he new flexible Weibull shows he second lowes AIC values. For he fourh and fifh examples where he Weibull disribuion can no fi well, he new flexible Weibull disribuion shows he second lowes AIC value for he fourh case and lowes AIC value for he fifh case. 5. Applicaion We use a real daa example o illusrae he efficiency of new flexible Weibull disribuion. The failure ime daa sudied in Murhy e al. (2004) and Aryal and Elbaal (2015) were used for applicaion. The

10 408 Sangun Park, Jihwan Park, Youngsik Choi Produc-limi esimae Nelson-Aalen esimae Survival probabiliy Cumulaive hazard Time Time Figure 8: Survival funcion and cumulaive hazard funcion for real daa example. Table 2: Maximum likelihood fi of Example 6 Model log L AIC K-S Weibull Modified Weibull Flexible Weibull Very flexible Weibull Chen e al New flexible Weibull AIC = Akaike Informaion Crierion; K-S = Kolmogorov-Smirnov. daa are as follows. Example , 0.058, 0.061, 0.074, 0.078, 0.086, 0.102, 0.103, 0.114, 0.116, 0.148, 0.183, 0.192, 0.254, 0.262, 0.379, 0.381, 0.538, 0.570, 0.574, 0.590, 0.618, 0.645, 0.961, 1.228, 1.600, 2.006, 2.054, 2.804, 3.058, 3.076, 3.147, 3.625, 3.704, 3.931, 4.073, 4.393, 4.534, 4.893, 6.274, 6.816, 7.896, 7.904, 8.022, 9.337, , , , , Figure 8 illusraes he survival funcion and cumulaive hazard funcion using produc-limi esimae and Nelson-Aalen esimae, respecively. We compare he efficiency of new flexible Weibull disribuion wih aforemenioned disribuions using log-likelihood value, AIC and K-S values (Table 2). The resuls in Table 2 indicae ha flexible Weibull disribuion shows he larges log-likelihood value and lowes AIC value; however, flexible Weibull disribuion has he poores fi in erms of K-S value. On he oher hand, new flexible Weibull disribuion shows second bes fi in erms of log-likelihood value and AIC value along wih lowes K-S value. 6. Conclusions Some well-known modified Weibull disribuions can be represened as a muliplicaion of wo cumulaive hazard funcions, bu some cumulaive hazard funcions do no saisfy he properies as a cumulaive hazard funcion. We consider exp () 1 and log( + 1), and sugges a new modified Weibull

11 A new flexible Weibull disribuion 409 disribuion called a new flexible Weibull disribuion which is heoreically rigorous and shows more flexibiliy. The hazard funcion of he new flexible Weibull disribuion can cover he monoone shape as well as non-monoone shape ha include bahub-shaped, modified bahub-shaped or upside-down bahub-shaped. We presened he parameer esimaion mehods and compared heir performance wih some modified Weibull disribuions for various ypes of disribuions and real daa applicaion. Acknowledgemen This research was suppored by Basic Science Research Program hrough he Naional Research Foundaion of Korea (NRF) funded by he Minisry of Educaion (NRF-2015R1A2A1A ). References Almalki SJ and Yuan J (2013). A new modified Weibull disribuion, Reliabiliy Engineering and Sysem Safey, 111, Aryal G and Elbaal I (2015). On he exponeniaed generalized modified Weibull disribuion, Communicaions for Saisical Applicaions and Mehods, 22, Bebbingon M, Lai CD, and Ziikis R (2007). A flexible Weibull exension, Reliabiliy Engineering and Sysem Safey, 92, Chen Z (2000). A new wo-parameer lifeime disribuion wih bahub shape or increasing failure rae funcion, Saisics and Probabiliy Leers, 49, Lai CD, Xie M, and Murhy DNP (2003). A modified Weibull disribuion, IEEE Transacions on Reliabiliy, 52, Lemone AJ, Cordeiro GM, and Orega EM (2014). On he addiive Weibull disribuion, Communicaions in Saisics-Theory and Mehods, 43, Murhy DNP, Xie M, and Jiang R (2004). Weibull Models, John Wiley & Sons Inc., Hoboken, NJ. Nadarajah S and Koz S (2005). On some recen modificaion of Weibull disribuions, IEEE Transacions on Reliabiliy, 54, Park S and Park J (2016). A general class of flexible Weibull disribuions, Communicaions in Saisics - Theory and Mehods. Available from: hp://dx.doi.org/ / Xie M and Lai CD (1995). Reliabiliy analysis using an addiive Weibull model wih bahub-shaped failure rae funcion, Reliabiliy Engineering and Sysem Safey, 52, Xie M, Tang Y, and Goh TN (2002). A modified Weibull exension wih bahub-shaped failure rae funcion, Reliabiliy Engineering and Sysem Safey, 76, Received July 4, 2016; Revised Sepember 10, 2016; Acceped Sepember 11, 2016