Compound distributions motivated by linear failure rate

S a is ics & Ope a ions Resea ch T ansac ions SORT 40 (1) Janua y-June 2016, 177-200 S a is ics & Ope a ions Resea ch T ansac ions © Ins i u d’Es ad´ıs ica de Ca alunya so @idesca .ca ISSN: 1696-2281 eISSN: 2013-8830 www.idesca .ca /so / Compound dis ibu ions mo i a ed by linea ailu e a e Na jes Gi i a 1, Sadegh Rezaei1and Sa alees Nada ajah2 Abs ac Mo i a ed by h ee ailu e da a se s (li e ime o pa ien s, ailu e ime o ha d d i es and ailu e ime o a p oduc ), we in oduce h ee di e en h ee-pa ame e dis ibu ions, s udy basic ma hema ical p ope ies, add ess es ima ion by he me hod o maximum likelihood and in es iga e ini e sample pe o mance o he es ima o s. We show ha one o he new dis ibu ions p o ides a be e i o each da a se han eigh o he dis ibu ions each ha ing h ee pa ame e s and h ee dis ibu ions each ha ing wo pa ame e s. MSC: 62E15. Keywo ds: Linea ailu e a e dis ibu ion, maximum likelihood es ima ion, Poisson dis ibu ion. 1. In oduc ion Sys ems o componen s ha ing linea ailu e a es a e common in eal li e. Examples include conc e e unde mul iaxial s a es o s ess (Donida and Men as i, 1982), com- posi e lamina es wi h ans e se shea (Reddy and Reddy, 1992) and load-sha ing sys- ems (Su a and Naik-Nimbalka , 2014). The e a e also many eal da a se s ha exhibi app oxima ely linea ailu e a es a leas in he uppe ails. We p esen h ee examples. The i s da a se , due o Dispenzie i e al. (2012), consis s o he numbe o days om isi o clinic un il dea h o 100 pa ien s. The da a esul om a s udy o he ela ionship be ween se um ee ligh chain and mo ali y. The 100 pa ien s we e selec ed andomly om a o al o 7874 pa ien s, including pa ien s who had no died. The pa ien s who had died we e diagnosed wi h monoclonal gammapo hy. 1Ami kabi Uni e si y o Technology, Teh an, IRAN, email: s ez[email p o ec ed] 2Uni e si y o Manches e , Manches e M13 9PL, UK Recei ed: July 2015 Accep ed: Ap il 2016 178 Compound dis ibu ions mo i a ed by linea ailu e a e Table 1: Summa y s a is ics o he h ee da a se s. S a is ic Da a se 1 Da a se 2 Da a se 3 minimum 0.0054 0.0053 0.0035 i s qua ile 0.3368 0.3977 0.318 median 0.4774 0.7770 0.4211 hi d qua ile 0.7412 0.9304 0.5581 maximum 0.9514 1.4040 0.6878 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 Failu e ime / 5000 Failu e a e unc ion Figu e 1: Kaplan-Meie es ima e o he ailu e a e unc ion o he pa ien da a o Dispenzie i e al. (2012). The second da a se om h ps://www.backblaze.com/ha d-d i e- es -da a.h ml is one hun- d ed ailu e imes in days o ha d d i es. The da a we e selec ed andomly om a o al o 52422 ha d d i es, which included ha d d i es which had no ailed. The da a we e collec ed by a la ge backup s o age p o ide o e wo yea s. On each day, he Sel - Moni o ing, Analysis, and Repo ing Technology (SMART) s a is ics o ope a ional d i es we e eco ded. When a ha d d i e was no longe ope a ional, i was ma ked as a ailu e and emo ed. The hi d da a se due o Hong and Meeke (2013) is one hund ed ailu e da a in weeks o a p oduc called P oduc D2 ha is used in o ices o esidences. P oduc D2 is “simila o a high-end copying machine connec ed o he In e ne and ins alled wi h a Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 179 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 1 2 3 4 5 6 Failu e ime / 1000 Failu e a e unc ion Figu e 2: Kaplan-Meie es ima e o he ailu e a e unc ion o he ha d d i e ailu e da a. sma chip o eco d he numbe o pages ha ha e been p in ed, as a unc ion o ime” (Hong and Meeke , 2013, page 136). The one hund ed da a we e selec ed andomly om a o al o 1800 obse a ions. All h ee da a se s a e p esen ed in he appendix. Kaplan-Meie es ima es o he ailu e a e unc ion (FRF) o he h ee da a se s a e shown in Figu es 1, 2 and 3. We can see ha he FRFs a e app oxima ely linea a leas in he uppe ails. The his og am o he h ee da a se s a e shown in Figu es 8, 9 and 10. Some summa y s a is ics o he h ee da a se s a e shown in Table 1. We suppose ha he pa ien ’s body o he ha d d i e o he p oduc D2 is made o a numbe o componen s say Nwo king independen ly in se ies. The assump ion o he se ies s uc u e is mo e easonable han a pa allel s uc u e because i is unlikely ha a pa ien ’s body will ail i and only i all i s componen s ail o ha a ha d d i e will b eak i and only i all i s componen s b eak o ha a p oduc will ail i and only i all i s componen s ail. I is mo e likely ha a pa ien ’s body will ail i and only i any o i s componen s ails o ha a ha d d i e will b eak i and only i any o i s componen s b eaks o ha a p oduc will ail i and only i any o i s componen s ails. Howe e , in p ac ice he componen s may no wo k independen ly. The dis ibu ion o he ailu e 180 Compound dis ibu ions mo i a ed by linea ailu e a e 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 Failu e ime / 100 Failu e a e unc ion Figu e 3: Kaplan-Meie es ima e o he ailu e a e unc ion o he ailu e da a o Hong and Meeke (2013). ime may no ha e a closed o m i we assume ha he componen s a e dependen , see (2) below and i s discussion. We shall suppose independence o simplici y. The numbe Nmay a y om one pa ien o ano he o one ha d d i e o ano he o one p oduc o ano he . I may depend on he ype o ha d d i e, ype o pa ien , ype o p oduc , weigh , leng h, and so on. So, we may ake Nas a andom a iable. The ailu e ime can be w i en as X=min(Y1,Y2,...,YN), whe e Y1,Y2,...,YNdeno e he ailu e imes o he Ncomponen s. S anda d models o Na e he geome ic, ze o unca ed Poisson, loga i hmic, ze o unca ed nega i e binomial and ze o unca ed binomial dis ibu ions. Fo simplici y, we shall conside only he i s h ee since each o hem has one pa ame e . The las wo dis ibu ions ha e wo pa ame e s each. Tha is, we ake N o ha e one o he ollowing p obabili y mass unc ions (PMFs): P (N=n) = (1−λ)λn−1 o 0 <λ<1 and n=1,2,...; P (N=n) = λn (eλ−1)n! Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 181 o λ>0 and n=1,2,...; o P (N=n) = −1 ln(1−λ) λn n o 0 <λ<1 and n=1,2,.... Since he ailu e a e o he h ee da a se s is app oxima ely linea a leas in he uppe ail (see Figu es 1, 2 and 3), we shall suppose Y1,Y2,... oo ollow a dis ibu ion ha has a linea FRF. The dis ibu ion cha ac e ized by a linea ailu e a e is ac ually known as he linea ailu e a e (LFR) dis ibu ion due o Bain (1974). I s p obabili y densi y unc ion (PDF) and cumula i e dis ibu ion unc ion (CDF) a e speci ied by Y(y;γ,β) = (β+γy)exp−βy−γ 2y2 and FY(y;γ,β) = 1−exp−βy−γ 2y2, espec i ely, o y>0, β≥0, γ≥0 and β+γ>0. I is easy o see ha he FRF is hY(y;γ,β) = β+γy, a linea unc ion o y. Bo h pa ame e s, βand γ, a e e e ed o as scale pa ame e s. The dis ibu ion o X=min(Y1,Y2,...,YN)can now be de i ed gi en he assump ions ha Nis ei he geome ic, Poisson o loga i hmic and Y1,Y2,... a e independen LFR andom a iables independen o N. In he gene al case, he CDF and he PDF o Xcan be de i ed as FX(x) = P [min(Y1,Y2,...,YN)<x] = 1−P [min(Y1,Y2,...,YN)>x] =1− ∞ X n=1 P [min(Y1,Y2,...,Yn)>x|N=n]P (N=n) =1− ∞ X n=1 P [Y1>x,Y2>x,...,Yn>x]P (N=n) =1− ∞ X n=1 P n[Y>x]P (N=n) = 1− ∞ X n=1 [1−FY(x)]nP (N=n) and X(x) = Y(x) ∞ X n=1 n[1−FY(x)]n−1P (N=n), 182 Compound dis ibu ions mo i a ed by linea ailu e a e espec i ely. In he case Nis geome ic, we ob ain X(x;λ,γ,β) = (1−λ)(β+γx)exp−βx−γ 2x2 h1−λexp−βx−γ 2x2i2, which we shall e e o as he linea ailu e a e geome ic (LFRG) dis ibu ion and w i e X∼LFRG(λ,γ,β) o 0 <λ<1, β≥0, γ≥0 and β+γ>0. In he case Nis ze o unca ed Poisson, we ob ain X(x;λ,γ,β) = λ1−e−λ−1(β+γx)exp−λ−βx−γ 2x2exphλexp−βx−γ 2x2i, (1) which we shall e e o as he linea ailu e a e Poisson (LFRP) dis ibu ion and w i e X∼LFRP(λ,γ,β) o λ>0, β≥0, γ≥0 and β+γ>0. In he case Nis loga i hmic, we ob ain X(x;λ,γ,β) = − λ(β+γx)exp−βx−γ 2x2 ln(1−λ)h1−λexp−βx−γ 2x2i, which we shall e e o as he linea ailu e a e loga i hmic (LFRL) dis ibu ion and w i e X∼LFRL(λ,γ,β) o 0 <λ<1, β≥0, γ≥0 and β+γ>0. These dis ibu ions do no ha e linea ailu e a es. Bu hX(y;λ,γ,β)∼hY(y;γ,β)∼γyas y→∞. So, he assump ion o linea ailu e a e o Y1,Y2,... gua an ees ha linea ailu e a e holds o X oo a leas in he uppe ail. The limi ing cases o he LFRG, LFRP and LFRL dis ibu ions as λ↓0 is he LFR dis ibu ion. The LFRG and LFRL dis ibu ions limi o a degene a e dis ibu ion as λ↑1. I Y1,Y2,...a e dependen andom a iables hen he CDF o Xcan only be exp essed as FX(x) = 1− ∞ X n=1 P [Y1>x,Y2>x,...,Yn>x]P (N=n).(2) This canno be educed o a closed o m unless he join dependence o (Y1,Y2,...,Yn) akes a e y simple o m. In he es o his sec ion, Sec ion 2 and Sec ion 3, we shall ocus on he LFRP dis ibu ion. The de ails o he LRFG and LRFL dis ibu ions can be de i ed simila ly. One o he mos popula models o coun s is he ze o unca ed Poisson dis ibu ion. Some o i s ecen applica ions can be ound in an de Heijden e al. (2003), Elhai e al. (2008), Gineb a and Puig (2010) and Xu and Hu (2011). Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 183 Figu e 4: P obabili y densi y unc ion o he LFRP dis ibu ion o (a) γ=0.5and β=1, (b) γ=1and β=0.5, (c) β=0.05 and λ=3, (d) γ=2and λ=1. Possible shapes o (1) a e shown in Figu e 4. We see ha bo h mono onically de- c easing and unimodal shapes a e possible. The mode o (1) is he oo o γ β+γx−β−γx=λ(β+γx)exp−βx−γ 2x2. Fu he mo e, X(0) = λβ/1−e−λand X(x)∼λγ 1−e−λ−1 xexp−λ−βx−γ 2x2 as x→∞. The lowe ail o he PDF has a ixed poin while i s uppe ail decays expo- nen ially. The CDF and FRF o X∼LFRP(λ,γ,β)a e FX(x) = 1 eλ−1neλ−exphλexp−βx−γ 2x2io and hX(x) = (β+γx)λexp−βx−γ 2x2 1−exph−λexp−βx−γ 2x2i,(3) 184 Compound dis ibu ions mo i a ed by linea ailu e a e Figu e 5: Failu e a e unc ion o he LFRP dis ibu ion o (a) γ=0.5and β=1, (b) γ=1and β=0.5, (c) β=0.05 and λ=3, (d) λ=3and γ=0.5. espec i ely, o x>0, λ>0, β≥0, γ≥0 and β+γ>0. Figu e 5 shows possible shapes o (3) o di e en pa ame e alues. We see ha he LFRP dis ibu ion can exhibi inc easing, dec easing and upside down ba h ub shapes o he ailu e a e. The LFR dis ibu ion can exhibi only inc easing o cons an ailu e a es. Reliabili y and su i al analysis o en encoun e upside down ba h ub ailu e a es. Examples can be ound in edundancy alloca ions in sys ems (Singh and Mis a, 1994) and mo ali y modelling (Sil a e al., 2010). The mode o he an i-mode o (3) is he oo o γ β+γx−β−γx=−λ(β+γx)exp−βx−γ 2x2nexphλexp−βx−γ 2x2i−1o−1. Fu he mo e, hX(0) = λβ/1−e−λand hX(x)∼γxas x→∞. The lowe ail o he FRF has a ixed poin . As al eady no ed, he uppe ail o he FRF o he LFRP dis ibu ion beha es in he same manne as ha o he LFR dis ibu ion. Ye he o me does exhibi upside down ba h ub ailu e a es while he la e does no . The q h quan ile o X∼LFRP(λ,γ,β)say xqde ined by FX(xq) = qis xq=−β γ+sβ2 γ2−2 γlnnln[eλ−q(eλ−1)] 1 λo. Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 185 In pa icula , he median o Xis Median(X) = −β γ+ u u β2 γ2−2 γln(lneλ−1 2(eλ−1)1 λ). Quan iles a e use ul o es ima ion and simula ion. Se e al o he dis ibu ions ha e been in oduced in he li e a u e by aking X= min(Y1,Y2,...,YN), whe e Nis a geome ic, ze o unca ed Poisson o a loga i hmic andom a iable: By aking N o be a geome ic andom a iable and Y1,Y2,... o be independen and iden ical Weibull andom a iables, Ba e o-Souza e al. (2011) in o- duced he h ee-pa ame e Weibull geome ic (WG) dis ibu ion gi en by he PDF (x) = (1−λ)βγ−βxβ−1exph−(x/γ)βi n1−λexph−(x/γ)βio2 o x>0, 0 <λ<1, β>0 and γ>0; By aking N o be a ze o unca ed Poisson andom a iable and Y1,Y2,... o be independen and iden ical Weibull andom a iables, Lu and Shi (2012) in oduced he h ee-pa ame e Weibull Poisson (WP) dis ibu ion gi en by he PDF (x) = λβγ−βxβ−1expn−(x/γ)β+λexph−(x/γ)βio exp(λ)−1 o x>0, λ>0, β>0 and γ>0; By aking N o be a loga i hmic andom a iable and Y1,Y2,... o be independen and iden ical Weibull andom a iables, Ciuma a and P eda (2009) in oduced he h ee-pa ame e Weibull loga i hmic (WL) dis ibu ion gi en by he PDF (x) = − (1−λ)βγ−βxβ−1exph−(x/γ)βi lnλn1−(1−λ)exph−(x/γ)βio o x>0, 0 <λ<1, β>0 and γ>0; By aking N o be a geome ic andom a iable and Y1,Y2,... o be independen and iden ical gene alized exponen ial andom a iables, Mahmoudi and Ja a i (2012) in oduced he h ee-pa ame e gene alized exponen ial geome ic (GEG) dis ibu ion gi en by he PDF (x) = (1−λ)βγ exp(−γx)[1−exp(−γx)]β−1 nλ[1−exp(−γx)]β−1o2 192 Compound dis ibu ions mo i a ed by linea ailu e a e 1. he biases o each pa ame e a e gene ally posi i e; 2. he biases o each pa ame e dec ease o ze o as n→∞; 3. he biases appea smalles o he pa ame e , λ; 4. he mean squa ed e o s o each pa ame e dec ease o ze o as n→∞; 5. he mean squa ed e o s appea smalles o he pa ame e , λ; 6. he mean squa ed e o s appea la ges o he pa ame e , β; 7. he biases and mean squa ed e o s o each pa ame e appea easonably small o all n≥60. We ha e p esen ed esul s o only one choice o (λ,β,γ), namely ha (λ,β,γ) = (1,1,1). Bu he esul s we e simila o a wide ange o o he choices. In pa icula , he biases and mean squa ed e o s o each pa ame e appea ed easonably small o all n≥60. The h ee eal da a se s in Sec ion 4 each has a sample size g ea e han o equal o six y. So, we can expec he es ima es in Sec ion 4 o be easonable. 4. Real da a applica ions He e, we e u n o he h ee da a se s o illus a e he applicabili y o he LFRP dis ibu- ion. The ollowing dis ibu ions we e i ed o each da a: he LFR, LFRG, LFRP, LFRL, WG, WP, WL, GEG, GEP and GEL dis ibu ions. We also i ed he Weibull and gamma dis ibu ions gi en by he PDFs (x) = βxβ−1 γβexp"−x γβ# and (x) = xβ−1 γβΓ(β)exp−x γ, espec i ely, o x>0, α>0 and β>0. Each dis ibu ion was i ed by he me hod o maximum likelihood. The pa ame e es ima es, s anda d e o s, −lnL, AIC alues and BIC alues a e gi en in Tables 2, 3 and 4. The s anda d e o s we e compu ed by in e ed he obse ed in o ma ion ma ices. We see ha he LFRP dis ibu ion yields he smalles −lnL, he smalles AIC and he smalles BIC o each da a se . I p o ides a signi ican ly be e i han he LFR dis ibu ion o each da a se , as judged by he likelihood a io es . The s anda d e o s o he LFRP dis ibu ion appea easonable, as hey a e smalle han he pa ame e es ima es. Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 193 Table 2: Pa ame e es ima es, s anda d e o s, log-likelihood, AIC and BIC o he wel e dis ibu ions i ed o he pa ien da a o Dispenzie i e al. (2012). Dis ibu ion b λSE b βSE bγSE −lnLAIC BIC LFR 0.348 0.176 5.071 0.739 7.747 19.494 24.704 LFRG 0.001 0.000 0.348 0.176 5.069 0.738 7.751 21.503 29.318 LFRP 1.894 0.851 1.132 0.661 5.591 1.212 4.960 15.921 23.736 LFRL 0.001 0.000 0.342 0.173 5.063 0.734 7.750 21.500 29.315 WG 0.999 0.000 1.839 0.156 0.561 0.032 11.838 29.676 37.491 WP 2.230 0.910 1.434 0.204 0.394 0.065 8.818 23.637 31.452 WL 0.001 0.000 1.848 0.157 0.563 0.032 11.841 29.682 37.498 GEG 0.999 0.000 2.012 0.285 2.925 0.307 21.182 48.365 56.180 GEP 3.850 1.032 1.095 0.326 3.947 0.377 11.689 29.377 37.193 GEL 0.001 0.000 2.011 0.285 2.923 0.307 21.184 48.368 56.183 Weibull 1.839 0.156 0.561 0.032 11.837 27.674 32.885 Gamma 2.068 0.272 0.245 0.036 19.387 42.774 47.985 Table 3: Pa ame e es ima es, s anda d e o s, log-likelihood, AIC and BIC o he wel e dis ibu ions i ed o he ha d d i e ailu e da a. Dis ibu ion b λSE b βSE bγSE −lnLAIC BIC LFR 0.296 0.138 2.530 0.393 42.043 88.087 93.297 LFRG 0.000 0.000 0.292 0.135 2.465 0.384 42.072 90.143 97.959 LFRP 1.753 0.691 0.776 0.375 2.841 0.572 38.849 83.698 91.514 LFRL 0.000 0.000 0.296 0.138 2.530 0.393 42.044 90.088 97.904 WG 0.999 0.000 1.751 0.152 0.774 0.046 46.993 99.986 107.801 WP 1.831 0.738 1.484 0.189 0.584 0.079 43.848 93.695 101.511 WL 0.001 0.000 1.772 0.154 0.774 0.045 46.984 99.967 107.783 GEG 0.999 0.000 1.842 0.257 2.017 0.216 55.885 117.769 125.585 GEP 3.569 1.080 1.016 0.327 2.664 0.256 47.407 100.814 108.629 GEL 0.000 0.000 1.876 0.263 2.035 0.217 55.887 117.774 125.589 Weibull 1.772 0.154 0.775 0.045 46.982 97.964 103.174 Gamma 1.902 0.249 0.369 0.055 54.304 112.608 117.818 The pa ame e es ima es and he log-likelihood alues o he LFRG and LFRL dis- ibu ions a e e y close o all h ee da a se s. This sugges s ha he likelihood su aces o he LFRG and LFRL dis ibu ions a ain hei maximum poin s along he bo de co - esponding o λ=0. We no ed ea lie LFRG and LFRL dis ibu ions educe o he LFR dis ibu ion as λ↓0. So, he i s o LFRG and LFRL dis ibu ions do no imp o e on he i o he LFR dis ibu ion o he h ee da a se s. 194 Compound dis ibu ions mo i a ed by linea ailu e a e Table 4: Pa ame e es ima es, s anda d e o s, log-likelihood, AIC and BIC o he wel e dis ibu ions i ed o he ailu e da a o Hong and Meeke (2013). Dis ibu ion b λSE b βSE bγSE −lnLAIC BIC LFR 0.028 0.061 9.349 0.968 −32.148 −60.296 −55.086 LFRG 0.000 0.000 0.047 0.081 9.523 1.000 −32.069 −58.138 −50.323 LFRP 5.023 1.719 1.361 1.458 15.188 3.681 −48.555 −91.111 −83.295 LFRL 0.000 0.000 0.019 0.052 9.389 0.967 −32.133 −58.267 −50.451 WG 0.999 0.000 3.149 0.256 0.482 0.016 −44.743 −83.485 −75.670 WP 4.940 1.837 1.703 0.313 0.287 0.054 −46.938 −87.876 −80.061 WL 0.000 0.000 3.146 0.255 0.483 0.016 −44.745 −83.489 −75.674 GEG 0.003 0.000 4.552 0.476 0.753 0.133 −29.354 −52.708 −44.893 GEP 8.160 1.966 1.859 0.584 7.352 0.588 −42.532 −79.064 −71.249 GEL 2.082×10−50.000 5.546 0.918 5.304 0.442 −25.126 −44.253 −36.437 Weibull 3.146 0.255 0.483 0.016 −44.745 −85.489 −80.279 Gamma 5.371 0.735 0.081 0.012 −31.814 −59.629 −54.418 Failu e ime / 5000 Fi ed PDFs 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 LFR LFRG LFRP LFRL WG WP WL GEG GEP GEL Weibull Gamma Figu e 8: Densi y plo s o he wel e dis ibu ions i ed o he pa ien da a o Dispenzie i e al. (2012). Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 195 Failu e ime / 1000 Fi ed PDFs 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 LFR LFRG LFRP LFRL WG WP WL GEG GEP GEL Weibull Gamma Figu e 9: Densi y plo s o he wel e dis ibu ions i ed o he ha d d i e ailu e da a. The densi y plo s o he i o he dis ibu ions o he h ee da a se s a e shown in Figu es 8 o 10. The i ed PDFs o he LFRP dis ibu ion cap u es he obse ed his og ams be e han o he s. Hence, we can say ha he LFRP dis ibu ion p o ides he bes i o a leas h ee eal da a se s. The pa ame e es ima es o he bes i ing LFRP dis ibu ion o he h ee da a se s can be in e p e ed as ollows: • he pa ien ’s body can be modelled as a se ies sys em ha ing an a e age o b λ/h1−e−b λi=2.2 componen s wi h he 95 pe cen con idence in e al (0.37,4.09), whe e he ailu e a e o each componen is linea wi h an in e cep o 1.132 and a slope o 5.591. Tha is, he ailu e a e o each componen a ime ze o is 1.132 and he ailu e a e inc eases by 5.591 o e e y uni inc ease in ime; • he ha d d i e can be modelled as a se ies sys em ha ing an a e age o b λ/h1−e−b λi=2.1 componen s wi h he 95 pe cen con idence in e al (1.26,2.98), whe e he ailu e a e o each componen is linea wi h an in e cep o 0.776 and 196 Compound dis ibu ions mo i a ed by linea ailu e a e a slope o 2.841. Tha is, he ailu e a e o each componen a ime ze o is 0.776 and he ailu e a e inc eases by 2.841 o e e y uni inc ease in ime; • he p oduc D2 can be modelled as a se ies sys em ha ing an a e age o b λ/h1−e−b λi=5.1 componen s wi h he 95 pe cen con idence in e al (−1.97,12.08), whe e he ailu e a e o each componen is linea wi h an in e cep o 1.361 and a slope o 15.188. Tha is, he ailu e a e o each componen a ime ze o is 1.361 and he ailu e a e inc eases by 15.188 o e e y uni inc ease in ime. No e ha λ/1−e−λis he expec ed alue o a ze o unca ed Poisson andom a iable. The s a ed con idence in e als we e ob ained by he del a me hod. Failu e ime / 100 Fi ed PDFs 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 LFR LFRG LFRP LFRL WG WP WL GEG GEP GEL Weibull Gamma Figu e 10: Densi y plo s o he wel e dis ibu ions i ed o he ailu e da a o Hong and Meeke (2013). 5. Conclusions We ha e p oposed h ee dis ibu ions mo i a ed by h ee ailu e da a se s: he linea ailu e a e geome ic, linea ailu e a e Poisson and linea ailu e a e loga i hmic dis- ibu ions. Each o hese dis ibu ions has h ee pa ame e s. Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 197 We ha e s udied ma hema ical p ope ies and es ima ion issues o he linea ailu e a e Poisson dis ibu ion. We ha e shown in pa icula ha i s ailu e a e unc ion can be dec easing, inc easing and upside down ba h ub shaped, mo e a ied han he ailu e a e unc ion o he linea ailu e a e dis ibu ion. Among he wel e dis ibu ions i ed o he h ee da a se s, he linea ailu e a e Poisson dis ibu ion ga e he bes i . The adequacy o i s was assessed in e ms AIC alues, BIC alues and densi y plo s. A u u e wo k is o es ima e he pa ame e s o he linea ailu e a e Poisson dis- ibu ion by he me hod o pe cen iles, he me hod o p obabili y weigh ed momen s, he me hod o leas squa es, he me hod o weigh ed leas squa es, he me hod o gen- e alized momen s, and o he me hods. Ano he u u e wo k is o p opose bi a ia e and mul i a ia e gene aliza ions o he linea ailu e a e Poisson dis ibu ion. Appendix: Th ee da a se s The i s da a is 0.1102 0.2390 0.4598 0.7146 0.2608 0.0838 0.8746 0.1578 0.3358 0.0198 0.7192 0.7916 0.4486 0.4080 0.6048 0.3686 0.4686 0.5418 0.3760 0.8684 0.1572 0.4860 0.0118 0.4732 0.5450 0.8982 0.5674 0.2602 0.4330 0.3608 0.3648 0.5124 0.1360 0.7548 0.8960 0.4816 0.0818 0.3268 0.9514 0.8650 0.3372 0.5438 0.5392 0.5750 0.3672 0.6694 0.3068 0.2536 0.3756 0.3962 0.4690 0.3416 0.6430 0.9104 0.4426 0.7280 0.7370 0.7666 0.6420 0.2000 0.3588 0.6632 0.8752 0.8934 0.6526 0.1370 0.5222 0.7746 0.9230 0.6422 0.3298 0.7286 0.0054 0.3754 0.2448 0.9466 0.3256 0.3726 0.0516 0.4496 0.7850 0.8670 0.0758 0.5174 0.7742 0.5464 0.6152 0.7594 0.8310 0.4036 0.8954 0.7970 0.3638 0.0142 0.7998 0.1658 0.4572 0.7540 0.9220 0.3688 Fo compu a ional s abili y wi h i ing dis ibu ions, we ha e di ided each obse a ion by 5000. The second da a is 1.293458333 0.251375000 1.265458333 1.404000000 1.280416667 1.201500000 1.193458333 0.340333333 1.101166667 1.059250000 1.360541667 1.245125000 1.098041667 1.049875000 1.167875000 1.271500000 1.182000000 0.925916667 0.963333333 1.119666667 0.867791667 0.845375000 0.803416667 0.323500000 1.165083333 1.065958333 1.103583333 1.035583333 1.173958333 0.886916667 0.789958333 0.671791667 0.782666667 0.534125000 0.691000000 0.813750000 0.773416667 0.629291667 0.520291667 0.635000000 198 Compound dis ibu ions mo i a ed by linea ailu e a e 0.695041667 0.712625000 0.428000000 0.423208333 0.615541667 0.254416667 0.160791667 0.125083333 0.416791667 0.215416667 0.214958333 0.185375000 0.228458333 0.206958333 0.228833333 0.190083333 0.205000000 0.007458333 0.192750000 0.227666667 0.155916667 0.179791667 0.018625000 0.169458333 0.066416667 0.005333333 0.115416667 0.080375000 0.495833333 0.854916667 0.498750000 0.902875000 0.967958333 0.786916667 0.920583333 0.943875000 0.807666667 0.761708333 0.733583333 1.043833333 0.893583333 0.746500000 0.736583333 0.880500000 0.889708333 0.780666667 0.668041667 0.861291667 0.711916667 0.718500000 0.863041667 0.908000000 0.833791667 0.671416667 0.826083333 0.823000000 0.784375000 0.667833333 0.669750000 0.835750000 Fo compu a ional s abili y wi h i ing dis ibu ions, we ha e di ided each obse a ion by 1000. The hi d da a is 0.222673061 0.257639905 0.328155859 0.515672484 0.583401130 0.642256077 0.621521735 0.587506929 0.594755485 0.316753044 0.550884304 0.312962380 0.516646945 0.546445582 0.600493703 0.297813235 0.332441913 0.333245894 0.364800151 0.429097225 0.627439232 0.313363071 0.579554283 0.391397547 0.125167305 0.541816854 0.665764686 0.398880874 0.402492151 0.423982077 0.428143776 0.341767913 0.514537781 0.686683383 0.333088363 0.249962985 0.226748439 0.286643595 0.645490088 0.584664074 0.397377064 0.609634794 0.353187577 0.536304985 0.406031202 0.586163204 0.648786836 0.516497130 0.318475607 0.494774308 0.436782434 0.245923132 0.618409876 0.255245760 0.464312202 0.454133994 0.387982016 0.218311879 0.526363495 0.418258490 0.272839591 0.151997829 0.492728139 0.290973052 0.471553883 0.363069573 0.668371780 0.501805967 0.600306622 0.477109810 0.515188714 0.283784543 0.600625759 0.299420135 0.368553098 0.653382502 0.687845701 0.379423961 0.279504337 0.407995757 0.685695223 0.259685231 0.514854899 0.501119729 0.003522425 0.672089253 0.630145059 0.310811342 0.384073475 0.388312955 0.268080935 0.437408445 0.634243302 0.239656858 0.391844012 0.347107733 0.499160234 0.325770026 0.290634387 0.371908794 Fo compu a ional s abili y wi h i ing dis ibu ions, we ha e di ided each obse a ion by 100. 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