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
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ISSN: 1696-2281
eISSN: 2013-8830
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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.
Na jes Gi i a , Sadegh Rezaei and Sa alees Nada ajah 199
Acknowledgmen s
The au ho s would like o hank he Edi o and he h ee e e ees o ca e ul eading and
commen s which g ea ly imp o ed he pape .
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