Ma ganpoo , Shahdie; Ranjba , Vahid; Alizadeh, Mo ad; Abdollahnezhad, Kamel
A icle
Gene alised Odd F eche Family o Dis ibu ions:
P ope ies and Applica ions
S a is ics in T ansi ion New Se ies
P o ided in Coope a ion wi h:
Polish S a is ical Associa ion
Sugges ed Ci a ion: Ma ganpoo , Shahdie; Ranjba , Vahid; Alizadeh, Mo ad; Abdollahnezhad, Kamel
(2020) : Gene alised Odd F eche Family o Dis ibu ions: P ope ies and Applica ions, S a is ics in
T ansi ion New Se ies, ISSN 2450-0291, Exeley, New Yo k, Vol. 21, Iss. 3, pp. 109-128,
h ps://doi.o g/10.21307/s a ans-2020-047
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STATISTICS IN TRANSITION new se ies, Sep embe 2020
Vol. 21, No. 3, pp. 109–128, DOI 10.21307/s a ans-2020-047
Recei ed – 16.05.2018; accep ed – 22.05.2020
Gene alised Odd F eche Family o Dis ibu ions:
P ope ies and Applica ions
Shahdie Ma ganpoo 1,Vahid Ranjba 2Mo ad Alizadeh3
Kamel Abdollahnezhad4
ABSTRACT
A new dis ibu ion called Gene alized Odd F éche (GOF) dis ibu ion is p esen ed and i s
p ope ies explo ed. Some s uc u al p ope ies o he p oposed dis ibu ion, including he
shapes o he haza d a e unc ion, momen s, condi ional momen s, momen gene a ing unc-
ion, skewness, and ku osis a e p esen ed. Mean de ia ions, Lo enz and Bon e oni cu es,
Rényi en opy, and he dis ibu ion o o de s a is ics a e gi en. The maximum likelihood
es ima ion echnique is used o es ima e he model pa ame e s, and finally applica ions o he
model o a eal da a se a e p esen ed o illus a e he use ulness o he p oposed dis ibu ion.
Key wo ds: F éche dis ibu ion, Wiebull dis ibu ion, s uc u al p ope ies, ailu e- ime,
maximum likelihood es ima ion.
1. In oduc ion
Recen ly, some a emp s ha e been made o define new amilies o dis ibu ions o ex-
end well-known models and a he same ime p o ide g ea flexibili y in modelling da a in
p ac ice. Se e al echniques could be employed o o m a la ge amily om an exis ing
dis ibu ion by inco po a ing ex a pa ame e s. These gene alized dis ibu ions gi e mo e
flexibili y by adding one "o mo e" pa ame e s o he baseline model. Fo example, Gup a
e al. (1998) p oposed he exponen ia ed-G class, which consis s o aising he cumula i e
dis ibu ion unc ion (cd ) o a posi i e powe pa ame e . Many o he classes can be ci ed
such as he Ma shall-Olkin-G amily by Ma shall and Olkin (1997), be a gene alized-G
amily by Eugene e al. (2002), he gamma-gene a ed amily by Zog a os and Balak ish-
nan (2009), Kuma aswamy G amily by Co dei o and de Cas o (2011), Gene alized be a
gene a ed dis ibu ions by Alexande e e al. (2015a), exponen ia ed gene alized-G amily
by Co dei o e al. (2013), a new me hod o gene a ing amilies o con inuous dis ibu-
ions by Alzaa eh e al. (2013), exponen ia ed T-X amily o dis ibu ions by Alzaghal e
al. (2013), he Lomax gene a o o dis ibu ions by Co dei o e al. (2014), he WeibullG
amily o p obabili y dis ibu ions by Bou guignon e al. (2014), be a Ma shall-Olkin by
Alizadeh e al. (2015a), Kuma aswamy odd log-logis ic by Alizadeh e al. (2015b), be a
odd log-logis ic by Co dei o e al. (2015), Kuma aswamy Ma shall-Olkin by Alizadeh e
1Goles an Uni e si y, Go gan, 49138-15739, I an. sh.ma [email p o ec ed].
2Goles an Uni e si y, Go gan, 49138-15739, I an, [email p o ec ed], [email p o ec ed].
ORCID: h ps://o cid.o g/0000-0003-3743-0330.
3Pe sian Gul Uni e si y, Busheh , 751691-3798, I an. [email p o ec ed].
ORCID: h ps://o cid.o g/0000-0001-6638-2185.
4Goles an Uni e si y, Go gan, 49138-15739, I an. [email p o ec ed].
110 S. Ma ganpoo : Gene alised Odd F eche Family ...
al. (2015c), ansmu ed exponen ia ed gene alized-G amily by Youso e al. (2015), gen-
e alized ansmu ed-G by No al e al. (2015), gene alized ansmu ed amily by Alizadeh
e al. (Alizadeh2015a), ano he gene alized ansmu ed amily by Me o ci e al. (2015),
Kuma aswamy ansmu ed-G by Afi y e al. (2016a), ansmu ed geome ic-G by A fi y
e al. (2016b), be a ansmu ed-H by Afi y e al. (2016c), Bu X-G by Youso e al.
(2016), he odd Lindley-G amily o dis ibu ions by Sil a e al. (2016), exponen ia ed
ansmu ed-G amily by Me o ci e al. (2016), odd-Bu gene alized amily by Alizadeh
e al. (2016a) he complemen a y gene alized ansmu ed Poisson amily by Alizadeh e al.
(2016b), logis ic-X by Tahi e al. (2016a), a new Weibull-G by Tahi e al. (2016b), he
wo-sided powe -G class by Ko kmaz and Genc (2016), he ype I hal -logis ic amily by
Co dei o e al. (2016a), he Zog a os-Balak ishnan odd log-logis ic amily o dis ibu ions
by Co dei o e al. (2016b), he gene alized odd log-logis ic amily by Co dei o e al.(2016c),
he be a odd log-logis ic gene alized amily o dis ibu ions by Co dei o e al. (2016d), he
Kuma aswamy odd log-logis ic amily o dis ibu ions by Alizadeh e al. (2016d) and a new
gene alized odd log-logis ic amily o dis ibu ions by Haghbin e al. (2017), he gene al-
ized odd log-logis ic amily o dis ibu ions: p ope ies, eg ession models and applica ions
by Co dei o e al. (2017), he odd powe Cauchy amily o dis ibu ions by Alizadeh e
al. (2018), a new amily o he con inuous dis ibu ions: he ex ended Weibull- amily by
Ko kmaz (2018a), he Ma shll-Olkin gene alized G Piosson o dis ibu ions by Ko kmaz
e al. (2018b) and a new amily o dis ibu ions wi h p ope ies, eg ession models and
applica ions by Youso e al. (2018), among o he s.
The a icle is ou lined as ollows: in Sec ion 2, we in oduce he GOF dis ibu ion and
p o ide plo s o he densi y and haza d a e unc ions. Shapes, quan ile unc ion, momen s,
and momen gene a ing unc ion a e also ob ained. Mo eo e , mean de ia ion, o de s a is-
ics, Lo enz and Bon e oni cu es and finally asymp o ic p ope ies a e p esen ed in his
sec ion. Es ima ion by he me hod o maximum likelihood and an explici exp ession o he
obse ed in o ma ion ma ix a e p esen ed in Sec ion 3. The simula ion s udy is p esen ed
in Sec ion 4. The applica ions o eal da a se s a e conside ed in Sec ion 5. Finally, Sec ion
6 o e s some concluding ema ks.
2. Gene alized Odd F eche Family o dis ibu ion
The cd o he Gene alized Odd F eche (GOF) Family o dis ibu ions is gi en by
F(x;a,b,ξ)=exp−(G(x,ξ)−a−1)b(1)
whe e ξ=(ξ1;ξ2;...)is a pa ame e ec o , and aand ba e posi i e pa ame e s. The
co esponding p obabili y densi y unc ion (pd ) is
(x;a,b,ξ)=abg(x,ξ)G(x,ξ)−a−1[G(x,ξ)−a−1]b−1exp−(G(x,ξ)−a−1)b(2)
Fo a=1 we ob ain Odd F eche amily. Some o he possible shapes o he densi y unc ion
(2) o gene alized odd F eche Wiebull dis ibu ion (GOFW), o he selec ed pa ame e
STATISTICS IN TRANSITION new se ies, Sep embe 2020 111
alues a e illus a ed in Figu e 1. As seen in Figu e 1, he densi y unc ion can ake a ious
o ms depending on he pa ame e alues.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.2 0.4 0.6 0.8
x
densi y
a=2.0, b=1.3, α=0.7, λ=1.5
a=2.0, b=1.3, α=0.5, λ=1.5
a=2.0, b=1.3, α=0.3, λ=1.5
a=2.0, b=1.3, α=0.1, λ=1.5
a=1.0, b=1.3, α=1.0, λ=1.5
a=0.3, b=1.3, α=1.0, λ=1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
Hazad a e unc ion
a=1.3, b=0.1, α=1.0, λ=2.0
a=1.0, b=0.5, α=2.0, λ=1.0
a=1.0, b=1.0, α=2.0, λ=2.0
a=0.5, b=1.5, α=2.0, λ=2.0
a=1.0, b=1.3, α=1.0, λ=1.5
a=1.0, b=2.5, α=2.0, λ=2.0
a=0.5, b=1.3, α=0.1, λ=1.0
Figu e 1: Di e en shapes o GOFW pd (le ) and Haza d unc ion ( igh )
2.1. Su i al and Haza d Ra e Func ions
A cen al ole is played in he eliabili y heo y by he quo ien o he pd and su i al
unc ion. We ob ain he su i al unc ion co esponding o (1) as
R(x)=1−exp−(G(x,ξ)−a−1)b
In eliabili y s udies, The haza d a e [h(x)], e e sed-haza d a e unc ion [ (x)] and
cumula i e haza d a e unc ion [H(X)] a e impo an cha ac e is ics and undamen al o
he design o sa e sys ems in a wide a ie y o applica ions. The e o e, we discuss hese
p ope ies o he GOF dis ibu ion. The h(x), (x)and H(x)o X ake he o m
h(x)=abg(x)G(x)−a−1[G(x)−a−1]b−1e−[G(x)−a−1]b
1−e−[G(x)−a−1]b
(x)=abg(x,ξ)G(x,ξ)−a−1[G(x,ξ)−a−1]b−1
and
H(x)=−log1−exp−(G(x,ξ)−a−1)b
Plo s o he h o he GOFW dis ibu ion o se e al pa ame e alues a e displayed in
Figu e 1.
112 S. Ma ganpoo : Gene alised Odd F eche Family ...
2.2. Mix u e ep esen a ions o he pd and cd
Se e al s uc u al p ope ies o he ex ended dis ibu ions may be easily explo ed using
mix u e o ms o Exp-G models. The e o e, we ob ain mix u e o ms o exponen ia ed-G
("Exp-G") o F (x) and (x). In his subsec ion, we p o ide al e na i e mix u e ep esen a-
ions o he pd and cd o X. Some use ul expansions o (1) can be de i ed by using he
concep o powe se ies and gene alized binomial expansion. We ha e
F(x)=exp−(G(x)−a−1)b=exp−1−G(x)a
G(x)ab
=
∞
∑
i=0
(−1)i
i!1−G(x)a
G(x)abi
=
∞
∑
i,j=0
(−1)i+j
i!bi
jG(x)ajG(x)−abi (3)
=
∞
∑
j,k=0
k
∑
l=0
wj,k,lG(x)aj+l(4)
whe e
wj,k,l=
∞
∑
i=0
(−1)i+j+k+l
i!bi
j−abi
kk
l
Fu he mo e, he co esponding GOF densi y unc ion is ob ained by di e en ia ing (4)
(x)=
∞
∑
j,k=0
k
∑
l=0
wj,k,l(aj+l)g(x)G(x)aj+l−1(5)
Using ela ion (3) we ob ain ano he o m o expansions o (1) as bellow, which is used in
es o he pape ,
F(x)=
∞
∑
i,j=0
(−1)i+j
i!bi
jG(x)a(j−bi)=
∞
∑
k=0
ekHk(x)(6)
whe e ¯
G(x)=1−G(x),
ek=
∞
∑
i,j=0
(−1)i+j+k
i!bi
ja(j−bi)
k(7)
and Hδ(x)=(1−G(x))δis he su i al unc ion o he Exp-G dis ibu ion wi h powe
pa ame e δ. Then he co esponding GOF densi y unc ion is ob ained by di e en ia ing
(6)
(x)=
∞
∑
k=0
ekhk(x)(8)
whe e hδ(x)=δg(x)G(x)δ−1.

STATISTICS IN TRANSITION new se ies, Sep embe 2020 113
2.3. Momen s and Momen Gene a ing Func ion
Some o he mos impo an ea u es and cha ac e is ics o a dis ibu ion can be s udied
h ough momen s (e.g. endency, dispe sion, skewness and ku osis). Now we ob ain o di-
na y momen s and he momen gene a ing unc ion (mg ) o he GOF dis ibu ion. The h
o dina y momen o Xis gi en by
μ
=E(X )=x (x)dx =
∞
∑
k=0
ekE(Y
k)(9)
whe e E(Y
k)=x kg(x)G(x)k−1dx; which can be compu ed nume ically o mos pa en
dis ibu ions. The skewness and ku osis measu es can be calcula ed om he o dina y
momen s using well-known ela ionships. One can also find he k h cen al momen o he
GOF dis ibu ion h ough he ollowing well-known equa ion
μk=E(X−μ)k=
k
∑
=0k
μ
(−μ)k− .(10)
Using (10), he a iance, skewness and ku osis measu es can be ob ained. Skewness mea-
su es he deg ee o he long ail and ku osis is a measu e o he deg ee o ail hea iness.
The skewness can be compu ed as
S=μ3
μ3/2
2
=μ
3−3μ
2μ
1+2μ3
1
(μ
2−μ2
1)3/2
and he ku osis is based on oc iles as
K=μ4
μ2
2
=μ
4−4μ
1μ
3+6μ2
1μ
2−3μ4
1
μ
2−μ2
1
.
When he dis ibu ion is symme ic S=0, and when he dis ibu ion is igh (o le ) skewed
S>0(o S <0).AsKinc eases, he ail o he dis ibu ion becomes hea ie . These mea-
su es a e less sensi i e o ou lie s and hey exis e en o dis ibu ions wi hou momen s.
The h momen o gene alized odd F eche Weibull (GOFW) dis ibu ion using ela ion
(8) is gi en by
μ
=∞
0x (x)dx =
∞
∑
k=0
kek∞
0x α
λ(x
λ)α−1e−(x
λ)α(e−(x
λ)α)k−1dx
=
∞
∑
k=0
kek∞
0x αxα−1
λαe−k(x
λ)αdx =λ A(λ,α, )(11)
whe e Γ(a)=∞
0xa−1e−xdx is gamma unc ion and
A(λ,α, )=
∞
∑
k=0
(ek
k /α)Γ(1+k1/α
λ).
114 S. Ma ganpoo : Gene alised Odd F eche Family ...
Using powe se ies, he momen gene a ing unc ion o GOFW is as bellow
MX( )=E(e X)=
∞
∑
n=0
n
n!E(Xn)=
∞
∑
n=0
n
n!λnA(λ,α,n)
I is o be highligh ed ha he equa ion (11) can be easily compu ed nume ically using
ma hema ical o s a is ical so wa e. Fo his pu pose, one can compu e his equa ion o a
la ge na u al numbe , say N, ins ead o infini y in he sums. The e o e, se e al quan i ies o
Xsuch as momen s, skewness and ku osis can be compu ed nume ically using (11). Plo s
o skewness and ku osis a e p esen ed in Figu e 2.
alpha
be a
skewness
a
b
skewness
a
alpha
skewness
b
be a
skewness
alpha
be a
ku osis
a
b
ku osis
a
alpha
ku osis
b
be a
ku osis
Figu e 2: The skewness and ku osis plo s o GOF dis ibu ion o selec ed a,b,α,β.
2.4. O de s a is ics
O de s a is ics make hei appea ance in many a eas o s a is ical heo y and p ac ice.
Suppose X1,...,Xnis a andom sample om any GOF dis ibu ion. Le Xi:ndeno e he i h
o de s a is ic. The pd o Xi:ncan be exp essed as
i:n(x)=K (x)Fi−1(x){1−F(x)}n−i=K
n−i
∑
j=0
(−1)jn−i
j (x)F(x)j+i−1,
whe e K=1/B(i,n−i+1). We use he esul o G adsh eyn and Ryzhik (2000) o a powe
se ies aised o a posi i e in ege n( o n≥1)
∞
∑
i=0
aiuin
=
∞
∑
i=0
dn,iui,(12)
STATISTICS IN TRANSITION new se ies, Sep embe 2020 115
whe e he coe ficien s dn,i( o i=1,2,...) a e de e mined om he ecu ence equa ion
(wi h dn,0=an
0)
dn,i=(ia0)−1i
∑
m=1
[m(n+1)−i]amdn,i−m.(13)
We can show ha he densi y unc ion o he i h o de s a is ic o any GOF dis ibu ion can
be exp essed as
i:n(x)=
∞
∑
,k=0
m ,kH +k+1(x),(14)
whe e H +k+1(x)s ands o he he su i al unc ion o he Exp-G dis ibu ion wi h powe
pa ame e +k+1.
m ,k=n!( +1)(i−1)!e +1
( +k+1)
n−i
∑
j=0
(−1)j j+i−1,k
(n−i−j)!j!.
He e, e is gi en by (7) and he quan i ies j+i−1,kcan be de e mined gi en ha j+i−1,0=
ej+i−1
0and ecu si ely we ha e:
j+i−1,k=(ke0)−1k
∑
m=1
[m(j+i)−k]em j+i−1,k−m,k≥1.
Equa ion (14) is he main esul o his sec ion. The e o e, se e al ma hema ical quan i-
ies o hese o de s a is ics like o dina y and incomple e momen s, ac o ial momen s, mg ,
mean de ia ions and o he s can be de i ed using his esul .
2.5. Mean De ia ions, Lo enz and Bon e oni Cu es
Mean de ia ion abou he mean and mean de ia ion abou he median as well as Lo enz
and Bon e oni cu es o he GOF dis ibu ion a e p esen ed in his sec ion. Bon e oni
and Lo enz cu es a e a widely used ool o analysing and isualizing income inequali y.
Lo enz cu e, L(p) can be ega ded as he p opo ion o o al income olume accumula ed
by hose uni s wi h income lowe han o equal o he olume y, and Bon e oni cu e, B(p)
is he scaled condi ional mean cu e, ha is, a io o g oup mean income o he popula ion.
2.5.1 Mean de ia ions
The amoun o sca e in a popula ion may be measu ed o some ex en by de ia ions
om he mean and median. These a e known as he mean de ia ion abou he mean and he
mean de ia ion abou he median, defined by
δ1(X)=∞
0|x−μ| (x)dx,and δ2(X)=∞
0|x−M| (x)dx.
116 S. Ma ganpoo : Gene alised Odd F eche Family ...
espec i ely, whe e μ=E(X)and M=Median(X)=Q(0.5)deno es he median and Q(p)
is he quan ile unc ion. The measu es δ1(X)and δ2(X)can be calcula ed using he ela-
ionships
δ1(X)=2μF(μ)−2μ
0x (x)dx,and δ2(X)=μ−2M
0x (x)dx
Finally o GOFW dis ibu ion we ha e
δ1(X)=2μF(μ)−2
∞
∑
k=0
kekμ
0xαxα−1
λαe−k(x
λ)αdx
=2μF(μ)−2λB(λ,α,μ)
whe e γ(s,x)=x
0 s−1e− d is lowe incomple e gamma unc ion and
B(λ,α,μ)=
∞
∑
k=0
ek
k1/αγ(2,μλα
k)
And
δ2(X)=μ−2λB(λ,α,M).
2.5.2 Bon e oni and Lo enz cu es
The Bon e oni and Lo enz cu es ha e applica ions in economics as well as o he fields
like eliabili y, medicine and insu ance. Le X∼GOFW(a,b,α,λ)and F(x)be he cd o
X, hen he Bon e oni cu e o he GOFW dis ibu ion is gi en by
B(F(x)) = 1
μF(x)x
0 ( )d ,
whe e μ=E(X). The e o e, om (15), we ha e
B(F(x)) = 1
μF(x)×λB(λ,α,x).
The Lo enz cu e o he GOFW dis ibu ion can be ob ained using he ela ion
L(F(x)) = F(x)B(F(x)) = λ
μB(λ,α,x).
2.6. Asymp o ic P ope ies
One o he main usage o he idea o an asymp o ic dis ibu ion is in p o iding app oxi-
ma ions o he cumula i e dis ibu ion unc ions o he s a is ical es ima o s.
STATISTICS IN TRANSITION new se ies, Sep embe 2020 123
In addi ion, PP plo o he GOFW dis ibu ion a e plo ed in Figu e 4. We also plo ed
he fi ed pd s and cd s o he conside ed models o he sake o isual compa ison, in Figu e
5. Figu e 4 and 5 sugges ha he GOFW fi s he skewed da a e y well.
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
PP Plo
Theo e ical Pe cen iles
Sample Pe cen iles
Figu e 4: The PP plo .
densi y
0.0 0.1 0.2 0.3 0.4 0.5
02468
Lindley
PL
GL
EPL
EEL
50 100 150 200
0.0 0.2 0.4 0.6 0.8 1.0
x
Fn(x)
Weibull
EW
KwW
BW
MCW
GOLLW
TIGEW
OFW
GOF
Figu e 5: Fi ed densi ies o dis ibu ions.

124 S. Ma ganpoo : Gene alised Odd F eche Family ...
6. Conclusion
In his pape , we p esen a new class o dis ibu ions called he Gene alized Odd F eche
(GOF) amily o dis ibu ions. The s a is ical p ope ies o he GOF dis ibu ion including
he haza d and e e se haza d unc ions, quan ile unc ion, momen s, incomple e momen s,
gene a ing unc ions, mean de ia ions, Bon e oni and Lo enz cu es, o de s a is ics and
maximum likelihood es ima ion o he model pa ame e s a e gi en. Simula ion s udies
we e conduc ed o examine he pe o mance o he new GOF dis ibu ion. We also p esen
applica ions o his new model o a eal li e da a se in o de o illus a e he use ulness o
he dis ibu ion.
REFERENCES
AFIFY A.Z., ALIZADEH, M., YOUSOF, H. M., ARYAL, G. and AHMAD, M. (2016a).
The ansmu ed geome ic-G amily o dis ibu ions: heo y and applica ions. Pak. J.
S a is ., 32(2), pp. 139–160.
AFIFY A.Z., CORDEIRO, G. M., YOUSOF, H. M., ALZAATREH, A. and NOFAL, Z.
M. (2016b). The Kuma aswamy ansmu ed-G amily o dis ibu ions: p ope ies and
applica ions. J. Da a Sci., 14(2), pp. 245–270.
AFIFY, A.Z., YOUSOF, H.M. and NADARAJAH, S. (2016c). The be a ansmu ed-H am-
ily o dis ibu ions: p ope ies and applica ions. S a is ics and i s In e ence, 10(3),
pp. 505–520.
ALEXANDER, C., CORDEIRO, G.M., ORTEGA, E.M.M. and SARABIA, J.M. (2012).
Gene alized be a gene a ed dis ibu ions, Compu a ional S a is ics and Da a Analysis,
56, pp. 1880–1897.
ALIZADEH, M., CORDEIRO, G. M., DE BRITO, E. and DEMÉTRIO C.G.B. (2015a).
The be a Ma shall- Olkin amily o dis ibu ions. Jou nal o S a is ical Dis ibu ions
and Applica ions, 23(3), pp. 546–557.
ALIZADEH, M., CORDEIRO, G.M., MANSOOR, M., ZUBAIR, M. and HAMEDANI,
G.G. (2015b). The Kuma aswamy Ma shal-Olkin amily o dis ibu ions. Jou nal o
he Egyp ian Ma hema ical Socie y, 23, pp. 546–557.
ALIZADEH, M., MEROVCI, F. and HAMEDANI, G.G. (2015c). Gene alized ansmu ed
amily o dis ibu ions: p ope ies and applica ions. Hace epa Jou nal o Ma hema -
ics and S a is ics, 46(4), pp. 645–667.
STATISTICS IN TRANSITION new se ies, Sep embe 2020 125
ALIZADEH M., EMADI M., DOOSTPARAST M., CORDEIRO G.M., ORTEGA E.M.M.
and PESCIM R.R., (2016) A new amily o dis ibu ions: he Kuma aswamy odd log-
logis ic, p ope ies and applica ions. Hace epe Jou nal o Ma hema ics and S a is ics,
44(6), pp. 1491–1512.
ALIZADEH, M., CORDEIRO, G.M., NASCIMENTO, A.D.C. LIMA M.D.S. AND OR-
TEGA, E.M.M. (2016a). Odd-Bu gene alized amily o dis ibu ions wi h some
applica ions. Jou nal o S a is ical Compu a ion and Simula ion, 83, pp. 326–339.
ALIZADEH, M., YOUSOF, H.M., AFIFY A.Z., CORDEIRO, G.M., and MANSOOR, M.
(2016b). The complemen a y gene alized ansmu ed Poisson-G amily. Aus ian
Jou nal o S a is ics, 47(4), pp. 60–80.
ALIZADEH, M., ALTUN, E., CORDEIRO, G. M., and RASEKHI, M. (2018). The odd
powe cauchy amily o dis ibu ions: p ope ies, eg ession models and applica ions.
Jou nal o S a is ical Compu a ion and Simula ion, 88(4), pp. 785–807.
ALZAATREH, A., LEE, C. and FAMOYE, F. (2013). A new me hod o gene a ing ami-
lies o con- inuous dis ibu ions. Me on, 71, pp. 63–79.
ALZAGHAL, A., FAMOYE, F. and LEE, C. (2013). Exponen ia ed T-X amily o dis i-
bu ions wi h some applica ions. In e na ional Jou nal o P obabili y and S a is ics,2,
pp. 31–49.
ANDRADE NLR, MOURA RMP, SILVEIRA A (2007) De e minação da Q7,10 pa a o Rio
Cuiab´a, Ma o G osso, B asil e compa ação com a azão egula izada após a implan-
ação do ese a ó io de ap o ei amen o múl iplo de manso. 24oCong esso B asilei o
de Engenha ia Sani á ia e Ambien al. Belo Ho izon e, Minas Ge ais B asil.
BOURGUIGNON, M., SILVA, R.B. and CORDEIRO, G.M. (2014). The WeibullG amily
o p obabili y dis ibu ions, Jou nal o Da a Science, 12, pp. 53–68.
CORDEIRO, G. M., DE CASTRO, M. (2011). A new amily o gene alized dis ibu ions.
Jou nal o S a is ical Compu a ion and Simula ion, 81, pp. 883–898.
CORDEIRO, GAUSS M., SARALEES NADARAJAH, and EDWIN MM ORTEGA, (2012).
The Kuma aswamy Gumbel dis ibu ion. S a is ical Me hods and Applica ions, 21.2,
pp. 139–168.
CORDEIRO, G. M., GOMES, A. E., DA-SILVA, C. Q. and ORTEGA, E. M., (2013). The
be a exponen ia ed Weibull dis ibu ion. Jou nal o S a is ical Compu a ion and Sim-
ula ion, 83, pp. 114–138.
126 S. Ma ganpoo : Gene alised Odd F eche Family ...
CORDEIRO, G. M., HASHIMOTO, E. M. and ORTEGA, E. M., (2014). McDonald
Weibull model. S a is ics: A Jou nal o Theo e ical and Applied S a is ics, 48, pp.
256–278.
CORDEIRO, G. M.,ALIZADEH, M., TAHIR, M. H., MANSOOR, M., BOURGUIGNON,
M., & HAMEDANI, G. G., (2015). The be a odd log-logis ic gene alized amily o
dis ibu ions, Hace epe Jou nal o Ma hema ics and S a is ics, 45(73), pp. 126–139.
CORDEIRO, G. M., ALIZADEH, M. and DINIZ MARINHO, P. R., (2016a). The ype I
hal -logis ic amily o dis ibu ions. Jou nal o S a is ical Compu a ion and Simula-
ion, 86, pp. 707–728.
CORDEIRO, G. M., ALIZADEH, M., ORTEGA, E. M. and SERRANO, L. H., V.(2016b).
The Zog a os- Balak ishnan odd log-logis ic amily o dis ibu ions: P ope ies and
Applica ions. Hace . J. Ma h. S a ., 7(1), pp. 211–234.
CORDEIRO, G. M., ALIZADEH, M., OZEL, G., HOSSEINI, B., ORTEGA, E. M. M. and
ALTUN, E., (2016c). The gene alized odd log-logis ic amily o dis ibu ions: p op-
e ies, eg ession models and applica ions, Jou nal o S a is ical Compu a ion and
Simula ion, 87, pp. 908–932.
CORDEIRO, G. M., ALIZADEH, M., TAHIR, M. H., MANSOOR, M., BOURGUIGNON,
M. and HAMEDANI G. G., (2016d). The be a odd log-logis ic gene alized amily o
dis ibu ions, Hace epe Jou nal o Ma hema ics and S a is ics, 45(6), pp. 1175–1202.
CORDEIRO, G. M., ALIZADEH, M., OZEL, G., HOSSEINI, B., ORTEGA, E. M. M.
and ALTUN, E., (2017). The gene alized odd log-logis ic amily o dis ibu ions:
p ope ies, eg ession models and applica ions. Jou nal o S a is ical Compu a ion
and Simula ion, 87(5), pp. 908–932.
EMLET, R. B., MCEDWARD, L. R. and STRATHMANN, R. R., (1987) Echinode m la al
ecology iewed om he egg, in: M. Jangoux and J.M. Law ence (Eds.) Echinode m
S udies,2, pp. 55–136.
EUGENE, N., LEE, C. and FAMOYE, F., (2002). Be a-no mal dis ibu ion and i s appli-
ca ions. Commun. S a . Theo y Me hods, 31, pp. 497–512.
GRADSHTEYN, I. S., RYZHIK, I. M., (2000). Table o in eg als, se ies, and p oduc s.
Academic P ess, San Diego.
GUPTA, R. C., GUPTA, P. L. and GUPTA, R. D., (1998). Modeling ailu e ime da a by
Lehmann al e na i es. Commun. S a . Theo y Me hods, 27, pp. 887–904.
STATISTICS IN TRANSITION new se ies, Sep embe 2020 127
HAGHBIN H., OZEL G., ALIZADEH, M. and HAMEDANI, G. G., (2017) A new gene -
alized odd log-logis ic amily o dis ibu ions. Communica ions in S a is ics - Theo y
and Me hods, 46(20), pp. 9897–9920.
KORKMAZ, M. C., GENC, A. I., (2016). A new gene alized wo-sided class o dis i-
bu ions wi h an emphasis on wo-sided gene alized no mal dis ibu ion. Communica-
ions in S a is ics - Simula ion and Compu a ion, 46, pp. 1441–1460.
KORKMAZ, M. C., (2018). A new amily o he con inuous dis ibu ions: he ex ended
Weibull-G amily. Communica ions Facul y o Sciences Uni e si y o Anka a Se ies
A1 Ma hema ics and S a is ics, 68(1), pp. 248–270.
KORKMAZ, M. C. , YOUSOF H. M., HAMEDANI, G. G. and ALI, M. M., (2018). The
Ma shall-Olkin Gene alized G Poisson Family o Dis ibu ions, Pakis an Jou nal o
S a is ics, 34(3), pp. 251–267.
LEHMANN, E. L., CASELLA, G., (1998) Theo y o Poin Es ima ion, Sp inge .
MARSHALL, A. W., OLKIN, I., (1997). A new me hods o adding a pa ame e o a amily
o dis ibu ions wi h applica ion o he Exponen ial and Weibull amilies. Biome ika,
84, pp. 641–652.
MEROVCI, F., ALIZADEH, M. and HAMEDANI, G. G., (2016). Ano he gene alized
ansmu ed amily o dis ibu ions: p ope ies and applica ions, Aus ian Jou nal o
S a is ics, 45, pp. 71–93.
MEROVCI, F., ALIZADEH, M., YOUSOF, H. M. and HAMEDANI, G. G., (2016). The
exponen ia ed ansmu ed-G amily o dis ibu ions: heo y and applica ions. Com-
mun. S a . Theo y Me h- ods, 46(21), pp. 10800–10822.
NOFAL, Z. M., AFIFY, A. Z., YOUSOF, H. M. and CORDEIRO, G. M., (2017). The gen-
e alized ansmu ed-G amily o dis ibu ions. Commun. S a . Theo y Me hods,46(8),
pp. 4119–4136.
SILVA, F. G., PERCONTINI, A., DE BRITO, E., RAMOS, M. W., VENANCIO, R. and
CORDEIRO, G. M., (2016). The odd Lindley-G amily o dis ibu ions, Aus ian
Jou nal o S a is ics, VV, 1–xx.
TAHIR, M. H., CORDEIRO, G. M., ALZAATREH, A., MANSOOR, M. and ZUBAIR,
M., (2016a). The logis ic- X amily o dis ibu ions and i s applica ions. Commun.
S a . Theo y Me hods,45(24), pp. 7326–7349.
128 S. Ma ganpoo : Gene alised Odd F eche Family ...
TAHIR, M. H., ZUBAIR, M., MANSOOR, M., CORDEIRO, G. M., ALIZADEH, M. and
HAMEDANI, G. G., (2016b). A new Weibull-G amily o dis ibu ions. Hace . J.
Ma h. S a .,45 (2), pp. 629–647.
YOUSOF, H. M., AFIFY, A. Z., ALIZADEH, M., BUTT, N. S., HAMEDANI, G. G. and
ALI, M. M., (2015). The ansmu ed exponen ia ed gene alized-G amily o dis ibu-
ions, Pak. J. S a . Ope . Res., 11, pp. 441–464.
YOUSOF, H. M., AFIFY, A. Z., HAMEDANI, G. G. and ARYAL, G., (2016). he Bu X
gene a o o dis ibu ions o li e ime da a. Jou nal o S a is ical Theo y and Applica-
ions,16(3), pp. 288–305.
YOUSOF, H. M., ALTUN, E., RAMIRES, T. G., ALIZADEH, M., and RASEKHI, M.,
(2018). A new amily o dis ibu ions wi h p ope ies, eg ession models and applica-
ions. Jou nal o S a is ics and Managemen Sys ems, 21(1), pp. 163–188.
ZOGRAFOS, K., BALAKRISHNAN, N., (2009). On amilies o be a and gene alized
gamma-gene a ed dis ibu ions and associa ed in e ence. S a is ical Me hodology,6,
pp. 344–362.