A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model

symme y
S
S
A icle
A Gene alized Rayleigh Family o Dis ibu ions Based on he
Modi ied Slash Model
Inmaculada Ba anco-Chamo o 1,*,† , Yu i A. I ia e 2,†, Yolanda M. Gómez 3,† , Juan M. As o ga 4,†
and Héc o W. Gómez 2,*,†


Ci a ion: Ba anco-Chamo o, I.;
I ia e, Y.A.; Gómez, Y.M.; As o ga,
J.M.; Gómez, H.W. A Gene alized
Rayleigh Family o Dis ibu ions
Based on he Modi ied Slash Model.
Symme y 2021,13, 1226. h ps://
doi.o g/10.3390/sym13071226
Academic Edi o : Jinyu Li
Recei ed: 3 June 2021
Accep ed: 4 July 2021
Published: 8 July 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2020 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Depa amen o de Es adís ica e In es igación Ope a i a, Facul ad de Ma emá icas, Uni e sidad de Se illa,
41012 Se illa, Spain
2Depa amen o de Ma emá icas, Facul ad de Ciencias Básicas, Uni e sidad de An o agas a,
An o agas a 1240000, Chile; [email p o ec ed]
3Depa amen o de Ma emá icas, Facul ad de Ingenie ía, Uni e sidad de A acama, Copiapó 1530000, Chile;
[email p o ec ed]
4Depa amen o de Tecnologías de la Ene gía, Facul ad Tecnológica, Uni e sidad de A acama,
Copiapó 1530000, Chile; [email p o ec ed]
*Co espondence: [email p o ec ed] (I.B.-C.); hec o [email p o ec ed] (H.W.G.)
† These au ho s con ibu ed equally o his wo k.
Abs ac :
Speci ying a p ope s a is ical model o ep esen asymme ic li e ime da a wi h high
ku osis is an open p oblem. In his pape , he h ee-pa ame e , modi ied, slashed, gene alized
Rayleigh amily o dis ibu ions is p oposed. I s s uc u al p ope ies a e s udied: s ochas ic ep esen-
a ion, p obabili y densi y unc ion, haza d a e unc ion, momen s and es ima ion o pa ame e s ia
maximum likelihood me hods. As me i s o ou p oposal, we highligh as pa icula cases a ple ho a
o li e ime models, such as Rayleigh, Maxwell, hal -no mal and chi-squa e, among o he s, which a e
able o accommoda e hea y ails. A simula ion s udy and applica ions o eal da a se s a e included
o illus a e he use o ou esul s.
Keywo ds:
gene alized Rayleigh dis ibu ion; EM algo i hm; ku osis; maximum likelihood es ima-
ion; slashed gene alized Rayleigh dis ibu ion
1. In oduc ion
Vodua [
1
] p oposed a wo-pa ame e dis ibu ion o he analysis o posi i e da a,
he well-known gene alized Rayleigh dis ibu ion. This model o e comes he p oblems
ound in p ac ice, which a e ela ed o a Rayleigh dis ibu ion and poin ed ou by au ho s,
such as Siddiqui [
2
] and Hi ano e al. [
3
]. Recall ha a andom a iable ( )
X
ollows
a gene alized Rayleigh dis ibu ion, deno ed as
X∼GR(θ
,
α)
, i i s p obabili y densi y
unc ion (pd ) is gi en by he ollowing:
(x;θ,α) = 2θα+1
Γ(α+1)x2α+1e−θx2,x>0, (1)
whe e
θ>
0 is a scale pa ame e ,
α>−
1 is a shape pa ame e and
Γ(α) = R∞
0uα−1e−udu
is he gamma unc ion. The cumula i e dis ibu ion unc ion (cd ) o he GR dis ibu ion
can be exp essed as ollows:
F(x;θ,α) = γ(α+1, θx2)
Γ(α+1)=P(α+1, θx2),x>0, (2)
Symme y 2021,13, 1226. h ps://doi.o g/10.3390/sym13071226 h ps://www.mdpi.com/jou nal/symme y
Symme y 2021,13, 1226 2 o 18
whe e
γ(α
,
x) = Rx
0uα−1e−udu
is he incomple e gamma unc ion and
P(α+
1,
θx2)
de-
no es he egula ized gamma unc ion [
4
]. The momen s o
X
a e easily de i ed by using
he esul s in [5] (Sec ion 3.478). In ac , he h dis ibu ional momen is as ollows:
µ =E(X ) = Γ
2+α+1
Γ(α+1)θ /2 , =1, 2, 3, ... (3)
Following he app oach p oposed in [
6
], li e ime models o accommoda e ou lie s
we e p oposed in [
7
]. We ocus on he app oach p oposed in [
8
], whe e an ex ension o he
gene alized Rayleigh dis ibu ion called he slashed gene alized Rayleigh dis ibu ion was
in oduced. This model has a ku osis coe icien , which exhibi s a wide ange o alues
han he ku osis coe icien in he GR dis ibu ion; he e o e, i is app op ia e o i he da a
se s wi h ou lie s. Recall ha a
T
ollows a slashed gene alized Rayleigh dis ibu ion,
deno ed as T∼SGR(θ,α,q), i i s s ochas ic ep esen a ion is gi en by he ollowing:
T=X
U1/q,q>0, (4)
whe e
X∼GR(θ
,
α)
and
U∼
U
(
0, 1
)
a e independen . The associa ed pd is gi en by he
ollowing:
T( ;θ,α,q) = q −(q+1)
Γ(α+1)θq/2 Γ2α+q+2
2Fθ 2,2α+q+2
2, 1, >0, (5)
whe e
θ>
0 is a scale pa ame e ,
α>−
1 is a shape pa ame e ,
q>
0 is a ku osis pa ame e
and
F(x
,
α
,
β) = Rx
0
βα
Γ(α) α−1e−β d
is he cd o a gamma dis ibu ion wi h shape pa ame e
α>
0 and a e pa ame e
β>
0. I
α=
0 and
θ= (
2
σ)−1
, he GR dis ibu ion educes
o he slashed-Rayleigh dis ibu ion. I
α=−
1
/
2 and
θ= (
2
σ2)−1
, he GR dis ibu ion
educes o slashed-hal no mal dis ibu ion [9].
In his wo k, we p opose a modi ied e sion o he SGR dis ibu ion, gi en in (4), which
can be used as an al e na i e model o he GR and SGR dis ibu ions.
We ollow he idea p oposed by [
10
], who in oduced an ex ension o he no mal
dis ibu ion called modi ied slash (MS) dis ibu ion. A
X
ollows a modi ied slash
dis ibu ion, deno ed as X∼MS(q), i i can be w i en as ollows:
X=Y
W1/q,q>0,
whe e
Y∼
N
(
0, 1
)
and
W∼Exp(
2
)
a e independen . I
q→∞
, hen he MS dis ibu ion
con e ges in law o a no mal dis ibu ion.
The pape is o ganized as ollows. In Sec ion 2 he s ochas ic ep esen a ion, he pd
and o dina y momen s o he p oposed model a e gi en. Sec ion 3is de o ed o he amily
o dis ibu ions belonging o he gene alized Rayleigh amily o dis ibu ions based on he
modi ied slash model. Among o he s, we can ind he e he modi ied slashed Rayleigh,
hal -no mal, Maxwell and chi-squa e dis ibu ions. Recall ha he uni ied ea men o
a amily o dis ibu ions is o in e es in s a is ics; see, o ins ance, [
11
–
13
]. In
Sec ion 4
,
he es ima ion o pa ame e s is discussed by conside ing he momen s and maximum
likelihood app oaches. Simila o [
14
], hei asymp o ic no mali y is es ablished. In
addi ion, we ca y ou a simula ion s udy o illus a e he pe o mance o he es ima o s.
In Sec ion 5, wo applica ions o eal da a se s a e p esen ed, whe e we illus a e ha he
p oposed dis ibu ion may p o ide a be e i han he gene alized Rayleigh and slashed
gene alized Rayleigh dis ibu ions. Finally, he main conclusions o his pape a e gi en.
2. The New Model
In his sec ion, some s uc u al p ope ies o he new model a e gi en. These a e he
s ochas ic ep esen a ion, pd , haza d a e unc ion and dis ibu ional momen s.
Symme y 2021,13, 1226 3 o 18
2.1. S ochas ic Rep esen a ion
De ini ion 1.
A andom a iable
T
ollows a modi ied slashed gene alized Rayleigh dis ibu ion,
deno ed as T ∼MSGR(θ,α,q), i i can be exp essed as ollows:
T=X
−1
2log(Z)1/q,q>0, (6)
whe e X ∼GR(θ,α)and Z ∼U(0, 1)a e independen .
Rema k 1.
1.
Le
T∼MSGR(θ
,
α
,
q)
. Then,
T
can be exp essed as
T=XY−1/q
, whe e
X∼GR(θ
,
α)
and Y ∼Exp(2)a e independen .
2.
I
q=
1 hen
T
ollows a canonic modi ied slashed gene alized Rayleigh (CMSGR) dis i-
bu ion, and we can w i e
T=−
2
Xlog−1(Z)
, whe e
X∼GR(θ
,
α)
and
Z∼
U
(
0, 1
)
a e
independen . This is deno ed as T ∼CMSGR(θ,α).
The s ochas ic ep esen a ion gi en in (6) can be used o gene a e alues o
T∼
MSGR(θ,α,q)by using he ollowing algo i hm:
1. Gene a e X∼GR(θ,α).
2. Gene a e Z∼U(0, 1).
3. Compu e T=X−1
2log(Z)−1/q.
2.2. P obabili y Densi y Func ion
To deal wi h he pd s o gene alized slash dis ibu ions, i is common o use hype geo-
me ic unc ions as can be seen in [
7
,
15
,
16
]. In his subsec ion, we i s in oduce he special
unc ion, which is used in he exp ession o he pd o he MSGR dis ibu ion, which is
gi en in P oposi ion 1.
De ini ion 2. Fo γ>0, a >0, >0and s >0, le us conside he ollowing:
I(γ,a, ,s) = Z∞
0xγ−1e−ax −sx dx (7)
whe e I(·,·,·,·)is de ined as in [17] (Equa ion 2.3.1.13).
P oposi ion 1. Le T ∼MSGR(θ,α,q). Then, he pd o T is gi en by he ollowing:
T( ;θ,α,q) = 2qθ−q
2
Γ(α+1) q+1Iα+q
2+1, 2
θq
2 q,q
2, 1, >0, (8)
whe e θ>0is a scale pa ame e , α>−1is a shape pa ame e and q >0is a ku osis pa ame e .
P oo .
F om (6), and by applying he change o a iable me hod, he pd o
T
is gi en by
he ollowing:
T( ;θ,α,q) = 4qθα+1
Γ(α+1) 2α+1Z∞
0w2α+q+1e−θ( w)2−2wqdw
=2qθ−q
2
Γ(α+1) q+1Z∞
0uα+q
2e−u−2u
θ 2q
2
du .
By conside ing I(·,·,·,·), he in eg al in oduced in De ini ion 2, (8) is ob ained.
Symme y 2021,13, 1226 4 o 18
Co olla y 1. Le T ∼MSGR(θ,α,q). Then, he cd o T is as ollows:
FT( ;θ,α,q) = 2q
Γ(α+1)θq
2Z
0u−(q+1)Iα+q
2+1, 2
θuq,q
2, 1du
whe e Γ(·)is he gamma unc ion and I(·,·,·,·)was de ined in (7).
As o he shape o he pd in he MSGR model, we highligh ha he pd can be
unimodal o mono onically dec easing. Plo s o he pd o ce ain alues o he pa ame e s
a e gi en in Figu e 1. In Figu e 1a,b, we can see ha he pa ame e
q
is a ku osis pa ame e
since i mainly a ec s he igh ail o he dis ibu ion. In Figu e 1c, we can app ecia e
ha he pa ame e
θ
is a scale- ype pa ame e ; he smalle he alue o
θ
, he mo e sp ead
ou he dis ibu ion [
18
]. Finally, in Figu e 1d, we can see ha he pa ame e
α
is a shape
pa ame e .
012345
0.0 0.2 0.4 0.6 0.8
(a): MSGR(θ=1, α=0.5, q)
Densi y unc ion
q
∞
1
2
3
8
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
(b): MSGR(θ=5, α=−0.5, q)
Densi y unc ion
q
∞
1
2
3
8
012345
0.0 0.2 0.4 0.6 0.8 1.0
(c): MSGR(θ, α=1.5, q=3)
Densi y unc ion
θ
5
4
3
2
1
012345
0.0 0.2 0.4 0.6 0.8 1.0
(d): MSGR(θ, α=1.5, q=3)
Densi y unc ion
α
−0.8
−0.5
0
1
2
Figu e 1. Plo s o he MSGR pd o ce ain alues o i s pa ame e s.
2.3. Reliabili y Analysis
Fo non-nega i e s, i is o in e es o know he eliabili y (o su i al) unc ion and
he haza d a e unc ion [
19
,
20
]. The eliabili y unc ion,
RT( ) =
1
−FT( )
, o a MSGR
dis ibu ion is gi en by he ollowing:
RT( ;θ,α,q) = 1−2q
Γ(α+1)θq
2Z
0u−(q+1)I(u)du, (9)
whe e I(u) = Iθ,α,q(u) = Iα+q
2+1, 2
θ
q
2uq,q
2, 1is de ined as in (7).
The haza d a e unc ion, de ined by hT( ) = T( )
RT( ), is gi en by he ollowing:
hT( ;θ,α,q) = −(q+1)I( )
Γ(α+1)θ
q
2
2q−Z
0u−(q+1)I(u)du
,
whe e I(u)is as in (9) and I( ) = Iα+q
2+1, 2
θ
q
2 q,q
2, 1, de ined in (7).
Figu e 2displays some plo s o he haza d a e unc ion o he MSGR dis ibu ion o
di e en alues o i s pa ame e s.
Symme y 2021,13, 1226 5 o 18
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4 5 6
MSGR(θ=1, α=−0.5, q)
Haza d a e
q
∞
1
5
10
15
0 1 2 3 4
0 1 2 3 4 5 6
MSGR(θ=1, α=2, q)
Haza d a e
q
∞
1
5
10
15
Figu e 2.
Plo o he haza d a e unc ion o he MSGR dis ibu ion o di e en alues o i s
pa ame e s.
Applica ions o in e es in enginee ing o he Rayleigh models can be seen in [21–24]
among o he s. In hese applica ions, i can be o in e es o apply ou p oposal.
2.4. Momen s
Nex , we p oceed o ob ain he momen s o he MGSR dis ibu ion. F om hese
ea u es, impo an cha ac e is ics o he model, such as he mean o expec ed alue,
a iance, skewness and ku osis coe icien s a e ob ained. These esul s will also be he
basis o apply he me hod o momen s in Sec ion 4.
P oposi ion 2.
Le
T∼MSGR(θ
,
α
,
q)
. Then, o
=
1, 2, ... and
q>
he
h dis ibu ional
momen is gi en by he ollowing:
µ =E(T ) = 2
qΓq−
qΓ
2+α+1
Γ(α+1)θ
2
. (10)
P oo .
Using he s ochas ic ep esen a ion gi en in Pa 1 o Rema k 1and he independence
o Xand Y, i ollows ha
µ =E(T )=E X
Y1
q! !=EX Y−
q=E(X )EY−
q,
whe e
EY−
q=
2
qΓq−
q
,
q>
and
E(X )=Γ(
2+α+1)
Γ(α+1)θ
2
a e dis ibu ional momen s o
he Exp(2)and GR(θ,α)dis ibu ions, espec i ely. .
Co olla y 2.
Le
T∼MSGR(θ
,
α
,
q)
. Then, he mean and a iance a e, espec i ely, he ollowing:
E(T) = 21/qΓq−1
qΓ(α+3/2)
Γ(α+1)θ1/2 ,q>1,
and
Va (T) = 22/q
Γ2(α+1)θΓ(α+1)Γ(α+2)Γq−2
q−Γ2(α+3/2)Γ2q−1
q,q>2.

Symme y 2021,13, 1226 6 o 18
Co olla y 3.
Le
T∼MSGR(σ
,
q)
. Then, he asymme y (
β1
) and ku osis (
β2
) coe icien s a e
he ollowing:
β1=1
A3/2(α,q)Γ2(α+1)Γ(α+5/2)Γq−3
q−3Γ(α+3/2)Γ(α+2)Γ(α+1)
×Γq−1
qΓq−2
q+2Γ3(α+3/2)Γ3q−1
q,q>3,
and
β2=1
A2(α,q)Γ3(α+1)Γ(α+3)Γq−4
q−4Γ(α+3/2)Γ(α+5/2)
×Γ2(α+1)Γq−1
qΓq−3
q+6Γ2(α+3/2)Γ(α+1)Γ(α+2)
×Γ2q−1
qΓq−2
q−3Γ4(α+3/2)Γ4q−1
q,q>4,
whe e A(α,q) = Γ(α+1)Γ(α+2)Γ(q−2
q)−Γ2(α+3
2)Γ2(q−1
q).
Rema k 2.
No ice ha
β1
and
β2
do no depend on
θ
because i is a scale pa ame e . Fu he mo e,
as q →∞ he β1and β2coe icien s end, espec i ely, o he ollowing:
β1GR =1
A3/2(α)hΓ2(α+1)Γ(α+5/2)−3Γ(α+3/2)Γ(α+2)Γ(α+1) + 2Γ3(α+3/2)i
and
β2GR =1
A2(α)hΓ3(α+1)Γ(α+3)−4Γ(α+3/2)Γ(α+5/2)Γ2(α+1)
+6Γ2(α+3/2)Γ(α+1)Γ(α+2)−3Γ4(α+3/2),
whe e
A(α) = Γ(α+
1
)Γ(α+
2
)−Γ2(α+
3
/
2
)
, which a e he asymme y and ku osis coe icien s
o he GR dis ibu ion. Figu e 3depic s plo s o he asymme y and ku osis coe icien s o he
MSGR model. In he igu e, we obse e ha he highes alues o asymme y and ku osis a e
associa ed wi h small alues o pa ame e
q
and la ge alues o pa ame e
α
. In addi ion, when
compa ing he asymme y and ku osis coe icien s o he GR and MSGR dis ibu ions, we obse e
ha
q
has a g ea impac on he inc ease in ku osis, which is why we e e o
q
as a ku osis
pa ame e . This poin is clea ly obse ed in Figu e 4.
Symme y 2021,13, 1226 7 o 18
ALPHA
0
5
10
15
20
q
510 15 20
ASYMMETRY
0
5
10
15
20
ALPHA
0
5
10
15
20
q
5
10
15
20
25
KURTOSIS
0
50
100
Figu e 3. Plo s o he asymme y and ku osis coe icien s o he MSGR model.
0 5 10 15 20
0 2 4 6 8 10
α
Asymme y
MSGR
q=6
q=5
q=4
q=3.5
GR
0 5 10 15 20
0 20 40 60 80
α
Ku osis
MSGR
q=7
q=6
q=5
q=4.5
GR
Figu e 4. Plo s o he asymme y and ku osis coe icien s o he GR and MSGR models.
3. The MSGR Family o Dis ibu ions
In his sec ion, we see ha well-known models a e associa ed wi h he MSGR amily
o dis ibu ions. Some o hese esul s ollow di ec ly om (8) unde ce ain epa ame e i-
za ions, while o he ones ollow aking he limi when
q→∞
. Simila o he echniques
used in [25], he limi dis ibu ion o (1) when q→∞is ob ained.
Lemma 1.
Le
Tq∼MSGR(θ
,
α
,
q)
. Then,
Tq
con e ges in dis ibu ion o
X∼GR(θ
,
α)
when
q→∞.
P oo . Le us conside he s ochas ic ep esen a ion o Tqgi en in Rema k 1:
Tq=X
Y1/q
whe e X∼GR(θ,α)and Y∼Exp(2).
Fi s , we p o e he con e gence in p obabili y o
Y1/q
o a cons an as
q→∞
. Since
Y∼Exp(2)and q>0, we ha e he ollowing:
E[Y1/q] =
Γ1+1
q
21/q,
Va [Y1/q] =
Γ1+2
q−Γ21+2
q
22/q.
Symme y 2021,13, 1226 8 o 18
By applying he Tchebyche inequali y, we ha e he ollowing:
PhY1/q−E[Y1/q]>ei≤Va [Y1/q]
e2∀e>0, (11)
and since Va [Y1/q]→0 as q→∞ hen
{Y1/q−E[Y1/q]}P
−→ 0.
Simila ly, he sequence o eal numbe s beha es as ollows:
E[Y1/q] =
Γ1+1
q
21/q−→ 1, q→∞.
The e o e,
Y1/q=Y1/q−E[Y1/q] + E[Y1/q]P
−→ 1 as q→∞.
By applying Slu sky’s heo em ([26] Co olla y 2.3.2), we ha e he ollowing:
Tq=X
Y1/q
d
−→ Xas q→∞
ha is, Tqcon e ges in dis ibu ion o X∼GR(θ,α)dis ibu ion.
Nex , we see ha well-known models a e pa icula cases o he
MSGR(θ
,
α
,
q)
amily
o dis ibu ions. Speci ically, o
α=
0 and
θ= (
2
σ)−1
, he
MSGR
model is educed o
he modi ied slashed Rayleigh dis ibu ion [
27
]. I
α=−
1
/
2 and
θ= (
2
σ2)−1
, hen he
MSGR
educes o he modi ied slashed hal -no mal dis ibu ion [
28
]. Fo
α=
1
/
2 and
θ=σ/
2, we ge he modi ied slashed Maxwell dis ibu ion. I
α=σ/
2
−
1 and
θ=
1
/
2,
we ob ain he modi ied slashed chi-squa e dis ibu ion. Taking he limi as
q→∞
in
p e ious models, he Rayleigh, hal -no mal [
29
], Maxwell [
30
] and chi-squa e dis ibu ion
a e ob ained [
31
]. De ails abou hese ela ed models a e gi en in he nex co olla ies. The
ela ionships among hese models a e summa ized in Figu e 5.
Co olla y 4
(Modi ied slashed Rayleigh model)
.
Le
T∼MSGR(
1
/(
2
σ)
, 0,
q)
. Then,
T
ollows a modi ied slashed Rayleigh dis ibu ion,
MSR(σ
,
q)
, whose densi y unc ion is gi en by he
ollowing:
T( ,σ,q) = 2q+2
2qσq
2
q+1I q
2+1, 2q+2
2σq
2
q,q
2, 1!, >0,
σ>0is a scale pa ame e , q >0is a ku osis pa ame e [27].
As
q→∞
hen
T
con e ges in dis ibu ion o a Rayleigh dis ibu ion whose pd is he
ollowing:
T( ) =
σe− 2
2σ, >0.
Co olla y 5
(Modi ied slashed hal -no mal model)
.
Le
T∼MSGR(
1
/
2
σ2
, 1
/
2,
q)
. Then,
T
ollows a modi ied slashed hal -no mal dis ibu ion,
MSHN(σ
,
q)
, whose pd is gi en by he
ollowing:
T( ;σ,q) = 2q+2
2qσq
√π q+1I q+1
2,2q+2
2σq
q,q
2, 1!, >0,
whe e σ>0is a scale pa ame e and q >0is a ku osis pa ame e [28].
Symme y 2021,13, 1226 9 o 18
As
q→∞
,
T
con e ges in dis ibu ion o a hal -no mal dis ibu ion whose pd is he ollowing:
T( ) = 2
σφ
σ, >0,
whe e φ(·)deno es he pd o a N(0, 1)dis ibu ion.
Co olla y 6
(Modi ied slashed Maxwell model)
.
Le
T∼MSGR(σ/
2, 1
/
2,
q)
. Then,
T
ollows a modi ied slashed Maxwell dis ibu ion, MSM(σ,q), whose pd is gi en by he ollowing:
T( ;σ,q) = 2
σq
24q
√π q+1I q+3
2,2q
2+1
σq
2 q,q
2, 1!, >0,
whe e σ>0is a scale pa ame e and q >0is a ku osis pa ame e .
As
q→∞
,
T
con e ges in dis ibu ion o a Maxwell dis ibu ion whose pd is he ollowing:
T( ) = 2
πσ3/2 2e−σ
2 2, >0.
Co olla y 7
(Modi ied slashed chi-squa e model)
.
Le
T∼MSGR(
1
/
2,
σ/
2
−
1,
q)
. Then,
T
ollows a modi ied slashed chi-squa e dis ibu ion, MSχ2
σ,q, whose pd is gi en by he ollowing:
T( ;σ,q) = 2q
2+1q
Γσ
2 q+1I σ+q
2,2q
2+1
q,q
2, 1!, >0,
whe e σ>0is a scale pa ame e and q >0is a ku osis pa ame e .
As
q→∞
,
T
con e ges in dis ibu ion o a gamma dis ibu ion wi h shape pa ame e
σ/
2
and a e pa ame e 1/2 as ollows:
T( ) = 1
2σ/2Γ(σ/2) σ/2−1e−
2, >0.
I σis a posi i e in ege , hen we ha e a chi-squa e dis ibu ion.
MSGR(θ,α,q)
θ=1/2;α=σ/2−1

q=1

α=0
α=1/2;θ=1/2σ2

α=1/2;θ=σ/2

q→∞

CMSCR(θ,α, 1)
α=1
**
MSR(θ,q)
q=1
uu
q→∞;θ=(2λ2)−1

MSHN(σ,q)
q→∞
33
MSM(σ,q)
q→∞

GR(θ,α)
α=ν/2−1;θ=1/2

α=−1/2;θ=(2λ2)−1
}}
α=1/2;θ=(2λ2)−1

α=1/2;θ=(2λ2)−1
__
MSχσ,q
q→∞

CMSR(θ, 1)
Rayleigh(λ)Maxwell(λ)Hal -no mal(λ)χ2
Figu e 5. Rela ionships among dis ibu ions in he MSGR amily.
Symme y 2021,13, 1226 16 o 18
Remission ime (Mon hs)
Densi y
0 20 40 60 80
0.00 0.02 0.04 0.06 0.08 0.10
Figu e 8.
Fi ed models by ML me hod o emission ime da a se : MSGR (solid line), SGR (dashed
line), GR (do ed line).
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●
●●
●
0 20 40 60 80
0 20 40 60 80
(a)
Theo ical Quan iles
Sample Quan iles
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●
●●
●
●●●
●●
●
0 20 40 60 80
0 20 40 60 80
(b)
Theo ical Quan iles
Sample Quan iles
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●●
●
●●
●
0 20 40 60 80
0 20 40 60 80
(c)
Theo ical Quan iles
Sample Quan iles
Figu e 9. QQ-plo s: (a) MSGR model, (b) SGR model, (c) GR model.
6. Concluding Rema ks
In his a icle, a new gene aliza ion o he GR dis ibu ion, which is able o model
posi i e da a wi h high ku osis, is in oduced. The p oposed dis ibu ion can be w i en as
he a io o wo independen andom a iables. These a e a GR dis ibu ion and a powe
o he exponen ial dis ibu ion. We de i ed some special cases ha can be seen as wo-
pa ame e ex ensions o he Rayleigh, hal no mal, Maxwell and chi squa e dis ibu ions.
The pa ame e s we e es ima ed by using momen s and maximum likelihood me hods.
The me hod o momen s es ima es we e used o s a he maximum likelihood es ima ion
h ough he New on–Raphson p ocedu e. The asymme y and ku osis coe icien s we e
also ob ained. The s udy o hese coe icien s illus a es he ac ha he MSGR model can
be used o desc ibe posi i e igh -skewed da a wi h high ku osis. Finally, he MSGR model
was i ed o wo eal da a se s. In bo h applica ions, he MSGR dis ibu ion p esen ed
a be e i han he GR and SGR models, which shows he po en ial applicabili y o ou
p oposal. To conclude, we poin ou ha hese models can be used no only in eliabili y
s udies, bu also in he ields o economics, o ins ance, o desc ibe inancial da a, whe e
models wi h hea y ails a e e y in e es ing.
Au ho Con ibu ions:
Concep ualiza ion, I.B.-C., Y.A.I. and Y.M.G.; Fo mal analysis, I.B.-C., Y.A.I.,
Y.M.G. and J.M.A.; In es iga ion, I.B.-C., Y.A.I., Y.M.G., J.M.A. and H.W.G.; Me hodology, I.B.-C.,
Y.A.I., Y.M.G. and H.W.G.; So wa e, Y.A.I. and Y.M.G.; Supe ision, I.B.-C., Y.M.G. and H.W.G.;
Valida ion, I.B.C., Y.A.I. and H.W.G. All o he au ho s con ibu ed signi ican ly o his esea ch a icle.
All au ho s ha e ead and ag eed o he published e sion o he manusc ip .

Symme y 2021,13, 1226 17 o 18
Funding: The esea ch o H.W. Gómez was suppo ed by SEMILLERO UA-2021 p ojec , Chile.
Da a A ailabili y S a emen : The da ase s a e a ailable in he e e ences gi en in Sec ion 5.
Acknowledgmen s:
The au ho s wan s o hank he e iewe s o hei hough ul commen s, which
con ibu e o imp o e he p esen pape .
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1. Vod˘a, V.G. In e en ial p ocedu es on a gene alized Rayleigh a ia e I. Apl. Ma . 1976,21, 395–412.
2.
Siddiqui, M.M. Some p oblems connec ed wi h Rayleigh dis ibu ions. J. Res. Na l. Bu . S and. Sec . D Radio P opag.
1962
,
66, 167–174. [C ossRe ]
3.
Hi ano, K. Rayleigh dis ibu ion. In Encyclopedia o S a is ical Sciences; Ko z, S., Johnson, N.L., Read, C.B., Eds.; Wiley: New Yo k,
NY, USA, 1986; pp. 647–649.
4.
Ab amowi z, M.; S egun, I.E. Handbook o Ma hema ical Func ions wi h Fo mulas, G aphs, and Ma hema ical Tables, 9 h ed.; Na ional
Bu eau o S anda s: Washing on, DC, USA, 1970
5. G adsh eyn, I.S.; Ryzhik, I.M. Table o In eg als, Se ies, and P oduc s; Academic P ess: San Diego, CA, USA, 2000.
6.
Gómez, H.W.; Quin ana, F.A.; To es, F.J. A new amily o slash-dis ibu ions wi h ellip ical con ou s. S a . P obab. Le .
2007
,
77, 717–725. [C ossRe ]
7.
Reyes, J.; Ba anco-Chamo o, I.; Gómez, H.W. Gene alized modi ied slash dis ibu ion wi h applica ions. Commun. S a . Theo y
Me hods 2020,49, 2025–2048. [C ossRe ]
8.
I ia e, Y.A.; Vilca, F.; Va ela, H.; Gómez, H.W. Slashed gene alized Rayleigh dis ibu ion. Commun. S a . Theo y Me hods
2017
,
46, 4686–4699. [C ossRe ]
9.
Olmos, N.M.; Va ela, H.; Gómez, H.W.; Bol a ine, H. An ex ension o he hal -no mal dis ibu ion. S a . Pap.
2012
,53, 875–886.
[C ossRe ]
10. Reyes, J.; Gómez, H.W.; Bol a ine, H. Modi ied slash dis ibu ion. S a is ics 2013,47, 929–941. [C ossRe ]
11. S acy, E. A gene aliza ion o he gamma dis ibu ion. Ann. Ma h. S a . 1962,33, 1187–1192. [C ossRe ]
12.
Rahman, M.S.; Gup a, R.P. Family o ans o med chi-squa e dis ibu ions. Commun. S a . Theo y Me hods
1992
,22, 135–146.
[C ossRe ]
13.
Hwang, L.C. Asymp o ic Non-De iciency o Some P ocedu es o a Pa icula Exponen ial Family Unde Rela i e LINEX Loss.
Seq. Anal. 2014,33, 345–359. [C ossRe ]
14.
Kud ya se , A.; Shes ako , O. Asymp o ically no mal es ima o s o he pa ame e s o he gamma-exponen ial dis ibu ion.
Ma hema ics 2021,9, 273. [C ossRe ]
15.
Reyes, J.; Ba anco-Chamo o, I.; Galla do, D.I.; Gómez, H.W. Gene alized Modi ied Slash Bi nbaum—Saunde s Dis ibu ion.
Symme y 2018,10, 724. [C ossRe ]
16. Zö nig, P. On Gene alized Slash Dis ibu ions: Rep esen a ion by Hype geome ic Func ions. S a s 2019,2, 26. [C ossRe ]
17.
P udniko , A.; B ychko , Y.A.; Ma iche , O.I. In eg als and Se ies, Vols. 1, 2 and 3; Go don and B each Science Publishe s:
Ams e dam, The Ne he lands, 1986.
18.
Casella, G.; Be ge , R.L. S a is ical In e ence, 2nd ed.; Duxbu y Ad anced Se ies; Duxbu y-Thomson Lea ning: Paci ic G o e, CA,
USA, 2002
19. Polo ko, A. Fundamen als o Reliabili y Theo y; Academic P ess: San Diego, CA, USA, 1968.
20. Lawless, J.F. S a is ical Models and Me hods o Li e ime Da a, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2003.
21.
Sa i, A.; Co si, C.; Mazzini, E.; Lambe i, C. Maximum likelihood segmen a ion o ul asound images wi h Rayleigh dis ibu ion.
IEEE T ans. Ul ason. Fe oelec . F eq. Con ol 2005,52, 947–960. [C ossRe ]
22.
Kalaisel i, S.; Logana han, A.; Vijaya agha an, R. Bayesian Reliabili y Sampling Plans unde he Condi ions o Rayleigh-In e se-
Rayleigh Dis ibu ion. Econ. Qual. Con ol 2014,29, 29–38. [C ossRe ]
23.
Dhaundiyal, A.; Singh, S. App oxima ions o he Non-Iso he mal Dis ibu ed Ac i a ion Ene gy Model o Biomass Py olysis
Using he Rayleigh Dis ibu ion. C a Technol. Ag ic. 2017,20, 78–84. [C ossRe ]
24.
Ku uoglu, E.; Ze ubia, J. Modeling SAR Images wi h a Gene aliza ion o he Rayleigh Dis ibu ion. IEEE T ans. Image P ocess.
2004,13, 527–533. [C ossRe ]
25.
Ba anco-Chamo o, I.; Mo eno-Rebollo, J.L.; Pascual-Acos a, A.; Enguix-González, A. An O e iew o Asymp o ic P ope ies o
Es ima o s in T unca ed Dis ibu ions. Commun. S a . Theo y Me hods 2007,36, 2351–2366. [C ossRe ]
26. Lehmann, E. Elemen s o La ge-Sample Theo y; Sp inge : New Yo k, NY, USA, 1999.
27.
I ia e, Y.A.; Cas illo, N.O.; Bol a ine, H.; Gómez, H.W. Modi ied slashed-Rayleigh dis ibu ion. Commun. S a . Theo y Me hods
2018,47, 3220–3233. [C ossRe ]
28.
Olmos, N.M.; Venegas, O.; Gómez, Y.M.; I ia e, Y.A. An Asymme ic Dis ibu ion wi h Hea y Tails and I s Expec a ion-
Maximiza ion (EM) Algo i hm Implemen a ion. Symme y 2019,11, 1150. [C ossRe ]
29. Hogg, R.; Tanis, E. P obabili y and S a is ical In e ence; MacMillan Publishing: New Yo k, NY, USA, 1993.
30.
Bekke , A.; Roux, J. Reliabili y cha ac e is ics o he Maxwell dis ibu ion: A Bayes es ima ion s udy. Commun. S a . Theo y
Me hods 2005,34, 2169–2178. [C ossRe ]
Symme y 2021,13, 1226 18 o 18
31.
Johnson, N.L.; Ko z, S.; Balak ishnan, N. Con inuous Uni a ia e Dis ibu ions, 2nd ed.; John Wiley & Sons Inc.: New Yo k, NY,
USA, 1995; Volume 2
32.
Demps e , A.; Lai d, N.; Rubim, D. Maximum likelihood om incomple e da a ia he EM algo i hm (wi h discussion). J. R. S a .
Soc. Se . B 1977,39, 1–38.
33.
R Co e Team. R: A Language and En i onmen o S a is ical Compu ing; R Founda ion o S a is ical Compu ing: Vienna, Aus ia, 2021.
34. Chhika a, R.; Folks, J. The in e se Gaussian dis ibu ion as a li e ime model. Technome ics 1977,19, 461–468. [C ossRe ]
35. Akaike, H. A new look a he s a is ical model iden i ica ion. IEE T ans. Au om. Con ol 1974,19, 716–723. [C ossRe ]
36. Schwa z, G. Es ima ing he dimension o a model. Ann. S a . 1978,6, 461–464. [C ossRe ]
37. Lee, E.T.; Wang, J. S a is ical Me hods o Su i al Da a Analysis; John Wiley & Sons: Hoboken, NJ, USA, 2003.