A Bimodal Extension of the Beta-Binomial Distribution with Applications

Ci a ion: Reyes, J.; Naje a-Zuloaga, J.;
Lee, D.-J.; A ue, J.; I ia e, Y.A. A
Bimodal Ex ension o he
Be a-Binomial Dis ibu ion wi h
Applica ions. Axioms 2024,13, 662.
h ps://doi.o g/10.3390/
axioms13100662
Academic Edi o : Simeon Reich
Recei ed: 24 Augus 2024
Re ised: 17 Sep embe 2024
Accep ed: 19 Sep embe 2024
Published: 25 Sep embe 2024
Copy igh : © 2024 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/).
axioms
A icle
A Bimodal Ex ension o he Be a-Binomial Dis ibu ion
wi h Applica ions
Jimmy Reyes 1, Josu Naje a-Zuloaga 2, Dae-Jin Lee 3, Jaime A ué 1and Yu i A. I ia e 1,*
1Depa amen o de Es adís ica y Ciencia de Da os, Facul ad de Ciencias Básicas, Uni e sidad de An o agas a,
An o agas a 1270300, Chile; jimmy. [email p o ec ed] (J.R.); [email p o ec ed] (J.A.)
2Depa men o Ma hema ics, Uni e si y o he Basque Coun y UPV/EHU, 48940 Leioa, Spain;
[email p o ec ed]
3School o Science and Technology, IE Uni e si y, 28046 Mad id, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : In his pape , we p opose an al e na i e dis ibu ion o model coun da a exhibi ing
uni/bimodali y. I a ises as a weigh ed e sion o he be a-binomial dis ibu ion, which is de ined by
a pa ame ic weigh unc ion ha admi s up o wo modes o he esul ing p obabili y mass unc ion.
Like he baseline be a-binomial dis ibu ion, he p oposed dis ibu ion pe o ms well in modeling
o e dispe sed binomial da a. S uc u al p ope ies o he new dis ibu ion a e s udied. Raw momen s
a e de i ed, which a e used o desc ibe he dispe sion beha io ela i e o he mean and he skewness
beha io . Pa ame e es ima ion is ca ied ou using he maximum likelihood me hod. A simula ion
s udy is conduc ed in o de o illus a e he beha io o he es ima o s. Finally, wo applica ions
illus a ing he use ulness o he p oposal a e p esen ed.
Keywo ds: be a-binomial dis ibu ion; bimodali y; coun da a; maximum likelihood; momen s;
o e dispe sion
MSC: 62E10; 62F10
1. In oduc ion
Coun da a ep esen s he numbe o imes a pa icula e en occu s in an in e al o
ime, space, o o he uni o measu emen . This ype o da a is commonly ound in a ious
a eas, such as medicine, economics, and enginee ing, o name a ew. Fo example, Böhning
e al.
[1]
analyzed coun da a om a den al epidemiological s udy unde he si ua ion o
addi ional ze os. Salman e al.
[2]
analyzed bank up cy coun da a om Swedish small
manu ac u ing i ms wi h he aim o in es iga ing he business ailu e isk ac o s o small
manu ac u ing i ms. Calab ia e al.
[3]
analyzed he eliabili y o epai able sys ems om
in-se ice ailu e coun da a.
The e a e many eal-wo ld scena ios whe e he p obabili y o success in binomial
expe imen s canno be conside ed cons an . Fo example, he p obabili y o consuming
alcohol ac oss he 7 days o a pa icula week a ies om one indi idual o ano he (see
Alanko and Lemmens
[4]
). Conside ing a be a dis ibu ion o he p obabili y o success in
a binomial dis ibu ion (which gi es ise o he be a-binomial dis ibu ion) is no o e ly
es ic i e since he be a dis ibu ion is e y lexible in e ms o he shapes o i s p obabili y
densi y unc ion.
A andom a iable
X
ollows he be a-binomial dis ibu ion, deno ed
X∼BB(n
,
α
,
β)
,
i i s p obabili y mass unc ion (p.m. .) is gi en by
P(X=x) = n
xB(x+α,n−x+β)
B(α,β),x=0, 1, 2, . . . , n,α,β>0, (1)
Axioms 2024,13, 662. h ps://doi.o g/10.3390/axioms13100662 h ps://www.mdpi.com/jou nal/axioms
Axioms 2024,13, 662 2 o 16
whe e B(a,b) = R1
0ua−1(1−u)b−1du,a,b>0, is he be a unc ion.
In Bayesian in e ence, he be a-binomial dis ibu ion is used o make p edic ions abou
he numbe o successes in u u e ials, aking in o accoun he unce ain y in he es ima e
o he p obabili y o success. In classical in e ence, he be a-binomial dis ibu ion can be
used o model da a wi h o e dispe sion in binomial expe imen s, i.e., when he obse ed
a iabili y is g ea e han ha expec ed unde a s anda d binomial dis ibu ion.
A e iew o he applicabili y and ex ensions o he be a-binomial dis ibu ion can be
ound in Wilcox
[5]
. The use o he be a-binomial dis ibu ion in he con ex o eg ession is
discussed in C owde
[6]
. De ails on he es ima ion o he pa ame e s o he be a-binomial
dis ibu ion can be ound in T ipa hi e al. [7].
Rega ding mo e ecen applica ions o he be a-binomial dis ibu ion, se e al s udies
can be ound in he li e a u e. To name a ew, Palm e al.
[8]
use he be a-binomial dis ibu-
ion in he o mula ion o he BBARMA (Be a-Binomial Au o eg essi e Mo ing A e age)
model, which can cap u e he empo al dynamics and au o eg essi e s uc u e in coun
da a. Chen e al.
[9]
use he be a-binomial dis ibu ion o p opose a GARCH model ha
cap u es he a ia ion in he numbe o new cases o c yp ospo idiosis in ec ion, ob aining
a use ul model o ime se ies da a ha p esen bounded coun s and high ola ili y. Jansen
and Holling
[10]
, unde a Bayesian app oach, use he be a-binomial dis ibu ion in he
me a-analysis o a e e en s.
Al hough he be a-binomial dis ibu ion is applied in a ious eal-wo ld se ings, i s
pe o mance is no good when empi ical dis ibu ions exhibi bimodali y, i.e., when he e
a e wo modes o peaks in he empi ical dis ibu ions. The p esence o bimodali y can
be explained by he exis ence o wo g oups o subpopula ions wi h unique cha ac e is-
ics o by he exis ence o la en a iables ha signi ican ly in luence he dis ibu ion o
he popula ion.
A e y popula me hodology in he li e a u e o inco po a e lexibili y in e ms o
asymme y and mul imodali y is ela ed o he de ini ion o weigh ed dis ibu ions p o-
posed by Fishe
[11]
and Rao
[12]
. Suppose ha
X
is a andom a iable wi h p obabili y
unc ion (x). The weigh ed andom a iable Xwhas PDF
Xw(x) = w(x) (x)
µw, (2)
whe e w(·)is a nonnega i e weigh unc ion and µw=E[W(X)] <∞.
A pa icula ly salien case o (2) is ob ained when
w(x) = x
, which de ines a leng h-
biased dis ibu ion. These dis ibu ions a ise na u ally in applied ields, such as eliabili y
and su i al analysis, when indi iduals o mechanical uni s a e sampled wi h unequal
p obabili y due o he expe imen al design o he exis ing unequal p obabili y o de ec ion.
On he o he hand, i is possible o ind in he li e a u e weigh unc ions ha can
lead o mul imodali y o he weigh ed dis ibu ions esul ing om (2). Fo example, i
w(x) =
1
+1−α(x−µ)
σ2
,
α∈R
, and
(x)
is he pd o he no mal dis ibu ion wi h
mean
µ∈R
and a iance
σ2>
0, hen (2) educes o he amily o bimodal dis ibu ions
called he alpha-skew-no mal dis ibu ion, see Elal-Oli e o
[13]
. Based on he same weigh
unc ion, Gómez-Déniz e al.
[14]
in oduces a bimodal e sion o he Poisson dis ibu ion.
Co és e al. [15]
p opose a pa ame ic weigh unc ion ha in ol es a powe unc ion o
exponen 4, which can lead o a p obabili y unc ion wi h up o h ee modes.
In his pape , we p opose an ex ension o he be a-binomial dis ibu ion app op ia e
o i o e dispe sed binomial da a ha may exhibi bo h unimodali y and bimodali y. The
p oposal a ises om (2), using he weigh unc ion p oposed by Elal-Oli e o
[13]
unde a
be a-binomial baseline dis ibu ion. In his way, he new dis ibu ion is aimed a expanding
he use o be a-binomial dis ibu ions o eal-wo ld scena ios whe e empi ical dis ibu ions
exhibi bimodali y.
The emainde o he pape is o ganized as ollows. In Sec ion 2, we de ine he bimodal
be a-binomial andom a iable and s udy some o i s p ope ies, such as he p obabili y
Axioms 2024,13, 662 3 o 16
mass unc ion, cumula i e dis ibu ion unc ion, and he aw momen s. The la e a e used
o desc ibe he beha io o he ela i e dispe sion wi h espec o he mean and he skewness
beha io o he dis ibu ion. In Sec ion 3, pa ame e es ima ion o he new dis ibu ion
using he maximum likelihood me hod is discussed. A simula ion s udy is ca ied ou
o e alua e he beha io o he es ima o s. In Sec ion 4, wo applica ion examples wi h
eal da a a e p esen ed o illus a e he use ulness o he p oposed dis ibu ion. Finally,
concluding ema ks a e p esen ed in Sec ion 5.
2. Bimodal Be a-Binomial Dis ibu ion
In his sec ion, we de i e he new dis ibu ion and s udy some o i s main p ope ies.
2.1. Bimodal Be a-Binomial Random Va iable
The ollowing p oposi ion p esen s he p.m. . o he new dis ibu ion.
P oposi ion 1. Le X ∼BB(n,α,β)and w(·)be a pa ame ic unc ion gi en by
w(x) = 1+1−q(x−µ)
σ2
,x=0, 1, . . . , n,
whe e
µ=nα
α+βand σ2=nαβ(α+β+n)
(α+β)2(α+β+1),
a e he mean and a iance o X, espec i ely. Then, he p.m. . o he weigh ed andom a iable Xwis
Xw(x;α,β,q) = P(Xw=x)
=1
2+q2(1+1−q(x−µ)
σ2)n
xB(x+α,n−x+β)
B(α,β),x=0, 1, . . . , n, (3)
such ha α,β>0, q ∈Rand B(·,·)is he be a unc ion.
P oo .
Fi s , we obse e ha
Xw(x)>
0 o all
x=
0, 1, 2,
. . .
,
n
when
α
,
β>
0 and
q∈R
.
Second, i can be seen ha
n
∑
x=0
Xw(x) = 1
(2+q2)
n
∑
x=0 1+1−q(x−µ)
σ2! X(x)
=1
2+q2 2−2q
σ
n
∑
x=0
(x−µ) X(x) + q2
σ2
n
∑
x=0
(x−µ)2 X(x)!
=1.
In consequence, i is concluded ha (3) is a alid p.m. .
De ini ion 1. Le
Xw
be a andom a iable wi h p.m. . gi en in (3), hen we say ha
Xw
ollows a
bimodal be a-binomial dis ibu ion. We deno e his as Xw∼BBB(n,α,β,q).
The name gi en in De ini ion 1 o e e o he new dis ibu ion is based on he bimodal
beha io ha he p.m. . can p esen . Figu e 1shows some plo s o he p.m. . o he bimodal
be a-binomial dis ibu ion o di e en alues o i s pa ame e s. In he igu e, i can be
seen ha he BBB p.m. can p esen a g ea a ie y o shapes depending on i s pa ame e s:
mono onic shape, symme ic/asymme ic unimodal shape, ba h ub shape, o asymme ic
bimodal shape.
A unc ion in he R p og amming language [
16
] o compu ing (3) is p o ided
in Appendix A.
Axioms 2024,13, 662 4 o 16
5 10 15 20
0.00 0.10
x
p.m. .
BBB(α = 4, β = 4, q=0)
5 10 15 20
0.00 0.10
x
p.m. .
BBB(α = 4, β = 4, q=3)
5 10 15 20
0.00 0.10
x
p.m. .
BBB(α = 4, β = 4, q= −3)
5 10 15 20
0.00 0.15
x
p.m. .
BBB(α = 1, β = 5, q=0)
5 10 15 20
0.00 0.06
x
p.m. .
BBB(α = 8, β = 1, q=0)
5 10 15 20
0.00 0.10 0.20
x
p.m. .
BBB(α = 3, β = 8, q=0)
Figu e 1. Plo s o he p.m. . o he bimodal be a-binomial dis ibu ion wi h
n=
20 and di e en
alues o α,βand q.
2.2. Two Rela ed Dis ibu ions
Co olla y 1. Le Xw∼BBB(n,α,β,q). Then,
1. Xw(x
;
α
,
β
,
q=
0
) = n
xB(x+α,n−x+β)
B(α,β)
,
x=
0, 1, 2,
. . .
,
n
,
α
,
β>
0, which is he
p.m. . o he be a-binomial dis ibu ion.
Axioms 2024,13, 662 5 o 16
2. I n =1, hen Xw(x;α,β,q) = θx(1−θ)1−x, x =0, 1, such ha
θ=


β
(2+q2)(α+β)h1+1+qpα/β2i,i x =0,
α
(2+q2)(α+β)h1+1+qpβ/α2i,i x =1.
Co olla y 1is a di ec consequence o (3) conside ing ixed alues o
q
and
n
. Pa 1
shows ha he be a-binomial dis ibu ion is a special case o he bimodal be a-binomial
dis ibu ion ob ained when
q=
0. The second pa shows ha he bimodal be a-binomial
dis ibu ion educes o he Be noulli dis ibu ion wi h pa ame e
θ
, whe e
θ
is a unc ion o
he pa ame e s α,β, and q.
2.3. Cumula i e Dis ibu ion Func ion
The cumula i e dis ibu ion unc ion (c.d. .) o
Xw∼BBB(n
,
α
,
β
,
q)
can be ob ained
s aigh o wa dly om P oposi ion 1.
Co olla y 2. Le
Xw∼BBB(n
,
α
,
β
,
q)
. Then, he cumula i e dis ibu ion unc ion (c.d. .) o
Xw
is gi en by
FXw(x) = P(Xw≤x)
=










0, i ⌊x⌋<0,
1
2+q2
⌊x⌋
∑
=0(1+1−q( −µ)
σ2)n
B( +α,n− +β)
B(α,β),i 0≤ ⌊x⌋<n,
1, i ⌊x⌋ ≥ n,
(4)
whe e ⌊x⌋=max{k∈Z|k<x}, x ∈R.
Figu e 2shows some plo s o he c.d. . o he bimodal be a-binomial dis ibu ion
o di e en alues o
α
,
β
, and
q
. As expec ed, he igu e shows ha he equencies
a e no dec easing as
x
inc eases. Howe e , wo sha p inc eases in equency can be
obse ed in wo di e en in e als o
x
, which is explained by he bimodal beha io o he
co esponding p.m. .
A unc ion in he R p og amming language o compu ing (4) is p o ided in Appendix A.
2.4. Momen s
The ollowing p oposi ion de i es he aw momen s o he be a-binomial dis ibu ion.
Essen ially, hese momen s a e exp essed as a unc ion o he aw momen s o he be a-
binomial dis ibu ion.
P oposi ion 2. Le Xw∼BBB(n,α,β,q). Then, he h aw momen o Xwis gi en by
E(Xw )=aµ −bµ +1+cµ +2, =1, 2, . . . , (5)
whe e
a=1
2+q22+2qµ
σ+q2µ2
σ2,b=1
2+q22q
σ+2q2µ
σ2,c=q2
(2+q2)σ2,
such ha
µj=EXj=
n
∑
x=0
xjn
xB(x+α,n−x+β)
B(α,β),j=1, 2, . . .
is he j h aw momen o he be a-binomial dis ibu ion.

Axioms 2024,13, 662 6 o 16
5 10 15 20
0.0 0.4 0.8
x
c.d. .
BBB(α = 4, β = 4, q=0)
5 10 15 20
0.0 0.4 0.8
x
c.d. .
BBB(α = 4, β = 4, q=3)
5 10 15 20
0.0 0.4 0.8
x
c.d. .
BBB(α = 4, β = 4, q= −3)
Figu e 2. Plo s o he c.d. . o he bimodal be a-binomial dis ibu ion wi h
n=
20,
α=
4,
β=
4 and
di e en alues o q.
P oo . By de ini ion o expec a ion, we ha e ha
E(X
w)=
n
∑
x=0
x Xw(x;α,β,q)
=1
2+q2
n
∑
x=0
x (1+1−q(x−µ)
σ2) X(x;α,β), (6)
Axioms 2024,13, 662 7 o 16
whe e
X(x
;
α
,
β)
is he p.m. . o he be a-binomial dis ibu ion. The e o e, a e some
algeb a, we see ha
E(X
w)=1
2+q22+2qµ
σ+q2µ2
σ2n
∑
x=0
x X(x;α,β)−1
2+q22q
σ+2q2µ
σ2n
∑
x=0
x +1 X(x;α,β)
+q2
(2+q2)σ2
n
∑
x=0
x +2 X(x;α,β),
and he esul is ob ained by ecognizing he aw momen s o he be a-binomial dis ibu ion
in he abo e exp ession.
Al e na i ely, in (6) we can w i e
h1−q(x−µ)
σi2=c21−q
σcx2
, whe e
c=
1
+qµ
σ
.
Then, using he binomial heo em, we ha e
1−q(x−µ)
σ2
=c22
∑
k=0
kxk, wi h k=qk
σkck.
Thus, we can w i e (7) as
E(Xw )=1
2+q2"n
∑
x=0
x X(x;α,β) + c22
∑
k=0
k∑
x=0
x +k X(x;α,β)#
=1
2+q2 µ +c22
∑
k=0
kµ +k!, =1, 2, . . . ,
whe e µ is he h aw momen o he be a-binomial dis ibu ion.
Co olla y 3. Le
Xw∼BBB(n
,
α
,
β
,
q)
. Then, he coe icien o a ia ion (
c
.
.
(Xw)
) and he
Fishe ’s skewness coe icien (pβ) o Xwa e gi en by
c. .(Xw) = qµ2−bµ3+cµ4−(aµ1−bµ2+cµ3)2
aµ1−bµ2+cµ4
and
pβ=aµ3−bµ4+cµ5−3(aµ1−bµ2+cµ3)(aµ2−bµ3+cµ4)+2(aµ1−bµ2+cµ3)3
haµ2−bµ3+cµ4−(aµ1−bµ2+cµ3)2i3/2 ,
whe e
µ1=Γ(α+1)Γ(α+β)n
Γ(α+β+1)Γ(α),
µ2=Γ(α+1)Γ(α+β)n(nα+n+β)
(α+β+1)Γ(α+β+1)Γ(α),
µ3=Γ(α+1)Γ(α+β)n3αβn+β2+3nβ+3n2α+n2α2+2n2−αβ
(2+α2+3α+2αβ +β2+3β)Γ(α+β+1)Γ(α),
µ4=Γ(α+1)Γ(α+β)A n
(6+α3+12αβ +6β2+6α2+3αβ2+3βα2+11α+11β+β3)Γ(α+β+1)Γ(α),
µ5=Γ(α+1)Γ(α+β)B n
(α+β+3)(α+β+2)(α+β+1)(α+β+4)Γ(α+β+1)Γ(α).
such ha a, b and c a e as in P oposi ion 2and
Axioms 2024,13, 662 8 o 16
A=−αβ +6n3−4αβ2+7nβ2+n3α3+12n2β+β3+18αn2β+7β2nα−4βnα2−5βnα+βα2
+6βn2α2−nβ−β2+6n3α2+11αn3,
B=15β3n+50n2β2+10n4α3+60n3β+50n4α−15nβ2+35n4α2+n4α4−30β2α2n−35αn2β
+25β2n2α2+15β3nα−10βα3n2+5βα3n+10βn3α3−45β2nα−11αβ3−5βnα+5βα2
−35βn2α2+60n3α2β−α3β−10n2β+75n2αβ2+110n3αβ +β4+11β2α2+24n4−5β3.
Figu e 3shows some cu es o he coe icien o a ia ion and he coe icien o
skewness o he bimodal be a-binomial dis ibu ion as a unc ion o
q
unde ixed alues o
α
and
β
. In he igu e, i can be seen ha he bimodal be a-binomial dis ibu ion (depending
on
q
) can p esen a g ea e o lesse ela i e dispe sion (and a g ea e o lesse skewness
le el) han he be a-binomial dis ibu ion (special case q=0).
−4 −2 0 2 4
0.2 0.4 0.6 0.8 1.0
q
Coe icien o a ia ion
α = 2, β = 2
α = 4, β = 2
α = 2, β = 4
−10 −5 0 5 10
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
q
Skewness
α = 2, β = 2
α = 4, β = 2
α = 2, β = 4
Figu e 3. Plo s o he coe icien o a ia ion and he skewness coe icien (as a unc ion o
q
) o he
bimodal be a-binomial dis ibu ion wi h n=20 and di e en alues o αand β.
Func ions in he R p og amming language o compu ing he
h momen (7) and o
he coe icien s o a ia ion and he coe icien o skewness o Co olla y 3a e p o ided
in Appendix A.
3. Pa ame e Es ima ion
In his sec ion, we discuss he maximum likelihood es ima o and conduc a simula ion
s udy o e alua e he pe o mance o he es ima o s.
3.1. Maximum Likelihood Es ima ion
Gi en a andom sample
X1
,
. . .
,
Xm
o he andom a iable
Xw∼BBB(n
,
α
,
β
,
q)
, he
log-likelihood unc ion o θ= (α,β,q)can be w i en as
Axioms 2024,13, 662 9 o 16
ℓ(θ;xi) = log
m
∏
i=1
Xw(xi;α,β,q)
=c+
m
∑
i=1
log n
xi+
m
∑
i=1
log(x1i) +
m
∑
i=1
log Γ(xi+α) +
m
∑
i=1
log Γ(n−xi+β), (7)
whe e
c=−mlog2+q2+mlog Γ(α+β)−mlog Γ(α)−mlog Γ(β)−mlog Γ(α+β+n)
,
x1i=1+[1−q(xi−µ)/σ]2and Γ(a) = R∞
0ua−1e−udu,a>0, is he gamma unc ion.
Then, he sco e unc ions a e gi en by
∂ℓ(θ;xi)
∂α =c1−2q1+qµ
σm
∑
i=1
x2i
x1i
+2q2
σ
m
∑
i=1
xix2i
x1i
+
m
∑
i=1
Ψ(xi+α), (8)
∂ℓ(θ;xi)
∂β =c2−2q1+qµ
σm
∑
i=1
x3i
x1i
+2q2
σ
m
∑
i=1
xix3i
x1i
+
m
∑
i=1
Ψ(n−xi+β), (9)
∂ℓ(θ;xi)
∂q=c3+2µ
σ21−q
σm
∑
i=1
1
x1i
−2
σ1−qµ
σ+q
σm
∑
i=1
xi
x1i
+2q
σ2
m
∑
i=1
x2
i
x1i
, (10)
whe e
c1=−mΨ(α) + mΨ(α+β)−mΨ(α+β+n)
,
c2=−mΨ(β) + mΨ(α+β)−mΨ(α+
β+n),c3=−2mq/(2+q2),Ψ(a) = ∂log Γ(a)/∂a, wi h a>0, is he digamma unc ion,
x2i=∂
∂αxi−µ
σ=−k1i
2hnα3β(α+β+1)(α+β+n)3i−1/2 and
x2i=∂
∂β xi−µ
σ=k2i
2hnα3β(α+β+1)(α+β+n)3i−1/2,
such ha
k1i=−α3xi+α3n−
2
xiα2n−xiα2β+
2
n2α2+
2
nα2β+n2αβ +nαβ2+αn2+
nαβ −αxin−xiαnβ+xiαβ2+αxiβ+xinβ+xiβ2+xiβ3+xiβ2n
and
k2i=−α3xi+α3n+
n2α2−xiα2n−α2xi+nα2+
2
nα2β−xiα2β+nαβ2−αxiβ−αxin+
2
nαβ +αn2+xiαnβ+
xiαβ2+xinβ+2xiβ2n+xiβ3.
Maximum likelihood (ML) es ima o
ˆ
θ= (ˆ
α
,
ˆ
β
,
ˆ
q)
o
θ= (α
,
β
,
q)
can be ob ained
by se ing (8)–(10) equal o ze o and sol ing he esul ing sys em o equa ions. Howe e ,
due o he analy ical complexi y o hese equa ions, es ima es mus be ob ained using
nume ical me hods.
The s anda d e o s o he ML es ima o s can be ob ained as he squa e oo s o he
elemen s o he diagonal o he ma ix
K−1(ˆ
θ) = −∂2ℓ(θ;xi)
∂θ∂θTθ=ˆ
θ−1
,
whe e ∂ℓ(θ;xi)/∂θ∂θTis he hessian ma ix.
Al e na i ely, ML es ima es can be ob ained by sol ing he op imiza ion p oblem
maxθℓ(θ
;
xi)
, subjec o
α>
0,
β>
0 and
q∈R
, whe e
ℓ(θ
;
xi)
is as in (7). Fo his,
we ecommend he use o he unc ion
s a :op im()
o he R p og amming language,
which also e u ns he nume ic Hessian unc ion. In pa icula , we conside he L-BFGS-B
me hod [
17
], which allows he imposi ion o box cons ain s on he pa ame e s. This means
ha i is possible o speci y lowe and uppe bounds o each pa ame e , which is e y
aluable in op imiza ion p oblems wi h high dimensions and speci ic cons ain s.
An R unc ion o compu ing (7) is p o ided in Appendix A.
Axioms 2024,13, 662 16 o 16
19.
Manoj, C.; Wijekoon, P.; Yapa, R.D. The McDonald gene alized be a-binomial dis ibu ion: A new binomial mix u e dis ibu ion
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22.
Ameijei as-Alonso, J.; C ujei as, R.M.; Rod íguez-Casal, A. Mode es ing, c i ical bandwid h and excess mass. Tes 2019,28, 900–919.
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Ameijei as-Alonso, J.; C ujei as, R.M.; Rod iguez-Casal, A. Mul imode: An R package o mode assessmen . a Xi 2018,
a Xi :1803.00472. [C ossRe ]
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25.
Rod íguez-A i, J.; Conde-Sánchez, A.; Sáez-Cas illo, A.; Olmo-Jiménez, M. A gene aliza ion o he be a–binomial dis ibu ion.
J. R. S a . Soc. Se . C Appl. S a . 2007,56, 51–61. [C ossRe ]
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people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .