1
The e sion o Reco d o his manusc ip has been published and is a ailable in Jou nal
o Applied S a is ics. Published online: 07 No 2023 DOI:
h ps://doi.o g/10.1080/02664763.2023.2277125
SMOOTHING LEVEL SELECTION FOR DENSITY ESTIMATORS BASED ON
THE MOMENTS
Rosa Ma ía Ga cía-Fe nándeza Depa men o Quan i a i e Me hods o Economics and
Business. Uni e si y o G anada
ORCID 0000-0002-0642-5216
Fede ico Palacios-González . Uni e si y o G anada
aRosa Ma ía Ga cía-Fe nández (co esponding au o )
Depa men o Quan i a i e Me hods o Economics and Business
Uni e si y o G anada
Facul y o Economics and Business Sciences, o ice C-112
Campus Uni e si a io de Ca uja, 18071 G anada. Spain.
Phone: 34 958248789.
E-mail: [email p o ec ed]
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ABSTRACT
This pape in oduces an app oach o selec he bandwid h o smoo hing pa ame e
in mul i esolu ion (MR) densi y es ima ion and nonpa ame ic densi y es ima ion.
I is based on he e olu ion o he second, hi d and ou h cen al momen s and he
shape o he es ima ed densi ies o di e en bandwid hs and esolu ion le els.
The p oposed me hod has been applied o densi y es ima ion by means o
mul i esolu ion densi ies as well as ke nel densi y es ima ion (MRDE and KDE
espec i ely). The esul s o he simula ions and he empi ical applica ion
demons a e ha he le el o esolu ion esul ing om he momen s me hod
pe o ms be e wi h mul imodal densi ies han he Bayesian In o ma ion C i e ion
(BIC) o mul i esolu ion densi ies es ima ion and he plug-in o ke nel densi ies
es ima ion.
KEYWORDS: Mul i esolu ion densi y es ima ion, ke nel densi y es ima ion,
bandwid h and le el o esolu ion
1.INTRODUCTION
This pape de elops a no el and s aigh o wa d app oach o selec he bandwid h
o smoo hing pa ame e in mul i esolu ion models
1
es ima ion and nonpa ame ic densi y
es ima ion. I is based on he momen s and he shape o he es ima ed densi ies o
di e en esolu ion le els and bandwid hs. The me hod is applied o densi y es ima ion
by means o mul i esolu ion densi ies [MRDE; see 12, 13 14] as well as ke nel unc ions
[KDE; see o ins ance, 18, 5].
Ou choice o conside ing he momen s o he selec ion o he smoo hing pa ame e was
guided by he obse ed changes in he shape o he es ima ed MRDE when he le el o
esolu ion, 𝑗 , a ies. I he le el o esolu ion is oo low, he densi y is smoo h bu he
bias is la ge. Con e sely, a la ge 𝑗 leads o a oughe densi y bu a small bias. Simila
esul s a e obse ed
2
when a densi y is i ed by he Ke nel me hod. Le us suppose a
bandwid h equals o ℎ=2−𝑗 wi h 𝑗∈𝑍. I 𝑗 is oo small hen ℎ is oo la ge and he esul
1
2
.
3
is a smoo h and biased densi y. As 𝑗 inc eases ℎ dec eases and he bias ends o diminish,
bu om a de e mined alue o 𝑗, o i s co esponding ℎ, an undesi able oughness
appea s. To sol e he p oblem, we need o selec a alue o 𝑗 , o i s co esponding
smoo hing pa ame e ℎ, so ha he bias will be easonably small wi hou incu ing an
excessi e oughness.
In bo h me hodologies, MRDE and KDE, he bias o he es ima ed densi y is
shown clea ly by i s dispe sion and shape, especially in he la ening o he i ed densi y.
O equi alen ly in an unde es ima ion o he ku osis ha e ol es owa ds mo e
easonable alues as he esolu ion le el 𝑗 inc eases. Tha is, he bias e olu ion is ela ed
o he cen al momen s o o de 2, 3 and 4 since hey a e used o compu e dispe sion,
asymme y and ku osis. When he esolu ion le el in he MRDE inc eases, o he
smoo hing pa ame e in he KDE dec eases, he la ening and he dispe sion ends o
s abilize indica ing ha he bias is small. F om a ce ain le el o esolu ion, he oughness
begins o inc ease indica ing which alue o 𝑗 o ℎ should be selec ed. This alue leads
o a su icien ly smoo h densi y wi h small bias. This is clea ly shown in he g aphs o
sec ions 3, 4 and 5 ha ep esen he e olu ion, as a unc ion o 𝑗, o he expec ed alue,
a iance, and skewness and ku osis coe icien s o MRDE and KDE.
The es o he pape is o ganized as ollows. Sec ion 2 in oduces he ma h
exp ession o calcula e he momen s o a mul i esolu ion densi y. Sec ion 3 shows, by
means o simula ions, how o selec he le el o esolu ion using he momen s o a MRDE.
Sec ion 4 ex ends he app oach o he Ke nel me hod. In bo h es ima ion me hods, we use
he Cubic Box Spline unc ion de ined in sec ion 2.1. Fo he MRDE me hod, his is he
scaling unc ion gene a ing he mul i esolu ion analysis s uc u e ha con ains he MRDE
and hei es ima es. In he KDE me hod his unc ion is used as he ke nel. Sec ion 5
con ains an applica ion o eal da a and sec ion 6 concludes.
Finally, we wan o poin ou ha he MRDE is a echnique de ised o massi e da a.
The e o e, e e y hing ha ollows mus be unde s ood in a con ex o la ge sample sizes.
2. MOMENTS CALCULATION FOR A MRDE
This sec ion shows he exp essions o he cen al and non-cen al momen s o a
mul i esolu ion densi y. The second, hi d and ou h cen al momen s will be used o
selec he le el o esolu ion o a MRDE. The ma h de elopmen is con ained in he
appendices A1-A6.
4
2.1. MULTIRESOLUTION DENSITIES
Le 𝜃(𝑥) be a symme ic densi y wi h mean ze o and compac suppo [−2,2],
known as Cubic Box Spline. I is gi en by
𝜃(𝑥)=
{
0 𝑖𝑓 𝑥≤−2
𝑝1(2+𝑥)𝑖𝑓 −2≤𝑥≤−1
𝑝2(2+𝑥)𝑖𝑓 −1≤𝑥≤ 0
𝑝2(2−𝑥)𝑖𝑓 0 ≤𝑥≤ 1
𝑝1(2−𝑥)𝑖𝑓 1≤ 𝑥≤ 2
0𝑖𝑓 𝑥≥ 2
,
whe e: 𝑝1(𝑥)=𝑥3
6 and whe e 𝑝2(𝑥)=−𝑥3
2+2𝑥2−2x+2
3 . (1)
Applying dila ions and ansla ions o he densi y 𝜃(𝑥), he densi ies 𝜆𝑗,𝑘(𝑥) a e buil as
ollows [see 12]: 𝜆𝑗,𝑘(𝑥)=𝑠𝜃(𝑠𝑥−𝑘), (2 )
whe e 𝑠=2𝑗. No e ha 𝜆0,0(𝑥)≡𝜃(𝑥).
Fo each le el o esolu ion 𝑗, he ollowing MR densi ies:
𝑓(𝑥)=∑𝑎𝑘
𝑘∈ℤ 𝜆𝑗,𝑘(𝑥), (3)
a e de ined.
In exp ession (3) 𝑎𝑘≥0 ∀𝑘∈ℤ and ∑𝑎𝑘𝑘∈ℤ =1. By de ini ion, all hese
unc ions belong o he 𝑉𝑗 space o he mul i esolu ion analysis s uc u e (MRA) de ined
by he scaling unc ion 𝜃(𝑥) [see 7, 23, 11]. Any densi y o squa ed in eg able, belonging
o he space o Hilbe 𝐿2(ℝ), has an app oxima ion in each 𝑉𝑗.
In he de ini ion o a mul i esolu ion s uc u e, dila ions and ansla ions o 𝜃 gi en by
𝜃𝑗,𝑘(𝑥)=𝑠12
⁄𝜃(𝑠𝑥−𝑘) a e used. I is ob ious ha 𝜆𝑗,𝑘(𝑥)=𝑠12
⁄𝜃𝑗,𝑘(𝑥) and using his
no a ion he MR densi y de ined by (3) can be ew i en as ollows:
𝑓(𝑥)=∑𝑎𝑘
𝑘∈ℤ 𝜆𝑗,𝑘(𝑥)=𝑠12
⁄∑𝑎𝑘
𝑘∈ℤ 𝜃𝑗,𝑘(𝑥) .
2.2 MOMENTS OF A RANDOM VARIABLE WITH MR DENSITY FUNCTION
The cen al momen o o de 𝑟 o a mul i esolu ion (MR) densi y is calcula ed as
ollows:
𝜇𝑓(𝑟)=1
𝑠𝑟∑(𝑟𝑖)𝜇0,0(𝑖)𝜇𝑎(𝑟−𝑖)
𝑟
𝑖=0 ,
5
whe e 𝜇𝑎(𝑟−𝑖)=∑𝑎𝑘(𝑘−𝑘)𝑟−𝑖
𝑘∈ℤ , 𝑘=∑𝑘×𝑎𝑘𝑘∈ℤ . By de ini ion, he exp ession
𝜇0,0(𝑟)=𝜇𝜃(𝑟) is he momen o o de 𝑟 o he densi y 𝜆0,0(𝑥)≡𝜃(𝑥) (see appendices
A1 and A2). Tha is:
𝜇0,0(𝑟)=𝜇𝜃(𝑟)=∫(𝑥−0)𝑟𝜃(𝑥)𝑑𝑥
2
−2 =∫𝑥𝑟𝜃(𝑥)𝑑𝑥
2
−2 ≡𝑚𝜃(𝑟).
The calculus o 𝜇𝜃(𝑟) is in p oposi ion 4 in appendix A1.
Gi en a MR densi y 𝑓𝑗(𝑥) he non-cen al momen o o de 𝑟 is de ined by:
𝑚𝑗(𝑟)= ∫𝑥𝑟𝑓𝑗(𝑥)𝑑𝑥
+∞
−∞ .
I can be ob ained as ollows (see appendix A4):
𝑚𝑗(𝑟)=∑𝑎𝑘
𝑘∈𝑍 𝑚𝜆𝑗,𝑘(𝑟),
whe e 𝑚𝜆𝑗,𝑘(𝑟)=∫𝑥𝑟𝜆𝑗,𝑘(𝑥)𝑑𝑥
+∞
−∞ =1
𝑠𝑟∑(𝑟𝑖)𝑚𝜃(𝑖)𝑘𝑟−𝑖
𝑟𝑖=0 .
2.3 ASYMPTOTIC PROPERTIES OF THE MOMENTS OF AN ESTIMATED MR
DENSITY
PROPOSITION 1
Gi en a sample 𝑥𝑖 𝑖=1,2,…,𝑛 and an es ima ed MR densi y o ha sample:
𝑓󰆹𝑗(𝑥)=∑𝑎𝑘𝜆𝑗,𝑘(𝑥)
𝑘∈𝑍 ,
o a alue o 𝑗 la ge enough is e i ied:
𝑓󰆹𝑗(𝑥)=1𝑛∑𝜆𝑗,𝑘(𝑥𝑖)(𝑥)
𝑛
𝑖=1 ,
whe e
𝑘(𝑥𝑖)={ max
𝑡∈𝑍≤𝑠𝑥𝑖𝑡𝑖𝑓 𝑠𝑥𝑖− max
𝑡∈𝑍≤𝑠𝑥𝑖𝑡≤0.5
max
𝑡∈𝑍≤𝑠𝑥𝑖𝑡+1 𝑖𝑓 𝑠𝑥𝑖− max
𝑡∈𝑍≤𝑠𝑥𝑖𝑡>0.5 .
P oo . See Appendix A6.
PROPOSITION 2
Le be
𝑚𝑟(𝑗)= ∫𝑥𝑟𝑓󰆹𝑗(𝑥)
+∞
−∞ 𝑑𝑥,

6
he non-cen al momen o he es ima ed MR densi y om he sample 𝑥𝑖 𝑖=1,2,…,𝑛.
I 𝑗 con e ges o in ini y, hen 𝑚𝑟(𝑗) con e ges o i s sample coun e pa . Tha is:
lim
𝑗→∞𝑚𝑟(𝑗)=1𝑛∑𝑥𝑖𝑟
𝑛
𝑖=1 .
P oo . See Appendix A6
Since each cen al momen o o de 𝑟 is a con inuous (polynomic) unc ion o he
non-cen al momen s o o de less han o equal o 𝑟, we can s a e ha as 𝑗 app oaches o
in ini y he cen al momen s o a MR densi y es ima ed using a sample o size 𝑛 also
con e ge o he cen al momen s o he sample.
3. MOMENTS METHOD FOR SELECTING THE RESOLUTION LEVEL TO
ESTIMATE A MR DENSITY
In his sec ion we in oduce an al e na i e me hod o he Bayesian In o ma ion
C i e ion [17] o selec he le el o esolu ion in he es ima ion p ocess o a MR densi y.
I is based on he cen al momen s o o de s wo, hi d, and ou and he symme y and
ku osis coe icien s. When a MR densi y is es ima ed, i he esolu ion le el is oo low,
he es ima e is a smoo h cu e bu i has excessi e bias. Con e sely, i he esolu ion le el
is oo high he es ima e bias is small bu he oughness is la ge. In p ac ice, he bias is
mainly shown in an excessi e dispe sion and a la densi y. Since he la ening can be
measu ed by he Fishe coe icien , he e olu ion o he bias, as he esolu ion le el
inc eases, should be e lec ed in he g adual dec ease o he cen al momen o o de 2
and he cen al momen o o de 4. Based on hese momen s, we will choose an
app op ia e esolu ion le el so ha he bias will be accep able and he oughness o he
es ima o will be no excessi e.
Since we a e going o es ablish compa isons wi h he BIC c i e ion, le us
in oduce i b ie ly in he con ex o he MR densi ies. Any es ima ion using a MR
densi y, o a ini e size sample 𝑛 and any esolu ion le el 𝑗, can be conside ed a ini e
mix u e o densi ies 𝜆𝑗,𝑘(𝑥) (see sec ion 2.1) wi h he o m:
𝑓󰆹𝑗(𝑥)=∑𝑎𝑘𝑖
𝑝
𝑖=1 𝜆𝑗,𝑘𝑖(𝑥) , ( 4)
whe e 𝑎𝑘𝑖 is he p opo ion o da a wi hin he in e al (𝑘𝑖−0.5
2𝑗,𝑘𝑖+0.5
2𝑗].
7
No e ha he exp ession (4), es ima o o (3), has a ini e numbe o addends while
(3) has in ini e addends. This is explained as ollows. Depending on he le el o
esolu ion, wo ex eme si ua ions can a ise. Fi s ly, 𝑗 can be so small ha he en i e
sample will be wi hin a single in e al(𝑘−0.5
2𝑗,𝑘+0.5
2𝑗]. In his case, he e is only one
coe icien 𝑎𝑘 dis inc om ze o and mix u e (4) degene a es in o a single addend.
Secondly, 𝑗 can be so la ge ha each obse ed alue will be in a di e en in e al exis ing
as many 𝑎𝑘𝑖 di e en om ze o as di e en alues a e obse ed in he sample. Tha is, i
𝑛 is he sample size and 𝑝 is he numbe o di e en alues obse ed in he sample, hen
he numbe o addends in he mix u e (4) is 𝑝 and i is e i ied ha 1≤𝑝≤𝑛.
Ob iously, 𝑝=𝑛 i all he sample alues a e di e en and each o hem belongs o a
single in e al o he o m
3
(𝑘𝑖−0.5
2𝑗,𝑘𝑖+0.5
2𝑗] 𝑘𝑖∈𝑍.
Since mix u e (4) con ains 𝑝 posi ion pa ame e s 𝑘𝑖 𝑖=1,2,…,𝑝 and 𝑝 mix u e
pa ame e s 𝑎𝑘𝑖 𝑖=1,2,…,𝑝 he e a e 𝑚=2∗𝑝 pa ame e s. We can op imize he
numbe o pa ame e s, which depends on 𝑗, by using he BIC c i e ion [17]. Tha is, we
will conside ha he bes alue is ha which minimizes he exp ession:
𝐵𝐼𝐶(𝑗)=−2𝐿𝑜𝑔(𝐿𝑗)+𝑚𝐿𝑜𝑔(𝑛),
whe e
𝐿𝑗=∏𝑓󰆹𝑗(𝑥𝑖),
𝑛
𝑖=1
is he sample likelihood o 𝑓󰆹𝑗(𝑥).
The p oposed me hod based on he momen s, is simple and equi es less p ocess
ime han he BIC. Howe e , bo h c i e ia complemen and ein o ce each o he , as will
be shown in he ollowing simula ions.
To p oceed, we ha e simula ed a sample o size 10000 by using wo gene a o
models: a no mal dis ibu ion and a mix u e o double exponen ial dis ibu ions. We ha e
i ed hese gene a o models using MR densi ies o di e en le els o esolu ion in each
case. Finally, we ha e calcula ed 𝐸𝑗[𝑋],𝜇𝑗(2),𝛾1=𝜇𝑗(3)
𝜇𝑗(2)3
2 (Fishe coe icien o
skewness o asymme y) and 𝛾2=𝜇𝑗(4)
𝜇𝑗(2)2−3 (Fishe coe icien o ku osis) o selec an
3
𝑗
8
app op ia e 𝑗. In he supplemen al ma e ial, we p o ide mo e de ails abou he calculus
and a mac o o apply he de eloped me hodology o he da a gene a ing models used in
his pape .
3.1. No mal dis ibu ion
Table 3.1 and Figu e 3.1 shows he alues 𝐸𝑗[𝑋],𝜇𝑗(2),𝛾1 and 𝛾2 o a MR
es ima ion o a 𝑁 (10,5) dis ibu ion. Each alue is di ided by i s empi ical o sample
coun e pa s, which a e compu ed om he sample da a wi hou i ing any densi y.
These indica o s s abilize o 𝑗 =0 (see Figu e 3.1). The BIC c i e ion p o ides he alue
𝑗 = −1. Figu e 3.2 displays he i ed densi ies (𝑓󰆹(𝑥)) o 𝑗=−1 and 𝑗=0, and he
da a gene a o model (𝑁(10,5); 𝑓(𝑥)).
Table 3.1. Ra ios be ween MRDE momen s and sample momen s
𝑗
E[X]
Va iance
Asymme y
Ku osis
-3
1.0014
2.0494
0.5553
1.987432
-2
1.0058
1.2919
-0.6275
-3.905073
-1
1.0011
1.0717
0.0905
-2.285609
0
1.0000
1.0160
1.0618
1.212460
1
0.9998
1.0033
1.0149
0.887280
2
1.0000
1.0010
1.0179
1.072510
3
1.0000
1.0006
0.9854
1.023095
4
1.0000
1.0001
0.9973
0.985362
Fig. 3.1. Ra ios be ween MRDE momen s and sample momen s
9
Fig. 3.2 Es ima ed densi ies 𝑓󰆹(𝑥) o 𝑗=−1, 𝑗=0 and da a gene a o model 𝑓(𝑥)
The es ima e o 𝑗=−1 is smoo h bu he bias is no iceable. Fo 𝑗=0 he bias
almos disappea s bu he e is a small oughness ha may be accep able. Acco ding o he
BIC c i e ion, he op imum le el o esolu ion is 𝑗=−1.
3.2. Mix u e o double exponen ial dis ibu ion
The densi y unc ion o a double exponen ial dis ibu ion wi h pa ame e s 𝜇 and
𝜃 is gi en by: 𝑓(𝑥)=𝑒−|𝑥−𝜇|
𝜃 ∀𝑥∈ℝ.
16
Fig. 4.3 Es ima ed densi ies 𝑓󰆹(𝑥) o 𝑗=−1, 𝑗=0 and da a gene a o model 𝑓(𝑥)
No e ha 𝑗 is con enien ly he same o bo h es ima es and i can be ob ained
om he MR o ke nel es ima ed momen s. KDE and MRDE es ima es a e simila since
bo h a e good app oxima ions o he same unknown densi y.
In he ollowing simula ion, he gene a o model is a mix u e o h ee double
exponen ial dis ibu ions whose pa ame e s a e in Table 3.2. Table 4.2 and Figu e 4.4
display he alues o 𝐸𝑗[𝑋],𝜇𝑗(2),𝛾1 , 𝛾2, di ided by hei empi ical coun e pa s,
using le els o esolu ion om 𝑗=−3 o 𝑗=4. As can be seen, s abili y is eached
ei he when 𝑗=−1 o 𝑗=0. Figu e 4.5 shows he es ima ions, 𝑓󰆹(𝑥)) o 𝑗=−1, 𝑗=0,
and he da a gene a o model (mix u e o double exponen ial dis ibu ions, 𝑓(𝑥)).
Table 4.2 Ra ios be ween KDE momen s and sample momen s
𝑗
E[x]
Va iance
Asymme y
Ku osis
-3
1.00020842
1.23072002
0.64137168
0.56500524
-2
1.0000844
1.05642867
0.92675066
0.90709002
-1
1.00058047
1.0155786
0.97057506
0.97393645
0
0.99986737
1.00355414
0.99758049
0.9933315
1
0.99996728
1.000518
0.99559745
0.99654088
2
0.99999484
1.00019494
1.00076368
1.00119058
3
1.00000043
1.00001854
0.99983838
1.00028379
4
1.00000388
1.00003893
0.99975579
0.99954511

17
Fig. 4.4. Ra ios be ween KDE momen s and sample momen s
Fig. 4.5 Es ima ed densi ies 𝑓󰆹(𝑥) o 𝑗=−1, 𝑗=0 and da a gene a o model 𝑓(𝑥)
An al e na i e way o compa e KDE and MRDE is making 2𝑗=1ℎ
⁄ in (10). Tha
is:
𝑓󰆹𝑗(𝑥)=1𝑛∑𝜆𝑗,𝑘(𝑥𝑖)(𝑥)
𝑛
𝑖=1 =1𝑛∑2𝑗𝜃(2𝑗𝑥−𝑘(𝑥𝑖))
𝑛
𝑖=1 =1
𝑛ℎ∑𝜃(𝑥ℎ−𝑘(𝑥𝑖))
𝑛
𝑖=1
=1
𝑛ℎ∑𝜃(𝑥−ℎ𝑘(𝑥𝑖)
ℎ)
𝑛
𝑖=1 . (11)
Obse e ha (5) and (11) a e qui e simila . Taking in o accoun ha by de ini ion:
18
𝑘(𝑥𝑖)−0.5<2𝑗𝑥𝑖≤𝑘(𝑥𝑖)+0.5,
and mul iplying by ℎ=2−𝑗 he h ee e ms o he abo e inequali y we ha e:
ℎ𝑘(𝑥𝑖)−ℎ0.5<𝑥𝑖≤ℎ𝑘(𝑥𝑖)+ℎ0.5.
The la e exp ession shows an inc easing app oxima ion be ween 𝑥𝑖 and
ℎ𝑘(𝑥𝑖). No e ha he ampli ude o he p e ious in e al is ℎ .
lim
ℎ→0ℎ𝑘(𝑥𝑖)=lim
𝑗→∞𝑘(𝑥𝑖)
2𝑗=𝑥𝑖.
The MRDE and he KDE had been compa ed in e ms o ime needed o un hei
densi y unc ion in [12]. None heless, he p e ious simula ions e eal some ac s ha a e
wo h highligh ing. The MR densi y is no a pa icula case o ke nel densi y. On he one
hand, when he mul i esolu ion densi ies a e es ima ed acco ding o (4), he esul s a e
simila o a modi ied ke nel in which each sample da a 𝑥𝑖 is subs i u ed in (5) by ℎ𝑘(𝑥𝑖)
o ob ain (6), wi h ℎ=2−𝑗. On he o he hand, we canno s a e ha a ke nel es ima o is
a mul i esolu ion ke nel es ima o . We could make he ke nel and he scaling unc ion
iden ical. Also, we can equal bo h dila ion ac o s by making ℎ=2−𝑗. Bu he ke nel
o ℎ=2−𝑗 will be a densi y o he 𝑉𝑗 space o he mul i esolu ion analysis s uc u e only
i he sample is o he o m 𝑥𝑖=𝑘𝑖
2𝑗 𝑖=1,2,..,𝑛, wi h 𝑘𝑖∈𝑍 ∀𝑖=1,2,…𝑛 and whe e
𝑗 is a ixed in ege de e mined by he sample. Any o he es ima ion wi h a di e en ℎ will
no longe be a unc ion o he mul i esolu ion s uc u e.
Despi e he abo e commen , we ha e o poin ou ha he e is a well-de eloped
heo y abou he gene alized ke nel es ima o s, de eloped om he wa ele s and he
mul i esolu ion analysis s uc u es (see o ins ance [22]). B oadly speaking, his
me hodology equi es he mo he wa ele o scaling unc ion o he mul i esolu ion
analysis s uc u e o gene a e o hogonal bases o he 𝑉𝑗 spaces o he MRA. This is no
he case o he cubic box spline since i gene a es non o hogonal Riesz Bases. Going
deepe in o his aspec is an in e es ing ques ion, bu i is ou o he scope o his wo k. 5.
REAL DATA APPLICATION
In his sec ion, we apply he p oposed me hod o he g oss income o he Spanish
households. The sample da a comes om he Spanish Su ey o Household Finances
(EFF) o he yea 2014, which was conduc ed by he Bank o Spain [1]. The EFF
p o ides in o ma ion on asse s, deb , income and spending. The sample size is 6,120
households. The household income is calcula ed as he sum o labo and non-labo
incomes o all household membe s in 2013. I is exp essed in hund ed housand eu os.
19
Table 5.1 and Figu e 5.1 show he e olu ion, acco ding o 𝑗, o 𝐸𝑗[𝑋],𝜇𝑗(2),𝛾1, and
𝛾2, di ided by hei empi ical coun e pa o le els o esolu ion om 𝑗=−13 o 𝑗=
−6.
Table 5.1 Ra ios be ween MRDE momen s and sample momen s
𝑗
E[X]
Va iance
Asymme y
Ku osis
-13
0.99717924
1.01153181
0.98738403
0.98380212
-12
0.9995666
1.00258459
0.99586171
0.99430889
-11
0.99999127
1.00037759
0.9990044
0.99853096
-10
1.00016918
0.99997439
0.99958258
0.99926656
-9
0.99990519
1.00016536
0.99988219
0.99980877
-8
0.99989802
0.99998004
1.0000367
1.00010499
-7
0.99995971
0.99999305
1.00003152
1.0000789
-6
0.99992814
1.00000293
0.9999922
0.99998317
Fig. 5.1. Ra ios be ween MRDE momen s and sample momen s
Acco ding o he BIC c i e ion he op imum is 𝑗=−13. Howe e , he momen s
s abilize o 𝑗=−9 o 𝑗=−8 (Figu e 5.1). Le us ocus on his di e ence by compa ing
he densi y es ima es o he wo esolu ion le els plo ed in Figu e 5.2.
0.98
0.985
0.99
0.995
1
1.005
1.01
1.015
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5
20
Fig. 5.2 Es ima ed mul i esolu ion densi y o 𝑗=−8 and 𝑗=−13
The densi y has a peak be ween 8000 and 9000 eu os ha canno be cap u ed
accu a ely by using 𝑗=−13. So, a highe le el o esolu ion, 𝑗=−8 o 𝑗=−9, is
needed. The bias o 𝑗= −13 is e iden when he wo densi ies a e compa ed. The
oughness o 𝑗 = −8 is clea ly app eciable. The BIC chooses he smoo hness o he
cu e, which leads o a e y skewed es ima ed densi y a ound he mode. A simila ac
has been shown in sec ion 3.2. We ha e obse ed empi ically ha oughness has only a
sligh e ec on he cumula i e dis ibu ion unc ion. None heless, he bias has a
ema kable impac on he concen a ion measu emen p oducing an unde es ima ion o
he Gini index and he Lo enz cu e. This is an impo an issue o be conside ed i we
s udy dis ibu ional aspec s o he dis ibu ion as concen a ion o inequali y h ough he
i ed densi y. A his poin , i should be no ed ha he ke nel me hod is equen ly applied
o s udy income dis ibu ion (see o ins ance [8,16,3,20]). Figu es 5.3 and 5.4 plo he
cumula i e dis ibu ion unc ions and he Lo enz cu es espec i ely. The cumula i e
dis ibu ion unc ions a e simila excep in he income in e al [0, 10000]. This di e ence
leads o an unde es ima ion o he Gini index
5
: o 𝑗=−13 he index equals o 0.4338
and o 𝑗=−8 i is equals o 0.5131. I also a ec s he Lo enz cu e (see igu e 5.4)
which is unde es ima ed o 𝑗=−13 . The e o e, he le el o esolu ion 𝑗 = −8
5
No e ha he alues o he Gini coe icien can di e om o he publica ions since ou
illus a ion is based on g oss income ins ead o ne income.
0
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0.00007
020000 40000 60000 80000 100000 120000 140000 160000
j = -13 j = -8
21
ob ained by he me hod o momen s is p e e able o he alue 𝑗 = −13 selec ed by he
BIC in he si ua ions se ou abo e.
Fig. 5.3. Cumula i e dis ibu ion unc ions Fig. 5.4. Lo enz cu es
Nex , we epea he es ima ion o ke nel densi ies and compa e he esul s. Table
5.2 Figu e 5.5 show he e olu ion, acco ding o 𝑗, o 𝐸𝑗[𝑋],𝜇𝑗(2),𝛾1 and 𝛾2 di ided
by hei sample coun e pa o le els o esolu ion om 𝑗=−13 o 𝑗=−5.
Table 5.2 Ra ios be ween KDE momen s and sample momen s
𝑗
ℎ = 1/2𝑗
E[X]
Va iance
Asymme y
Ku osis
-13
8192
1
1.00759984
0.98870759
0.98497182
-12
4096
1
1.00189996
0.99715681
0.99621088
-11
2048
1
1.00047499
0.99928794
0.9990507
-10
1024
1
1.00011875
0.99982191
0.99976255
-9
512
1
1.00002969
0.99995547
0.99994063
-8
256
1
1.00000742
0.99998887
0.99998516
-7
128
1
1.00000186
0.99999722
0.99999629
-6
64
1
1.00000046
0.9999993
0.99999907
-5
32
1
1.00000012
0.99999983
0.99999977

22
Fig. 5.5 Ra ios be ween KDE momen s and sample momen s
The app op ia e le el o esolu ion acco ding o he momen s is 𝑗=−10 o 𝑗=−9.
Fig. 5.6 Es ima ed ke nel densi y o 𝑗=−9 and 𝑗=−13
The esul s a e simila ein o cing he idea o applying he analysis pe o med on
he MRDE. The le el o esolu ion selec ed o he MRDE was 𝑗=−9. Fo he KDE
we ha e op ed o ℎ= 1
2−9=29=512 . The plug-in me hod o selec he op imum ℎ
[19] p o ides he esul ℎ=459.277. This alue co esponds o 𝑗=−log2459.277=
−8.8432 which ounded o he nea es in ege would gi e 𝑗 = −9. I we use he
MRDE o 𝑗=−13 he plug-in me hod p o ides he esul ℎ=3140.324. Tha is
𝑗=−log23140.324 =−11.6167 whose nea es in ege is 𝑗=−12. This is less
0.98
0.985
0.99
0.995
1
1.005
1.01
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4
E[X] Va iance Asymme y Ku osis
0
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
020000 40000 60000 80000 100000 120000 140000 160000
(x), j=-13 (x), j=-9
23
conse a i e han he BIC bu s ill conse a i e. In any case, he esul ing 𝑗 is he same
o i is e y close o ha used o he plugged MRDE.
Gene alizing, i we use an es ima ed MR o a gi en 𝑗 as plugged densi y, he plug-
in me hod p o ides a alue o ℎ equal o −log2ℎ whose nea es in ege is he alue o 𝑗
u ilized o es ima e he MRDE. I is as e and easie o use he me hod o he momen o
KDE and de e mine he ℎ=1
2𝑗 ha we will use in he es ima ion. I we wan mo e
conse a i e esul s, ega ding he smoo hness o he i , we can educe he alue o 𝑗 by
one o wo uni s, paying special a en ion o he inc easing bias.
6. CONCLUSIONS
This pape in oduces an app oach o selec he bandwid h o smoo hing pa ame e
in semipa ame ic and nonpa ame ic densi y es ima ion. I is based on he e olu ion o
he expec ed alue, he a iance, he symme y and ku osis coe icien s o he es ima ed
densi ies o di e en bandwid hs. Using hese alues, di ided by hei empi ical
coun e pa , we selec a esolu ion le el so ha he bias will be accep able and he
oughness o he es ima o will be no excessi e.
This me hod has been applied o he densi y es ima ion by means o
mul i esolu ion densi ies as well as Ke nel densi y es ima ion. In his way, we ha e
expanded he a ailable c i e ia o smoo hing pa ame e selec ion.
The esul s o he simula ions and he empi ical applica ion indica e ha he le el
o esolu ion esul ing om he momen s me hod is mo e lexible o i a mul imodal
dis ibu ion han hose esul ing om he BIC o MRDE and he plug-in o KDE. The
BIC chooses he smoo hness o he cu e which leads o a skewed es ima ed densi y
a ound he modes. The me hod o he momen s a ibu es mo e impo ance o he use o
highe esolu ion le els and hence oughe es ima es o a oid he bias ha he i ing
p oduces. This p ocedu e is ecommended o analyse some dis ibu ional aspec s such as
he concen a ion o income. As i has been shown in he empi ical applica ion, he bias
can p oduce an unde es ima ion o he concen a ion o he dis ibu ion.
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32
Mo eo e , he in e als: [𝑘−2
2𝑗,𝑘+2
2𝑗),
ha e cen e 𝑘
2𝑗 and adius 1
2𝑗−1 , so o 𝑗 la ge enough he p e ious in e als will ha e a
adius so small ha each sample elemen , 𝑥𝑖, belongs o a di e en in e al. In his case,
he coe icien s g ea e han ze o a e hose associa ed wi h in e als ha con ain a sample
elemen , ha is: 𝑎𝑘(𝑥𝑖)=1𝑛 𝑖=1,2,…,𝑛 .
Hence he p oposi ion is ue.
P oo o P oposi ion 2.
𝑚𝑟= ∫𝑥𝑟𝑓󰆹𝑗(𝑥)
+∞
−∞ 𝑑𝑥= ∫𝑥𝑟∑𝑎𝑘𝜆𝑗,𝑘(𝑥)
𝑘∈𝑍
+∞
−∞ 𝑑𝑥=∑𝑎𝑘∫𝑥𝑟
+∞
−∞ 𝜆𝑗,𝑘(𝑥)𝑑𝑥
𝑘∈𝑍 .
Fo a 𝑗 la ge enough, acco ding o p oposi ion 1, we ha e:
𝑎𝑘={0𝑖𝑓 𝑘≠𝑘(𝑥𝑖) ∀𝑖=1,2,…,𝑛
1𝑛𝑖𝑓 𝑘=𝑘(𝑥𝑖) ∀𝑖=1,2,…,𝑛 ,
which allows us o w i e:
𝑚𝑟=1𝑛∑∫𝑥𝑟
+∞
−∞ 𝜆𝑗,𝑘(𝑥𝑖)(𝑥)𝑑𝑥
𝑛
𝑖=1 . (25)
Howe e ,
∫𝑥𝑟
+∞
−∞ 𝜆𝑗,𝑘(𝑥𝑖)(𝑥)𝑑𝑥=∫𝑥𝑟
+2
−2 𝑠𝜃(𝑠𝑥−𝑘(𝑥𝑖))𝑑𝑥, (26)
wi h 𝑠=2𝑗.
I we make he change o a iable 𝑦=𝑠𝑥−𝑘(𝑥𝑖) in (25), we ha e:
∫𝑥𝑟
+2
−2 𝑠𝜃(𝑠𝑥−𝑘(𝑥𝑖))𝑑𝑥=∫(𝑦+𝑘(𝑥𝑖)
𝑠)𝑟
+2
−2 𝜃(𝑦)𝑑𝑦 . (27)
I 𝑗 ends o in ini e, we ha e:
lim
𝑗→∞𝑦𝑠=0 lim
𝑗→∞𝑘(𝑥𝑖)
𝑠=𝑥𝑖 . (28)
The i s equali y in (28) is i ially ue and he second is also ue since by de ini ion:
𝑥𝑖∈[𝑘(𝑥𝑖)−2
𝑠,𝑘(𝑥𝑖)−2
𝑠),

33
and when 𝑗 ends o in ini y, he adius o he in e al con e ges o ze o and i s cen e is
he poin 𝑘(𝑥𝑖)
𝑠 .
Assuming ha (28) is ue we can w i e:
lim
𝑗→∞∫(𝑦+𝑘(𝑥𝑖)
𝑠)𝑟
+2
−2 𝜃(𝑦)𝑑𝑦=∫(𝑥𝑖)𝑟
+2
−2 𝜃(𝑦)𝑑𝑦=𝑥𝑖𝑟∫𝜃(𝑦)𝑑𝑦
+2
−2 =𝑥𝑖𝑟 . (29)
Whe ewi h, unde (25) and (29), we ha e:
lim
𝑗→∞𝑚𝑟=1𝑛∑𝑥𝑖𝑟 .
𝑛
𝑖=1
Tha is, he non-cen al momen s o he es ima ed MRDE con e ge o he non-cen al
momen s o he sample.