Smoothing level selection for density estimators based on the moments

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] 2 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. REFERENCES [1] Banco de España. Su ey o Household Finances (EFF) 2014: Me hods, Resul s and Changes since 2011, Analy ical a icle, 24, Janua y 2007. 24 [2] R. Cao, A. Cue as and W. Gonzalez Man eiga, A compa a i e s udy o se e al smoo hing me hods in densi y es ima ion, Compu . S a . 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UK e idence om ke nel densi y es ima es, Economics Le e s 49 (1995), pp. 407-413. h ps://doi.o g/10.1016/0165-1765(95)00698-F [9] M.C. Jones, On co ec ing o a iance in la ion in ke nel densi y es ima ion. Compu . S a is . Da a Anal. 11 (1991), pp. 3-15. [10] J.S. Ma on and S.J. Shea he , P og ess in da a-based bandwid h selec ion o ke nel densi y es ima ion, Compu . S a . 11 (1996), pp. 337–381 [11] S. Malla , A wa ele ou o signal p ocessing. Academic P ess. New Yo k, 1998 h ps://doi.o g/10.1016/B978-012466606-1/50008-8 [12] F. Palacios-González and R.M. Ga cía-Fe nández, A lexible amily o densi y Func ions, S a is ics: A Jou nal o Theo e ical and Applied S a is ics, 49 (2014a), pp. 680–704. h ps://doi.o g/10.1080/02331888.2014.883398 25 [13] F. Palacios-González and R.M. Ga cía-Fe nández, Mix u es o Mix u es Based on Mul i esolu ion Analysis Theo y, Communica ions in S a is ics, Simula ion and Compu a ion, 43 (2014b), pp. 723–742 h ps://doi.o g/10.1080/03610918.2012.714031 [14] F. Palacios-González, and R.M. Ga cía-Fe nández, A as e algo i hm o es ima e mul i esolu ion densi ies, Compu a ional S a is ics, 35 (2020), pp. 1207-1230 h ps://doi.o g/10.1007/s00180-020-00952-w [15] B.U. Pa k and J.S. Ma on, Compa ison o da a-d i en bandwid h selec o s. J. Am. S a . Assoc. 85 (1990), pp. 66–72 h ps://doi.o g/10.1080/01621459.1990.10475307 [16] G. M. Pi au, and R. Zelli, Tes ing o changing shapes o income dis ibu ion: I alian e idence in he 1990s om ke nel es ima es, Empi ical Economics, 29 (2004), pp. 415-430. h ps://doi.o g/10.1007/s00181-003-0175-3 [17] G. Schwa z, Es ima ing he dimension o a model, Annals o S a is ics, 6 (1978), pp. 461-464. [18] B.W.Sil e man, Densi y es ima ion o s a is ics and da a analysis. Monog aphs on S a is ics and Applied P obabili y, ol. 26. Chapman and Hall, London, 1986. [19] D. W. Sco , R. Tapia and J.R. Thompson, Ke nel densi y es ima ion e isi ed. Nonlinea Analysis- heo y Me hods & Applica ions 1 (1997), pp. 339-372. h ps://doi.o g/10.1016/S0362-546X(97)90003-1 [20] W. Shaoping , L. Ang, W. Kuangyu and W. Ximing, Robus ke nels o ke nel densi y es ima ion, Economics Le e s, 191 (2020), pp. 109138. doi.o g/10.1016/j.econle .2020.109138. [21] S.J. Shea he and M.C. Jones, A eliable da a-based bandwid h selec ion me hod o ke nel densi y es ima ion, J. R. S a . Soc. Se . B, 53 (1991), pp. 683–690. h ps://doi.o g/10.1111/j.2517-6161.1991. b01857.x [22] S.H. Huang, Densi y Es ima ion by Wa ele -Based Rep oducing Ke nels, S a is ica Sinica 9 (1999), pp. 137-151. 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.