Minimum density power divergence estimator for diffusion processes

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1 A Ist Stat Math 3) 65:3 36 DOI.7/s Mmum desty power dvergece estmator for dffuso processes Sagyeol Lee Jumo Sog Receved: 3 March 7 / Revsed: Aprl / ublshed ole: July The Isttute of Statstcal Mathematcs, Tokyo Abstract I ths paper, we cosder the robust estmato for a certa class of dffuso processes cludg the Orste Uhlebeck process based o dscrete observatos. As a robust estmator, we cosder the mmum desty power dvergece estmator MDDE) proposed by Basu et al. Bometrka 85: , 998). It s show that the MDDE s cosstet ad asymptotcally ormal. A smulato study demostrates the strog robustess of the MDDE. Keywords Dffuso processes The Orste Uhlebeck process Mmum desty power dvergece estmator Dscretely observed sample Robustess Itroducto Let us cosder the dffuso process: dxt = ax t,θ)dt + σ dw t, X = x, ) where θ, σ ), a covex compact subset of R p R +, a s a kow real valued fucto defed o R R p, ad W s a -dmesoal stadard Weer process. The dffuso process has log bee popular aalyzg radom pheomea face, S. Lee B) Departmet of Statstcs, Seoul Natoal Uversty, Seoul 5-74, Korea e-mal: jpgrslee@yahoo.com; sylee@stats.su.ac.kr J. Sog Departmet of Computer Scece ad Statstcs, Jeju Natoal Uversty, Jeju , Korea 3

2 4 S. Lee, J. Sog egeerg, physcal ad medcal sceces. artcularly, the dffuso process gve by ) has wde applcatos ad cludes the Orste Uhlebeck process as a specal case. Statstcal ferece for ths process has bee studed by may authors, for stace, Flores-Zmrou 989), ad the asymptotc propertes for varous parameter estmators are well summarzed rakasa Rao 999, pp ad 53 58), Yoshda 99), Kessler 997), At-Sahala ) ad may other researchers studed the estmato problem for a more geeral class of dffuso processes. I those artcles, the estmato procedure s coducted based o dscretely observed sample. Statstcal ferece for cotuous samples ca be foud Kutoyats 4). I ths paper, we cosder a robust estmato of θ, σ ) the model ) based o dscretzed observatos. I the statstcal lterature, t s well kow that the estmators based o Gaussa lkelhood are severely flueced by outlers or extreme values. Thus, t ca be guessed that the smlar stuatos happe the estmato procedure, based o Gaussa approxmato method, for dffuso processes [for e.g., the Euler estmator rakasa Rao 999, p 55) ad Kessler 997) estmator]. I fact, our smulato study, the Euler estmator s observed to be severely damaged by outlers cf. Tables, 3). I order to costruct a robust estmator for process ), we adopt the dea of Basu et al. 998) BHHJ for abbrevato) whch troduces a robust estmato procedure to mmze a desty-based dvergece measures, called the desty power dvergeces: f +α z) + α ) gz) f α z) + α g+α z) dz, α >, d α g, f ) := ) gz) log gz) log f z)) dz, α =, where f ad g are desty fuctos. For a famly of parametrc dstrbutos F θ : θ R m possessg destes f θ ad for a dstrbuto G wth desty g, they defed the mmum desty power dvergece fuctoal T α ) by d α g, ftα g)) = m θ d αg, f θ ). Note that f G belogs to F θ, T α g) = θ for some θ. I ths case, gve radom sample X,...,X wth ukow desty g, the mmum desty power dvergece estmator MDDE) s defed by ˆθ α, = arg m θ V α, θ; X t ), t= where f +α θ z)dz + ) α f α θ X t ), α >, V α θ; X t ) := log f θ X t ), α =. 3

3 Mmum desty power dvergece estmator for dffuso processes 5 BHHJ showed that ˆθ α, s weakly cosstet for T α g) ad asymptotcally ormal, ad demostrated that the estmator has strog robust propertes agast outlers ad the msspecfcato of uderlyg models wth lttle loss asymptotc effcecy relatve to the maxmum lkelhood estmator. Ths approach ca be exteded to regresso models. Let f θ y x) be a parametrc famly of regresso models dexed by the parameter θ ad let gy x) be the true desty for Y gve X = x. Substtutg f ad g ) by f θ x) ad g x) respectvely, a famly of the x-codtoal versos of desty power dvergeces s obtaed as follows: arg m θ t= f +α θ y x t ) dy + ) α t= fθ αy t x t ), α >, t= log f θ Y t x t ), α =. 3) Ths dea wll be adopted later for our ow purpose. Compared to other exstg desty-based dvergece methods, such as Bera 977), Tamura ad Boos 986) ad Smpso 987), whch use the Hellger dstace, ad Basu ad Ldsay 994) ad Cao et al. 995), ths method s kow to have mert of ot requrg ay smoothg methods. I ths case, oe ca avod drawbacks ad dffcultes lke the selecto of badwdth that ecessarly follow from the kerel smoothg method. The orgazato of ths paper s as follows. I Sect., we costruct the robust estmator usg 3) ad address the asymptotc propertes of the proposed estmator. I Sect. 3, we perform a smulato study ad compare the proposed estmator wth the Euler estmator. The proofs for the results Sect. are provded Sect. 4. Fally, some auxlary lemmas are preseted Sect. 5. Ma results Let θ,σ ) be the true parameter for the dffuso process ). Suppose that X t,, are observed, where t = h, h, ad h. Applyg the Euler s approxmato to ), we have that X t = X t + ax t,θ )h + σ Z, h +,, where Z, = h W t W t ) ad, = t t ) ax s,θ ) a X t,θ ds. Let G deote the sgma feld geerated by W s : s t. If gorg, s actually, oe ca check that max, = o p h ). cf. Lemma ), by otcg that Z,,...,Z, are d N, ), we ca see that for large, X t G behave 3

4 6 S. Lee, J. Sog lke depedet r.v. s followg NX t + ax t,θ ), σ h ). Hece, vewg the observatos as regresso data pars X t, X t ) ad applyg 3) to them ths case the famly of parametrc dstrbutos s the ormal dstrbutos), we ca defe the MDDE as ˆθ α, ˆσ α ) = arg m θ,σ V, α θ, σ ), where V, α θ, σ ) = [ σ ) α + α) + α ) exp α X t a X t X t ) ) ] X t,θ h /σ h, α >, ) ) X t a X t,θ h /σ h + log σ, α =. Remark The MDDE wth α = cocdes wth the Euler estmator rakasa Rao 999, p 55). Also, ˆθ s the same as the least squares estmator ˆθ LSQ [see also Flores-Zmrou 989] ad ˆθ LF that maxmze the dscrete approxmate lkelhood fuctos, respectvely: ˆθ LSQ = arg m θ X t ) ) X t a X t,θ h, ad ˆθ LF = arg max θ ) ) ) a X t,θ X t X t a X t,θ h. Below we establsh the cosstecy ad asymptotc ormalty of the MDDE. For ths task, we set = f x,θ): f C + x ) C for some C, where C does ot deped o θ, ad assume the codtos as follows: A) There exsts a costat C such that for ay x, y, ax,θ ) ay,θ ) C x y. A) The process X from ) s ergodc for θ,σ ) wth ts varat measure μ such that x k dμ x) < for all k. A3) sup t E X t k < for all k. A4) The fucto a s cotuously dfferetable wth respect to x for all θ ad the dervatves belog to. 3

5 Mmum desty power dvergece estmator for dffuso processes 7 A5) The fucto a ad all ts x-dervatves are three tmes dfferetable wth respect to θ for all x. Moreover, these dervatves up to the thrd order wth respect to θ belog to. A6) If ax,θ)= ax,θ ) for μ a.s. all x, the θ = θ. A7) S := θ ax,θ ) θ T ax,θ )dμ x) s postve defte, where θ a = a/ θ. Here are the ma results of ths paper. Theorem Assume that A) A6) hold. For each α, fh, h ad h q for some q >, the ) ˆθ α, ˆσ α θ,σ ) probablty. Remark The codto that h q forsomeq > s ot ecessary for the case of α =. However, the other cases, ths codto s essetal to obta the weak cosstecy result cf. Lemma ). Theorem Assume that A) A7) hold ad θ,σ ) s the teror of. For each α,fh, h ad h, the h ˆθ α θ ) ) N ˆσ α p+, α ) dstrbuto, σ ) where α = σ +α +α ) 3 S +α) 3 +α) +α ) +α ) +α) +α +α α ). Remark 3 I actual practce, oe may rase a questo of how to choose a optmal α. I the stuato of o outlers, oe may select α to mmze the asymptotc varace Theorem.e., the case of α = ). However, wth ther exstece, the outlers wll damage ths procedure, so a optmal α s very hard to choose. Covetoally, takg accout of ths dffculty, oe employs a fxed α rather tha seek for a sutable α, whch rages [.5,.5] sce too a large α, whch would have strog robust propertes, may result a bg loss effcecy whe the porto of outlers s ot very large as speculated. We wll see ths pheomeo from the smulato result preseted the ext secto. So far, we have cosdered the case that the dffuso coeffcet s a costat. However, the MDDE ca be exteded to the followg dffuso process: dx t = ax t,θ)dt + σ bx t )dw t. 4) Sce by trasformg X t to Y t = GX t ), where G satsfes the relatoshp x Gx) = /bx), ad usg Itô s formula, we obta dy t = μy t,θ,σ)dt + σ dw t, 5) 3

6 8 S. Lee, J. Sog where μy,θ,σ)= a G y), θ ) b G y) ) σ xb G y), σ ), ad subsequetly, the estmato procedure for the process 4) s reduced to that for the dffuso process ). For a more geeral class of dffuso process, we may defe the MDDE usg the Euler approxmato ad the cotrast fuctos 3). However, there are some techcal dffcultes obtag the same asymptotc propertes of the estmator. We leave ths task as a future study. 3 Smulato study I ths secto, we compare the performace of the MDDE wth α.5,.,...,.95 ad the Euler estmator EE) for the Orste Uhlebeck process: dxt = θ X t dt + σ dw t X =. 6) I our smulato, the case θ,σ ) =, ) s cosdered, ad the sample X o,t s obtaed dscretely wth samplg terval h =.55. The comparso s based o the umbers defed by d = d α := r r ˆθ, ) α θ ) + ˆσ α d, σ, dr := d of EE, where r s the umber of repettos. I fact, d s the average dstace betwee the true parameter ad ts estmates, so the smaller d or d R ) dcates that the estmator s more effcet. Frst, we hadle the case that the observato s ot cotamated by outlers. Based o repettos, the mea, stadard devato, d ad d R are calculated for = 5, 8,. Fgure plots the calculated d α s, where d s the oe for the EE. The results preseted Table ad Fg. show that the EE outperforms the MDDE, d alpha Fg. d wthout outlers. Square, damod ad tragle represet for the case of = 5, 8,, respectvely 3

7 Mmum desty power dvergece estmator for dffuso processes 9 Table Mea SD) ad ddr) wthout outlers EE MDDE θ.9.46).9.48).89.53).87.64).84.83).8.).8.33) σ.983.).983.).983.).983.).983.3).983.6).983.7) ddr).96.).97.5).3.9).3.54).39.).35.86).37.49) 8 θ.68.).7.9).67.3).66.).66.34).65.54).64.7) σ.987.6).987.6).987.7).987.7).987.8).987.).987.) ddr).6.).59.99).64.).73.44).87.).37.76).34.4) θ.8.98).8.98).8.99).84.4).84.5).87.3).89.46) σ.988.5).988.5).988.5).988.5).988.6).987.7).987.8) ddr).44.).45.).47.).54.39).69.).89.86).36.55) 3

8 S. Lee, J. Sog ad the MDDE wth α close to performs smlarly to the EE. It s also see that the performace of the MDDE wth α ot close to, say α =.5.5, s ot very poor. Tables ad 3 ad Fgs., 3, 4, 5, 6, ad 7 summarze the results for the case that outlers are volved the data. Here, we cosder the stuato that the sample X o,t from 6) s cotamated by the outlers X c,t d N,σ V ) ad the observed r.v. s follow the scheme X t = p ) X o,t + p X c,t, where p are d Beroull r.v. s wth success probablty p. It s assumed that p, X o,t ad X c,t are all depedet. The mea, stadard devato, d ad d R based o X t are calculated out of repettos for =, p =.5 ad.. From Tables ad 3, we ca see that the d R teds to get smaller as ether σv or p creases except for some cases such as α =.5,.5 ad p =.. The bold phased fgures deote the α s that gve a mmal d R s. Fgures ad 5 show the plots of d R whe observatos are cotamated by 5 ad % outlers, respectvely. It ca be see that the MDDE s wth the α lyg.5.3 produce very small d R s. Fgures 3 ad 6 show the plots of the EE s ad the MDDE s producg the smallest d optmal MDDE s), ad Fgs. 4 ad 6 are the correspodg hstograms. The fgures show that the EE s scatter wdely whereas the optmal MDDE s le ear the true parameter. From these results, we ca coclude that the MDDE possesses much more robust propertes tha the EE. It ca be see that the α s yeldg mmal d vares wth the cases. Ths dcates that choosg a optmal α s ot a easy task actual usage. Covetoally, α [.,.] s recommeded sce the MDDE wth the α stll keeps the effcecy whe there are o outlers ad are robust agast outlers. The same ca be appled to our case, but our smulato study suggests to us that a broader rage of α s, say, [.5,.5] may be employed costructo of the MDDE. 4 roofs We wll provde the proof for the case of α> sce the proofs of the case of α = are smlar to that of α>. I what follows, we deote a θ) = ax t,θ), Z, = Z, = θ, σ ), = θ,σ ). Moreover, C > deotes a uversal costat. roof of Theorem Note that Uσ, σ ) := ) α σ + α + ) + α σ ) α σ, σ >, has a mmal value at σ = σ. Smlarly to the proof of Theorem of Kessler 997), Theorem s proved f we verfy that 3

9 Mmum desty power dvergece estmator for dffuso processes Table Mea SD), d ad dr wth 5 % outlers σ V EE MDDE θ ) ) )..67).79.6).67.6).58.6).5.63).3.7)..8) σ.6.869).99.68).5.8).6.).5.9).48.9).47.9).47.).5.).57.) d dr θ ) ).398.7).97.54).63.5).53.53).46.55).4.57).5.69).8.8) σ ).55.84).4.4).53.).45.9).44.9).44.9).45.9).5.).59.) d dr θ ) ) ).99.4).7.39).6.4).55.4).49.43).35.5).7.6) σ ).33.48).3.8).45.9).4.9).4.9).4.9).4.9).49.).58.) d dr

10 S. Lee, J. Sog Table 3 Mea SD), d ad dr wth % outlers σ V EE MDDE θ ) ) 3.3.8) ).35.7).89.).53.5).9.4)..4).94.7) σ ) ) ).77.5).39.7).8.5).6.5).3.5).35.6).45.7) d dr θ ) ) ) ).88.6).4.6).6.4).99.4).88.5).75.) σ 4.5.6) ).536.9).45.3)..7).7.6)..6).6.6).34.7).47.8) d dr θ ) ).8 3.8).38.89).3.7).93.66).7.66).56.67).45.7).35.75) σ ) ) ).6.8).9.4).9.3).4.4)..4).3.5).46.6) d dr

11 Mmum desty power dvergece estmator for dffuso processes 3 d_r..3.5 d_r Fg. d R versus α wth 5 % outlers. Square, damod ad tragle represet for the case of σv =,, 3, respectvely sgma EE) sgma MDDE) theta EE) theta MDDE) Fg. 3 lottg ˆθ, ˆσ)of the EE ad the optmal MDDE wth 5 % outlers whe σv = 3 Frequecy 6 Frequecy 4 8 Frequecy 6 Frequecy theta EE) theta MDDE) sgma EE) sgma MDDE) Fg. 4 Hstogram of the EE ad the optmal MDDE wth 5 % outlers whe σv = 3 ) ) ut V α, θ, σ ) Uσ, σ ) uformly. h V, α θ, σ ) h V, α θ,σ) + α) ) α+ σ + α σ uformly. ) 3 σ K θ, σ ) = K, θ, σ ) := α σ σ Z A θ) h + α + α σ σ Z h ax,θ) ax,θ ) dμ x) A θ) σ h + α A θ) σ + α σ, h 3

12 4 S. Lee, J. Sog d_r..4.8 d_r Fg. 5 d R versus α wth % outlers. Square, damod ad tragle represet for the case of σv =,, 3, respectvely sgma EE) sgma MDDE) theta EE) theta MDDE) Fg. 6 lottg ˆθ, ˆσ)of the EE ad the optmal MDDE wth % outlers whe σ V = 3 Frequecy 4 8 Frequecy 6 Frequecy 4 8 Frequecy theta EE) theta MDDE) sgma EE) sgma MDDE) Fg. 7 Hstogram of the EE ad the optmal MDDE wth % outlers whe σv = 3 where A θ) = A, θ) := a θ ) a θ). I vew of Lemma Sect. 5, we have sup max θ, σ ) =o p h γ ) for γ<. Thus, we have that sup 3 + ) α exp α ) α σ ) X t X t a θ)h σ h exp α σ σ Z )

13 Mmum desty power dvergece estmator for dffuso processes 5 = sup + ) α ) sup + α C α exp sup σ ) α e α σ ) σ σ Z e K θ,σ ) ) α max e K θ,σ ) ) max K θ, σ ) = o ). Usg ths ad Lemma 4, we have that uformly, + ) α σ ) ) α X exp α t X t a θ)h σ h + ) ) α + α σ ) α σ σ, whch establsh ). Next, we verfy ). Usg Taylor s theorem, we ca wrte that e K θ,σ ) e K θ,σ ) = α σ σ Z A θ) h + α A θ) σ h + α A θ) σ α σ σ 4 Z A θ) h + K θ, σ ) α σ σ 4 Z A θ) h + K θ,σ)! e ζ, + K θ, σ ) 3 3! σ = α σ σ Z A θ) h + α + K θ,σ)! e ζ, α σ σ Z e ζ, + K θ, σ ) 3 e ζ,, 3! ) A θ) h + H, where ζ, K θ,σ), ζ, K θ, σ ), ad H θ, σ ) = H, θ, σ ) := α A θ) σ K θ, σ ) α σ ) σ 4 Z A θ) h. 3

14 6 S. Lee, J. Sog Owg to Lemma, wehavethat ad sup sup sup max θ, σ ) =o p h r ), r <.5, sup max θ, σ ) 3 =o p h r ), r <.5, sup max θ,σ) =o p h r 3 ), r 3 <, max eζ, exp sup max K θ,σ) = O ), max eζ, exp sup max K θ, σ ) = O ). 7) 8) By 7) ad 8), we have that sup max R θ, σ ) := sup max H, θ, σ ) + K θ,σ)! e ζ, + K θ, σ ) 3 e ζ, 3! = o h r ) for r <.5. 9) Thus, by Lemmas 4 ad 5 ad 9), we have that uformly θ, σ ), V, α h θ, σ ) V, α h θ,σ) = + α ) α+ α σ ) σ σ Z A θ) e α σ σ Z + + α) σ σ Z A θ) e α σ σ Z + R θ, σ ) e α σ σ h α h Z ) α+ + α) + α σ ) 3 σ σ ax,θ) ax,θ ) dμ x). ) α+ Ths completes the proof. roof of Theorem Let ˆ α = ˆθ α, ˆσ α ), lα ) = V α, ), L α = h θ l α ) ) σ l α ) 3, ad C α ) = h θθ T l α ) h σθ T l α ) h θσ lα ) ) l α σ ),

15 Mmum desty power dvergece estmator for dffuso processes 7 where =, θθ T θ θ T θσ = θ σ. Usg Taylor s theorem, we have that = l α ) + from whch we ca wrte that l α L α = C α, + u ˆ α ) ) du We ted to show that L α + u ˆ α ) ) ˆθ du α θ ) ˆσ α σ, h ˆθ α θ ) ) ˆσ α. σ ) d L N,,α ), ) where,α = σ α+ +α) +α) 3 S ), +α) +α ) +α) +α +α α ad S s the oe defed A7). Let U α, ) := σ ) α + α + ) exp α α A θ)h + σ Z h ) σ h ), ) ad ζ = ζ, := h θ U, α ) ) σ U, α, ) the, due to Lemma 6, ) ca be verfed f we show that E ζ G ), ) E ζ ζ T ) G,α, 3) E ζ 4 ) G 4) 3

16 8 S. Lee, J. Sog [cf. Theorems 3. ad 3.4 of Hall ad Heyde 98)]. Note that ad θ U, α ) = + α h σ α+ Z θ a θ ) e α Z, 5) α σ U, α ) = + α σ α+ θ U, α θ T U, α + α) ) = h σ U α, ) ) = α + α σ α+ σ α+ + α) + σ α+ θ U, α σ U, α ) = α + α Utlzg the argumets: + + α ) σ α+ Z e α Z, 6) Z θa θ ) θ T a θ ) e αz, α + α σ α+ Z ) e αz, σ α+ θ U α, ) ) Z e α Z + α) h σ α+ Z Z ) θa θ ) e αz. E Z e αz / G =, E Z e αz G = + α) 3, ) E Z e α Z G = α + α) 3, E Z ) e αz G = + α ) + α) 5, ad ) E Z Z 3 e αz G =, by smple calculus, we ca readly check ) ad 3). Sce 4) holds due to the facts: E θ U, α ) 4 ) G h C α + X t C, ad ) s establshed. 3 E σ U α, ) 4 G C α,

17 Mmum desty power dvergece estmator for dffuso processes 9 Now, owg to ), the theorem s asserted f we verfy that C α, + u ˆ α ) ) du +α S σ α+ α + +α) 3. 7) For ths task, t suffces to show that C α, ) +α S σ α+ α + +α) 3, 8) ad sup Cα, + ) C α, ), 9) ε where ε s ay postve real sequece decayg to. We frst deal wth 8). ut J, ) = A θ) θθ T a θ) + α σ ) σ Z θ a θ) θ T a θ), J, ) = θθ T a θ) + α σ σ Z A θ)h 3 + σ Z h + A θ) h +A θ) h + θ a θ) θ T a θ), J 3, ) = 4 α σ σ Z K ) + 4 α K ), ad J 4, ) = σ Z K ) h + A θ)h + ) α + α σ ) σ Z K ). It s ot dffcult to see that J, ) ) Cα + Z + Also, vew of Lemma, we ca see that ) X t C. ) sup sup sup max,) =o p h r ), r <.5, max 3,) =o p h r ), r <.5, max 4,) =o p h r ), r <. ) 3

18 3 S. Lee, J. Sog Meawhle, we have that θθ T l α + α ) = σ α+ e α = + α σ α+ + α σ α+ + α σ α+ σ σ Z e K ) σ θθ T a θ)z h + J, )h + J, ) σ θθ T a θ)z h e α σ σ Z J, ) α σ σ θθ T a θ)z A θ) R, ), σ l α ) = α + α σ α+ + α σ α+ e α + + α σ α+ θσ lα ) = + α σ α α σ α+3 + α σ α+3 σ σ Z where ζ K ), ad + + α σ α+ h e α σ σ Z + α 3 + α) σ σ Z + α σ 4 σ 4 Z α) α K ) α J 3, ) e α σ σ Z R, ) e α σ σ Z K ) e ζ, θ a θ) α + α σ ) σ Z σ Z h e α θ a θ)j 4, ) e α σ σ Z θ a θ)r 3, ) e α σ σ Z K ) e ζ, M ) = σ θθ T a θ)z h + J, )h + J, ), R, ) = J, ) J, )h + J, ) ) α σ σ Z A θ) h +M ) α σ σ Z A θ) h K, ) + K ) eζ σ R, ) = + α 3 + α) σ Z + α K ) + α R 3, ) = σ α + α σ ) σ Z Z h + J 4, ). 3 σ σ Z e α σ σ Z, e α σ σ Z, σ 4 σ 4 Z 4 + J 3, ),

19 Mmum desty power dvergece estmator for dffuso processes 3 The, usg 8), ) ad Lemma, we ca have that sup sup sup max,) =o p h r ), r <.5, max,)k ) e ζ =o p h r ), r <.5, max 3,)K ) e ζ =o p h r ), r <. ) Therefore, by Lemmas 4 ad 5, ) ad ), h θθ T l α ) + α σ α+ + α σ ) 3/ σ S ax,θ ) ax,θ)) θθ T ax,θ)μ x), σ l α ) α + α σ α+ + α σ α+ + α) + α σ σ 3 + α) σ σ θσ h lα ) + α σ ) 3 σ + 3α σ 4 σ 4 ) + α σ ) 5 σ, uformly. Thus 8) s verfed. Sce the above lmts are cotuous by A) ad A5), we ca demostrate that 9) holds. Ths completes the proof. 5 Some Lemmas Lemma Suppose that A) holds. The, for k, E k G C k h 3 k ) + X t k. Sce the result of Lemma s well kow, we omt the proof. Lemma Suppose that f : R R belogs to. The f A) ad A3) hold ad h q for some q >, sup ) max f j X t, Z k l hm where j, k, l,,, ad m >.5 l. = o h r ) for r <.5 l + m, 3

20 3 S. Lee, J. Sog roof For ay teger p wth.5l + m r)p q, we have that max E ) + X t C Z k l p h m r)p Ch.5l+m r)p = o), whch establshes the lemma. Lemma 3 Suppose that A) ad A3) hold. The, for f : R R, f ) X t, = O ). 3) If addto, h, A) holds ad f s dfferetable wth respect to x ad whose dervatves also belog to, the f ) X t, f x,)dμ x) 4) uformly. Itseasytoprove3), so we omt the proof. The argumet 4) sduetolemma 8ofKessler 997). Lemma 4 Suppose that A) A3) hold. The f f : R R s dfferetable wth respect to x ad wth dervatves belogg to, f f f uformly. ) X t, e α σ σ Z ) X t, Z e α σ σ Z ) X t, Z 4 e α σ σ Z + α σ ) σ + α σ ) 3 σ 3 + α σ ) 5 σ f x,)dμ x), 5) f x,)dμ x), 6) f x,)dμ x) 7) roof Let h ) = ) X f t, e α σ σ Z. 3

21 Mmum desty power dvergece estmator for dffuso processes 33 I vew of Lemma 9 of Geo-Catalot ad Jacod 993), the covergece result for each s esured by the facts: E h ) G = + α σ ) / σ f X t,) + α σ ) / σ f x,)dμ x), ad E h )) G f X t,)= o ). To establsh the uform covergece, vew of Theorem of Ibragmov ad Has msk 98, p 378)), t suffces to verfy that d E h ) C for all, 8) ad E h ) d h ) C d for all,, 9) where d s the dmeso of. We oly prove 9) sce 8) ca be proved essetally the same way. Note that sup h ) C α ) + Z + X t C. The, usg Cauchy s equalty ad Jese s equalty, we have that d h ) h )) d E h ) ) C α d E + Z C + E C α d, ) X t C where les betwee ad. Ths asserts 9). I a smlar fasho, we ca verfy 6) ad 7). 3

22 34 S. Lee, J. Sog Lemma 5 Uder the codtos Lemma 4, fh, the ) f X t h, Z e α σ σ Z = o ) 3) uformly. roof Let From the facts: ad h ) = h f X t,)z e α σ σ Z. 3) E h ) G =, E h ) G h ) f X t, = o ), we ca see that 3) holds for each. Thus, as the proof of Lemma 4, the lemma s proved f we show that 8) ad 9) hold for 3). By applyg Burkhoder s equalty to a sequece of martgale dffereces h ), G, oe ca easly see that 8) holds. Also, applyg Burkhoder s equalty to a sequece of martgale dffereces h ) h ), G ad usg the facts: sup h ) C α Z + Z 3 ) + X t C h, Cauchy s equalty ad Jese s equalty, we have that E d h ) h )) d E h ) h )) d d E h ) d E C α d h d d. C α d h d Z + Z 3 C + ) X t C Ths etals 9) ad completes the proof. 3

23 Mmum desty power dvergece estmator for dffuso processes 35 Lemma 6 Let U, α ) be the oe defed ). Uder A), A3) ad A4), f h, the h θ V α, ) h σ V α, ) θ U, α ) = o ), 3) σ U, α ) = o ). 33) roof We oly deal wth 3) sce 33) ca be prove smlarly. Note that θ V, α ) = + α ) σ σ α+ Z h + θ a θ ) e α Z e K,, where K, = K, ). It follows from 5) that θ V, α ) h C α h C α h θ U, α ) ) + X t C σ Z h + ) K, e ζ ) σ Z h ) + X t C Z K, h e ζ + e K, C α e max K, h ) + X t C Z K, h +, ) where ζ K,. Usg the facts that E K, G ) C + X t C h ad ) E K, 4 G ) C + X t C h 4,wehave h ) E + X t C Z K, ) G h + C h C h ) + X t C E Z )E K, G ) h + E G ) ) + X t C h.5 = O h ) 3

24 36 S. Lee, J. Sog ad ) E + X t C h Z K, ) G h + C ) + X t C h h 3 = O h ). Sce e max K, = O ), vewoflemma9ofgeo-catalot ad Jacod 993), 3) s establshed. Ths completes the proof. Ackowledgmets We would lke to thak the Assocate Edtor ad referees for ther helpful commets. Refereces At-Sahala, Y. ). Maxmum lkelhood estmato of dscretely sampled dffusos: A closed-form approxmato approach. Ecoometrca, 7, 3 6. Basu, A., Harrs, I. R., Hjort, N. L., Joes, M. C., et al. 998). Robust ad effcet estmato by mmzg a desty power dvergece. Bometrka, 85, Basu, S., Ldsay, B. G. 994). Mmum dsparty estmato for cotuous models: Effcecy, dstrbutos ad robustess. Aals of the Isttute of Statstcal Mathematcs, 48, Bera, R. 977). Mmum hellger dstace estmates for parametrc models. Aals of Statstcs, 5, Cao, R., Cuevas, A., Frama, R., et al. 995). Mmum dstace desty-based estmato. Computatoal Statstcs ad Data Aalyss,, Dacuha-Castelle, D., Flores-Zmrou, D. 986). Estmato of the coeffcet of a dffuso from dscrete observatos. Stochastcs, 9, Flores-Zmrou, D. 989). Approxmate dscrete tme schemes for statstcs of dffuso processes. Statstcs,, Geo-Catalot, V., Jacod, J. 993). O the estmato of the dffuso coeffcet for multdmesoal dffuso processes. Aales de l Isttut Her ocaré-robabltés et Statstques, 9, 9 5. Hall, p., Heyde, C. 98). Martgale lmt theory ad ts applcatos. New York: Academc ress. Ibragmov, I. A., Has msk, R. Z. 98). Statstcal estmato-asymptotc theory. New York: Sprger. Kessler, M. 997). Estmato of a ergodc dffuso from dscrete observatos. Scadava Joural of Statstcs, 4, 9. Kutoyats, Y. 4). Statstcal ferece for ergodc dffuso processes. New York: Sprger. Lee, S., Na, O. 5). Test for parameter chage based o the estmator mmzg desty-based dvergece measures. Aals of the Isttute of Statstcal Mathematcs, 57, rakasa Rao, B. L. S. 999). Statstcal ferece for dffuso type processes. Lodo: Arold. Smpso, D. G. 987). Mmum hellger dstace estmato for the aalyss of cout data. Joural of the Amerca Statstcal Assocato, 8, Tamura, R. N., Boos, D. D. 986). Mmum hellger dstace estmato for multvarate locato ad covarace. Joural of the Amerca Statstcal Assocato, 8, Yoshda, N. 99). Estmato for dffuso processes from dscrete observatos. Joural of Multvarate Aalyss, 4, 4. 3