MINIMUM HELLINGER DISTANCE ESTIMATION OF A STATIONARY MULTIVARIATE LONG MEMORY ARFIMA PROCESS

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1 Joura o Maheaca Scece: Avace a Appcao Voue 5 8 Page 3-36 Avaae a hp://cecavace.co. DOI: hp://.o.org/.864/aa_793 MIIMUM HIG DISTAC STIMATIO OF A STATIOAY MUTIVAIAT OG MMOY AFIMA POCSS K. STAISAS MBK a OUAGIA HII aoraor o Maheac a ew Techooge o Iorao aoa Poechc Iue Fé HOUPHOUT-BOIGY Yaououkro P. O. Bo 9 Ivor Coa e-a: o_h@ahoo.r Arac I h oe we eere he u Heger ace eaor (MHD) o a aoar uvarae og eor AFIMA (Auo regreve racoa egrae ovg average) proce. We eah uer oe aupo he ao ure covergece o he eaor a apoc ora.. Irouco Grager a Joeu [4] a Hokg [6] have propoe he AFIMA ( p q) oe o ee a e ere whch pree a characer o hor or og eor oowg. For he proce hor eor. For he proce og eor. Th og eor Maheac Suec Cacao: 6F 6H. Kewor a phrae: u Heger ace aoar uvarae AFIMA proce eao og eor. eceve Feruar 6 8; eve March Scec Avace Puher

2 4 K. STAISAS MBK a OUAGIA HII characerze a ow eca o he auocorreao uco or he u o uhe auocorreao. The proce o-aoar or a aoar or. I pe o AFIMA procee he oo o og eor ha eewe cue he auhor uch a B a H [] or ear procee wh og eor r a H [] or rog epee u-eoa Gaua procee. aura Maora [9] propoe u ace a ew eho or eag he paraeer o aoar a o-aoar AFIMA ( p q) proce or.75. Kaagaé a H [7] a [8] eae he u Heger ace eho a aoar uvarae AFIMA proce a he qua au kehoo approach a o-aoar uvarae AFIMA proce. I h paper we geeraze he reu o Kaagaé a H [7] o he uvarae cae. We coer a -eoa AFIMA aoar proce ( () ()) aer vero o he proce we eah he coece a apoc ora ug he u Heger ace. The reao or choog h eao echc e he ac ha hee eaor oae are ece a rou (c. Bera []). The paper orgae a oowg. Aer oe oe aou he eaor Seco we pree a uvarae AFIMA oe. Seco 3 evoe o he eao o paraeer cug he coec o he eaor a apoc ora. I Seco 4 we eah he a reu o h work. We eoe he vecor o paraeer o ere copoe o ( ) a ar coece. The u Heger ace eaor o ee arg H Θ

3 where MIIMUM HIG DISTAC STIMATIO 5 H ( ) he Heger ace ee H. () The u Heger ace ze he Heger ace ewee a. (.) a heoreca proa e (). a rao uco o () a (). a o-paraerc kere e eaor o () ee : () K () () K (3) where K : a kere uco a ( ) a equece o awh a :.. Muvarae AFIMA Moe The uvarae AFIMA oe wa rouce Sowe [3]. We coer a -eoa AFIMA aoar proce ( ) oowg whch geerae A( ) D( ) ( ) B( ) ( () () ) (4) where M eoe he rapoe o he ar M. he ackwar h operaor who o a eee o a e ere aocae he prevou oervao a X X

4 6 K. STAISAS MBK a OUAGIA HII { () () } are whe oe procee ha oow he ora aw o ea zero a covarace Deoe r{ ( )} δ( ) k. k K he pove ee covarace ar. The epreo D() ee (4) repree a agoa ( ) -ar o he racoa erece operaor o ackwar h ee ( ) D( ) ( ) wh ( ) Γ k k k Γ( ) k! a. Γ(). he gaa uco uch ha Γ ( )!. e A (). a B (). e ar pooa o egree p a q repecve ee a hereaer : A( ) I A A p B ( ) I B B q where I repree he ( ) e ar. The e A ( ) a e B ( ) are repecve he characerc pooa o he ar pooa A (). a B (.). We aue ha he roo o characerc pooa are a oue he u k. Oak [] a Hokg [6] howe ha he proce vere or a aoar or. Takg o accou he coo o he pooa he proce (4) vere a caua a a a repreeao o a auoregreve proce o e orer a oowg: p q

5 MIIMUM HIG DISTAC STIMATIO 7 () () D A B () () (). D A B (5) e C A B he equa (5) ca e wre a oow: () () () D c c c c a () () where B A Θ e he vecor paraeer o ere. For p a Θ. q a copac e. [ ] { }. q p are -ar aocae wh he ere ere eveope o he ar pooa D C power o uch a r Z or. r

6 8 K. STAISAS MBK a OUAGIA HII e or e he oervao. The ovao ( () () ) are o oervae he are eae () ( ) (). () ( ) 3. Paraeer ao To eah he coec a aw o he paraeer we ee he oowg aupo: Aupo (A) () or. For a ( u ) we have: () K ( u) u u K ( u) u or ; (3) uu K ( u) u u K( u) u or ; (4) There e C uch ha up K ( u ) K ( u) C. Aupo (A) u For each Θ a each he uco a are couou ereae a a a e aoue couou wh repec o he eegue eaure pove a eghourhoo o he org.

7 MIIMUM HIG DISTAC STIMATIO 9 Aupo (A3) For each he uco or q a or k q k q are he couou a ee ( ). Aupo (A4) α where α wh a ow varg uco. ( a) a. For each Θ up a k. Aupo (A5) k For Θ pe ha { / } a e o pove eegue eaure. Aupo (A6) We uppoe ha here a coa M uch a up M. Theore. Suppog ha Aupo (A)-(A6) are ae. The coverge ao ure o or a. We eoe : g g g g g V g g g. whe hee quae e.

8 K. STAISAS MBK a OUAGIA HII Theore. Suppog ha Aupo (A)-(A6) are ae. I Coo C: The copoe o g a or o hee copoe are couou uco o. Coo C: g g he he ruo o g are a he a o-guar ( ) -ar ( ) ( ) where g 4 g K ( u) u. 4. Proo o he Theore We ee he oowg ea o prove he Theore. ea. Suppog ha Aupo (A) a (A) are ae. The where a.. whe. Proo. B raguar equa we have up ( a) ( ) ( c) ( a) up ; () up ; () c up. We eorae ea hree ep.

9 MIIMUM HIG DISTAC STIMATIO Sep. The covergece o (a) o zero aer vero o he proce (4). Coerg he coo o he pooa uco A() a B() he proce (4) vere a ca e rewre a a repreeao o a auoregreve proce o e orer. We coer wo e uco (). a (). repecve o a. B Aupo (A) we have where C up () () () () () () repree he re whe rucag he ere ro. We aap he oo o ver accorg o Grager a Aere [3] a eow. e eae he oowg epreo:.

10 K. STAISAS MBK a OUAGIA HII Deoe χ. χ For a rea u a u u (6) he χ hu χ. e

11 MIIMUM HIG DISTAC STIMATIO 3 We w ow ocu o he epreo o () a (e).. e e coer a vecora ere vecora pace.. B raguar equa we have hereore. J Ug equa (6) we have J J coeque (7) where a are he coece o he vecor.

12 K. STAISAS MBK a OUAGIA HII 4 Ug he ae argue a () we oa e (8) where a are he coece o he vecor. Oak [] characerze he ver o he proce a uco ee a oowg:. or / or og or / Oak [] how ha he orer o ague o he u o quare o hee coece. / ; / o (9) B (7) (8) a (9) we have 3 o o C. 3 α α o

13 MIIMUM HIG DISTAC STIMATIO 5 The up whe o α α 3 4 up whe. o α α 4 Hece he covergece o (a) o α α. 4 o α α 4 where Sep. We w ow prove he ao ure covergece o () ug he Prakaa-ao equa []. B Prakaa-ao equa [] we have og og P ep. 8 c M e coer a equece ( ) ee S where a equece o awh aag Aupo (A4). e chooe ( ) uch ha α α ; wh α. α α α. The or he geera er equece pove.

14 K. STAISAS MBK a OUAGIA HII 6 e eae he o a S S α α S. α S Takg a δ K here a pove coa c uch a. up c K We ca rewre he Prakaa-ao equa a oowg: 8 ep M c P. K B Aupo (A6) we have 8 M c Q a oow: ep α α Q P. ep up α α Q P ()

15 MIIMUM HIG DISTAC STIMATIO 7 We w ow oae he e epreo. up 4 4 α P B Aupo (A4) we oa 3 3 α α α 3 µ Q where. 3 α µ There e a equece V uch ha V β wh β or a cera rak Q β µ uer he cora. µ Q The ep β µ Q hereore. ep β µ Q

16 K. STAISAS MBK a OUAGIA HII 8 B equa () we have up 4 4 β α P up 4 4 β α P. up 4 4 α P Ug he Bore-Cae ea. whe a.. up 4 Hece he ao ure covergece o o zero. Sep 3. The covergece o he a (c). B (3) we have K K z z K. K

17 MIIMUM HIG DISTAC STIMATIO 9 The Taor epao a eghourhoo o a uer 3 o (A) we oa o k k k k k k K K [ ] K o K up k k k [ ()] o K k where. k k Uer (A) a (A4). a. up whe. The covergece o (a) () a (c) pe ea. We ee he oowg ea o prove he Theore. ea. Bera [] a H [5] coer F he e o a ee wh repec eegue eaure o. We ee he ucoa Θ F : T a oowg e e F g we poe { } : Θ Θ g H g H g A where H he Heger ace.

18 3 K. STAISAS MBK a OUAGIA HII I A(g) reuce o a uque eee he T(g) ee a he vaue o h eee. ewhere he chooe a arrar u uque eee o A(g) a ca T(g). Proo. See Bera [] a H [5] or proo. Proo (Ao ure covergece). Theore a coequece o ea a. We have up up up B ea The Sce coeque up. a..whe. P or a. ( ) ( ) H a.. whe. B ea couou a ao ure whe. T equa o Θ he he ucoa T he Heger opoog. Thereore ( ) T T

19 MIIMUM HIG DISTAC STIMATIO 3 Th acheve he proo o he Theore. Proo o Theore (Apoc ora). The oowg ea 3 a ea 4 were repecve prove Bera [] a Wu a Meczuk [4]. ea 3. e uppoe ha Aupo (A) a (A5) a he coo C a C o Theore are ae a ha e eror o Θ. So or a e equece { } coverge o he Heger erc we have ( ) T V ( ) A g where A a o-guar [ ( p q) ] ar whoe copoe o e o zero whe. A ea 4. e uppoe ha Aupo (A) (A) a (A4) are ae he K u u [ ]. he ruo o Proo. e ocu ow o he proo o Theore reerrg o he aove ea. B he ea 3 we have ( ) T V ( ) A g.

20 3 K. STAISAS MBK a OUAGIA HII Sce T ( ) a upg he equao aove we have V A g. The copoe o whe. The A ( ) B o p () where e eae he aw o B V. B o euce he aw o ( ) where V a V where ea orhogoa. B Aupo (A) ( ) a he oowg agerac equa we ca rewre B. ( ) B V. ()

21 MIIMUM HIG DISTAC STIMATIO 33 ruv () we have B V C () where V C ( ). Ug equa 3 3 wh γ γ a pog δ we ca ake C aoue vaue a oowg: V C ( ). V ( C 3 ) δ B ea 3 ( ) δ V. 4 up a.. wh

22 34 K. STAISAS MBK a OUAGIA HII he ( ) a.. wh. V couou a oue (or e). B appg Va heore o he equece ( ) W V ( ) C proa whe. e coer he r er o he rgh o quao () V. (3) Thereore ea 4 he ruo o (3) ( ) where V V K ( u) u 4 V V K ( u)u g 4 g K ( u) u. Hece he reu. Ackowegee We hak he aoou reeree or a careu reag a coe whch hepe o prove he qua o he oe.

23 MIIMUM HIG DISTAC STIMATIO 35 eerece []. Bera Mu Heger ace eae or paraerc oe A. Sa. 5(3) (977) DOI: hp://.o.org/.4/ao/ [] A.. B a O. H aeur u u e ace e Heger e proceu éare à ogue éore Cope eu Mahéaque 348(7-8) () DOI: hp://o.org/.6/.cra... [3] C. W. J. Grager a A. Aere O he ver o e ere oe Sochac Proce. App. 8() (978) DOI: hp://o.org/.6/34-449(78)969-8 [4] C. W. J. Grager a. Joeu A rouco o og-eor e ere oe a racoa erecg J. Te Sere Aa. () (98) 5-9. DOI: hp://o.org/./ [5] O. H O he eao o oear e ere oe Sochac a Sochac epor 5(3-4) (995) 7-6. DOI: hp://.o.org/.8/ [6] J.. M. Hokg Fracoa erecg Boerka 68() (98) DOI: hp://o.org/.93/oe/ [7] A. Kaagaé a O. H ao par e u e ace e Heger u proceu AFIMA Cope eu Maheaque 35(3-4) () DOI: hp://o.org/.6/.cra..7.5 [8] A. Kaagaé a O. H The qua au kehoo approach o aca erece o a o-aoar uvarae AFIMA proce ao Oper. Soch. qu. (3) (3) DOI: hp://o.org/.55/roe-3-4 [9]. Maora Mu ace eao o aoar a o-aoar AFIMA procee coo. J. () (7) DOI: hp://o.org/./ x.7.. [] A. r a O. H Heger ace eao o rog epee ueoa Gaua procee Ieraoa Joura o Sac a Proa (3) (3) DOI: hp://o.org/.5539/p.v3p7

24 36 K. STAISAS MBK a OUAGIA HII [] M. Oak O he ver o racoa erece AIMA procee Boerka 8(3) (993) DOI: hp://o.org/.93/oe/ [] B.. S. Prakaa ao oparaerc Fucoa ao ew York: Acaec Pre 983. [3] F. Sowe A Decopoo o Bock Toepz Marce wh Appcao o Vecor Te Sere GSIA Mograph Carege Meo Uver 989. [4] W. B. Wu a J. Meczuk Kere e eao or ear proce The Aa o Sac 3(5) () g