Estimation for Nonnegative First-Order Autoregressive Processes with an Unknown Location Parameter

Transcription

1 Appe Maheacs hp://oorg/0436/a03a94 Pubshe Oe Deceber 0 (hp://wwwscrporg/oura/a) Esao for Noegave Frs-Orer Auoregressve Processes wh a Uow Locao Paraeer Arew Bare Wa McCorc Uversy of Georga Ahes USA Ea: wpcc@ugaeu Receve Sepeber 0 0; revse Ocober 0 0; accepe Ocober 7 0 ABSTRACT Coser a frs-orer auoregressve processes where he ovaos are oegave rao varabes wh reguar varao a boh he rgh epo fy a he uow ef epo θ We propose esaes for he auocorreao paraeer a he uow ocao paraeer θ by ag he rao of wo sape vaues chose wh respec o a eree vaue crera for a by ag he u of over he observe seres where represes our esae for The o srbuo of he propose esaors s erve usg po process echques A suao suy s prove o eae he sa sape sze behavor of hese esaes Keywors: Noegave Te Seres; Auoregressve Processes; Eree Vaue Esaor; Reguar Varao; Po Processes Irouco I ay appcaos he esre o oe he pheoea uer suy by o-egave epee processes has crease A ecee preseao of he cassca heory cocerg hese oes ca be fou for eape Brocwe a Davs [] Recey avacees such oes have shfe focus o soe specaze feaures of he oe eg heavy a ovaos or oegavy of he oe I hs paper we eae he behavor of raoa esaes uer coos eag o o-gaussa s For eape he saar approach o paraeer esao wh he AR() process s hrough he Yue-Waer esaor; LS where () A sghy ffere approach presee Mahew a McCorc [] use ear prograg o oba esaes for a uer cera opzao cosras Whe here are ay esabshe ehos o esae he auocorreao coeffce a AR() oe here are us a few approaches o esag he uow ocao paraeer a AR() oe Oe was eoe McCorc a Mahew [] where hey cosere rage rage where rage a proves he e of he aa a a respecvey for I hs paper we eae esao quesos a asypoc properes of aerave esaes for a respecvey reag o he oe where 0 0 a s a sequece of oegave rao varabes whose ovao srbuo F s assue o be reguary varyg a fy wh e a reguary varyg a wh e where eoes he uow bu posve ef epo As a resu of o resrcg he ovaos o be boue o a fe rage we ca frs esae he auoregressve paraeer hrough reguar varao a fy a he esae he posve bu uow ocao paraeer hrough reguar varao a he ef epo Whe we have eoe a few esabshe esao proceures oe oabe ecepo was ha of au ehoo Ahough ypcay racabe a rcae he e seres seg whe he ovaos he AR() oe are epoea he au ehoo proce- Copyrgh 0 ScRes

2 34 A BARTLETT W MCCORMICK ure ha a aor corbuo o he esao of posve heavy ae e seres Wh hese coseraos Rafery [3] eere he g srbuo of he au ehoo esae for he auocorreao coeffce As a resu he esaor () was cosere The reazao of hs esaor was he seppg soe for he wor oe hs paper aog wh Davs a McCorc [4] whch frs cosere hs aerave esaor a use a po process approach o oba he asypoc srbuo of he aura esaor Ths was oe he coe ha he ovaos srbuo F vares reguary a 0 he ef epo a sasfy soe oe coo The wor presee hs paper s a eeso of he wor oe Davs a McCorc [4] cug he foowg corbuos o epee e seres wh heavy-a ovaos The frs corbuo voves he eveope of esaes for he auocorreao coeffce a uow ocao paraeer uer reguar varao a boh epos wh a rae of covergece where s sowy varyg fuco The seco corbuo voves usg a eree vaue eho eg po processes o esabsh he asypoc srbuo of he propose esaors a wea covergece for he asypocay epee o srbuo A a observao s ha our esao proceure s especay easy o pee for boh a Tha s he auoregressve coeffce he causa AR() process s esae by ag he u of he rao of wo sape vaues whe esao for he uow ocao paraeer was acheve hrough zg over he observe seres Ths auray ovaes a coparso bewee he esao proceure presee hs paper a he saar ear prograg esaes eoe above sce wh a oegave AR() oe he ear prograg esae reuces o he esae propose where aey eoes he AR() process Ths coparso aog wh he coparso bewee Mahew a McCorc s [] opzao eho a Bare a McCorc [5] eree vaue eho was perfore hrough suao a s presee Seco 3 The resus fou appear o eosrae a favorabe perforace for our eree vaue eho over he 3 aerave esaors The a proofs hs paper rey heavy o po process ehos fro eree vaue heory The essea ea s o frs esabsh he covergece of a sequece of po processes base o spe quaes a he appy he couous appg heore o oba covergece of he esre sascs More bacgrou forao o po processes reguar varao a wea covergece ca be fou Resc [6] Aso a ce survey o ear prograg esao proceures a oegave e seres ca be fou Aě [7] Aě [8] a Daa a McCorc [9] whereas ore appcaos o oeg he pheoea wh heavy ae srbuos a esug esao ssues ca be fou Resc [0] The res of he paper s orgaze as foows: asypoc resus for he auocorreao paraeer uow ocao paraeer a o srbuo of are presee Seco whe Seco 3 s cocere wh he sa sape sze behavor of hese esaes hrough suao Asypocs The foowg po process resu s fuaea Sce he resu aes o use of a ARMA srucure we prese for ore geera ear oes subec o usua suaby coos o he coeffces I ha regar for hs resu we assue ha 0 s he saoary ear process gve by wh c 0 c 0 for soe 0 Furher- ore for hs resu we ay rea our assupos o he ovao srbuo a we requre ha has a reguary varyg a srbuo e P 0 for a sowy varyg fuco a he ovao srbuo s a baace P P p a q as P P Defe po processes b a e eoe PRM o 0 0 where has Rao-Noy ervave wh respec o Lebesgue easure p 0 q 0 Le 0 be a array wh a epee of Defe 0 c Copyrgh 0 ScRes

3 A BARTLETT W MCCORMICK 35 Our basc resu s o show ha M p 0 equppe wh he opoogy of vague covergece whch s cose saee a spr o Theore 4 Davs a Resc [] I vew of he cooay of he wo resus we prese oy he eee chages o he Davs a Resc proof o accooae he curre seg Ase fro eepg rac of he e whe pos occur e arge ups he fferece he po processes cosere here wh hose Davs a Resc [] s he cuso of ars e he seco copoe of he po b Ths copcao uces a aoa wea epeece he pos whch s aresse Lea hrough a sragh forwar bocg argue Frs we esabsh wea covergece of are po processes of a oraze vecor of ovaos For a posve eger efe a po process b Le e 0 0 eoe he saar bass vecors for Defe a assocae are po process wh he frs copoe pace o a as by b e I he foowg Lea we show ha a are asypocay sgushabe he foowg sese Le 0 0 a E 0 Coser he cass of recages R : R c E a R Lea As es o fy p B B0foraB Proof Foowg he proof presee Proposo of Davs a Resc [] suppose ha B s such ha for soe Be As oe Davs a Resc [] for a oe has c 0 Observe ha B c b e c () Sary e c c0 a 0 The B c c c c E () where y y y E accorg o y c a y c Noe ha E E P c c b b o Thus fro ()-(3) we oba E B B b as P c o o (3) (4) The he resu foows as Davs a Resc [] Proposo copeg he proof Lea Le a be he po pro cesses o he space E efe by where 0 a b Y Y s a sequece wh Y a s epee of M p E The Proof We epoy a bocg argue o esabsh hs resu Le r be a sequece of egers such ha r r as a r o Le h r a Defe bocs s Is r s rs J rs rs for s h a J rh The s cear ha for s Wre s h I 0 I 0 s a b s b I J s Copyrgh 0 ScRes

4 36 A BARTLETT W MCCORMICK The Le B E h h s s be a so uo of recages (5) B B (6) where B c R wh R y Le F F eoe he ea easure of whch s PRM o E To copee he proof we frs show ha for a ses B of he for gve (6) ha P B 0 ep B (7) The above resu foows fro he easy verfabe reaos: a h B P B B 0 P h 0; r P s B s B O for ; P s B r P s B O ; r (8) (9) (0) P B B o ; () h B p 0as () Iee vew of (5) a () (7) s equvae o showg h P s B 0 ep B (3) s a he above reao hos by (8) (0) a () vz h P B 0 P B s s h r ep B B or I s eae ha for a recage h B c y E we have E B B (4) Therefore he resu s see o ho by (7) a (4) by appcao of Theore 47 Kaeberg [] Lea 3 Le a be po processes o he space E 0 a b e The M p E Proof We beg by appyg he argue use Theore of Davs a Resc [] wh he ofcao ha he reeva coposo of aps of po processes s gve by u v v u v u v ue v ue v ue v Each ap beg couous he coposo s a co uous ap fro M p 0 o M p 0 wh each space beg equppe wh he opoogy of vague covergece Therefore by he couous appg heore a Lea we oba b e e (5) Fay we copee he proof by Lea a (5) argug as Davs a Resc [] We are ow reay o prese our fuaea resu Theore Le a be he po processes o he space E 0 efe by where a b c 0 s a array wh The M p E PRM a s 0 a epee of Rear Apar fro coserg a e coorae a resrcg he process o a AR() process he above Theore a Theore 3 Mahew a Mc- Copyrgh 0 ScRes

5 A BARTLETT W MCCORMICK 37 Corc [] coser esseay he sae po process resu However her resu gave a wrog po process Ths error s correce he curre paper Proof Observe ha he ap z z z c z 0 uces a couous ap o po processes gve by z z z z c z 0 Thus we oba fro Lea 3 ha (6) c b c 0 0 z The resu ow foows fro (6) by he sae argue Davs a Resc [] o fsh her Theore 4 Reurg o he AR() oe uer scusso hs paper a he esae gve () we oba he foowg asypoc resu Theore Le 0 be he saoary souo o he AR() recurso a Uer he assupos ha 0 a he ovao srbuo F has reguary varyg rgh a wh e a fe posve ef epo EW P b e for a 0 where W 0 a b F Proof For 0 efe a subse Q : 0 The oe ha for he po processes b we have Q b 0 b Appyg Theore he case of a AR() process so ha c 0 we have 0 Noe ha as a subse of E 0 he se Q s a boue couy se wh respec o he po process so ha P b P Q 0 W P P 0 where W 0 s a sequece epee of Le (7) W The by Proposo 56 Resc [0] we have ha f G eoes he srbuo of W he s a Posso rao easure o E wh ea easure G where 0 Usg (7) we ca wre W 0 ep P P Q Q EW he resu foows Coroary Uer coos gve Theore Sce Q as Proof Sce b F we have p Bu hs pes as sce a s o-creasg Le us ow efe our esae of : I where we efe he e se I : a a b where 0 s a fe vaue Lea 4 Uer he assupos ha F s reguary varyg wh e a s posve ef epo a F s reguary varyg wh e a fy s rgh epo a he p a 0 as I where a F Furherore for ay y 0 P a y y P a y e Proof Sce we have ab b s a gh sequece Therefore sce by Theore a sce 0 we have a a b wh I Copyrgh 0 ScRes

6 38 A BARTLETT W MCCORMICK a I p 0 The frs saee ow foows sce a a I I For he seco saee observe 0 P a y P a y I P a y a I a 0 P a b a y P a b P a y o P b a y 0 (8) The resu for he seco saee ow foows fro (8) a he frs par of he ea Fay he efcao of he srbuo s we ow A usefu observao foows fro hs ea whch we sae as a coroary Coroary Uer he assupos of Lea 4 for ay y 0 Pb a y Proof By Lea 4 we have P b a y I P b a y 0 The resu he foows fro 0 E b a y a y I P a y P a y I Coroary aows a spfcao eerg he o asypoc behavor of by aowg us o repace wh The e ea w prove aoher usefu spfcao hs e o For a posve eger efe 0 Lea 5 Le U a U be efe as U U b b The for ay 0 a P U U 0 Proof We frs oe for ay posve M ha U P U U P U U U P M P U M U U I orer o cacuae Tha s we wre where P we paro U U so ha 0 Defe po processes a b where 0 have he srbuoa properes gve Theore Appyg Theore wh c for a c equa o 0 oherwse we oba a The eg for 0 u uv : u0 va v we have Seg P b P 0 0 V P a V Copyrgh 0 ScRes

7 A BARTLETT W MCCORMICK 39 we have where 0 P P V P 0 uv : u0 vauv Sce s Posso rao easure wh ea easure H where V H a we oba E V P b ep E V Ne sce U U b we have for arge ha b P M P M U U ep M M E V Therefore sce E V V P M 0 U U Ne oe ha fro he aw for he au obae above by repacg wh a by ag recprocas we erve he aw for u P U b ep EW where W has he srbuo of for ay eger 0 PU ep Thus for ay ha 0 we have for supsup PU copeg he proof M (9) Thus M arge eough Wh Lea 5 ha we ca ow focus our ae- o o he g o srbuo of U Ths w be accopshe by a bocg argue To ha e for a fe posve eger e r a efe bocs for r by J r r q a J r q r where q s a posve eger greaer ha Furherore e a Now we efe he eves J 0 r r J : or a y b J : or a y b 0 where r We beg by showg ha he eves are eggbe Lea 6 For ay y 0 a Proof Observe ha P 0 0 P P b b 0 (0) P a y y () Thus for soe cosa c a ay P =0 esabshg he ea Defe eves A a c B by A J : b B J a y a : Copyrgh 0 ScRes

8 40 A BARTLETT W MCCORMICK The foowg resu proves he asypoc behavor of he probaby of hese eves Lea 7 For ay 0 y 0 we have as y PA EW a PB Proof Sce he eves A are epee we have P J b P J b Usg Lea 6 we have ha P b P J O b P J O b Fro (9) we showe ha P U b ep EW Hece usg hs aw o oba Thus b P J b ep EW we PA EW as () Sary usg he resu of Lea 4 we oba y PB as (3) Hece he ea hos Lea 8 For soe cosa c P A B cp A P B Rear Sce he caray of J epes o r whch epes o a he eves A a B epe o a P B epe o a P A The cocuso of hs ea proves ha for a a here s a cosa epee o o paraeers for whch he equay sae here hos Proof To cacuae he erseco we efe he foowg ses a : a K K J : a K K J Now for a suffcey arge a y b b 0 a y I he foows fro (0) () a epeece ha If P b 0 O we have a y P a y b a y z P F z O b Therefore for soe cosa c P a y K b c I orer o hae se K observe fro cosruco of he bocs J a se K ha f K he he eves b a y a are epee Thus f we efe as a epee copy of he Copyrgh 0 ScRes

9 A BARTLETT W MCCORMICK 4 P a y Kb P a K b : P A B P A P B c y where B J a y a where we use Lea 7 he as sep Thus we have ha for soe cosa c P A B K b P a K b c OP A PB y a y whch copees he proof vew of Lea 7 Lea 9 For ay 0 y 0 P a b ep EW y y Proof Frs fro Lea 6 we have ha P c o P a y b as Ne by () (3) a Lea 8 we oba ha as es o fy P EW y Therefore we oba EW y ep EW y c P Hece P a b ep EW y y Theore 3 Le eoe he saoa ry AR() process such ha he ovao srbuo F sasfes F s RV afy a F s RV a If he fo r ay 0 y 0 we have P b a y EW y e where W 0 Proof Le us frs observe ha for 0 PU > a > yp U U > PU > a > y PU > a > y Thus by Lea 9 we oba ep EW y sup P U U f PU a y sup PU a y ep EW y Leg e o fy he above a he e o 0 we oba fro Lea 5 a EW EW ha PU a y ep EW y The heore ow foows fro hs a Coroary 3 Suao Suy I hs seco we assess he reaby of our eree vaue esao eho hrough a suao suy Ths cue a coparso bewee our esao proceure a ha of hree aerave esao proceures for boh he auocorreao coeffce a he uow ocao paraeer uer wo ffere ovao srbuos Aoay he egrḙe of approao for he eprca probabes of a o s respecve g srbuo was repore To suy he perforace of he esaors a I re Copyrgh 0 ScRes

10 4 A BARTLETT W MCCORMICK specvey we geerae 5000 repcaos for he o- buo a wh e of reguar varao a egave e seres 0 r wo ffere reguar varyg a wh a fe e sape szes (500000) where fo s a AR() whereas case ) he ovao srbuo F s process sasfyg he fferece equao reguar varyg a a wh o resrco o or for a Frs we eae he suao resus for 09 The auoregressve paraeer s ae o be he uer F for each of he s ffere vaues corage fro 0 o guaraeeg a oegave e seres sere by copug 5000 esaes usg a he uow ocao paraeer s posve whe he ovaos a rage are ae o be cz f z where a Fz proves he e of he aa a a z f z respecvey for a LS For hs ovao srbuo e c a be o egave cosas such ha c he hs s- f 0 rbuo s reguary varyg a boh epos wh - e of reguar var ao a fy a e of re- guar varao a For hs suao suy wo f 3 srbuos were cos ere: ) F c 0 ) F c where The eas a saar e- Now observe case ) he ovao srbuo F vaos (wre beow pareheses) of hese ess a Pareo srbuo wh a reguar varyg a sr- aes are repore Tabe aog wh he average Tabe Coparso of esaors for = 09 uer F 95% CI Avg Legh M es Ma es Rage es LS es a rage LS <0000 <0000 < (<0000) (0005) (0006) (009) <0000 <0000 < (<0000) (00009) (00009) (00) (0006) (00083) (00083) (005) (00008) (00064) (00064) (0086) (00078) (0046) (0039) (0076) (00056) (007) (0009) (007) (006) (00673) (006) (0038) (0033) (00367) (00384) (003) (00909) (0050) (0044) (008) (004) (00456) (007) (008) (0378) (047) (0073) (0069) (00834) (063) (0037) (0033) Copyrgh 0 ScRes

11 A BARTLETT W MCCORMICK 43 egh for a 95 perce eprca cofece ervas wh eac coverage Sce he a purpose of hs seco s o copare our esaor o Bare a McCorc [5] esaor a McCorc a Mahew [] esaor rage a Davs a Resc s [3] esaor LS he cofece ervas were recy cosruce fro he eprca srbuos of a rage a og LS respecvey To evauae a copare he perforace of four ocao esaors s ffere scearos for a are presee Tabe uer F Whe 5000 esaes for each esaor; I rage rage a e LS e rage were ob- ae The epoe se he e se I : a a b was se o 09 The eas a saar evaos (wre beow pareheses) of hese esaes are repore Tabe aog wh he average egh for a 95 perce eprca cofece ervas For coveece he eprca s- rbuos of rage q e rage w a q e LS w were respecvey use where he orazg cosas q a w are obae hrough Equaos (3)-(36) of McCorc a Mahew [] Rear I he case ha 0 e coverge a Tabe Coparso of esaors for = uer F rage e e M es 95% CI Avg Legh Rage es e (rage) es < (00087) (04068) (06775) (07657) < (0000) (03605) (0438) (04960) (00879) (03566) (06656) (07503) (00503) (0399) (043) (04780) (04697) (07553) (06733) (0763) (04045) (0343) (0476) (048) NA NA 57 8 (-) (0940) (0537) (03706) NA NA (-) (094) (0438) (0565) (03863) (09569) (0598) (0385) (063) (09308) (0446) (070) (03604) (09443) (057) (0398) (0956) (0900) (04398) (0837) e (LS) es Copyrgh 0 ScRes

12 44 A BARTLETT W MCCORMICK a faser rae ha e a he case ha 3 e coverges a a faser rae ha e Lasy sce he McCorc a Mahew [] paper has he resrco ha Var oy whe ca he esaors e a e be fary copare whereas oy whe s our esaor appcabe Now observe for he seece vaues beg cosere Tabe shows ha our esaor perfors a eas as we as he hree oher aerave esaors Ths s parcuary rue uer he heaver a oes e whe 0 I hs rege our esae shows e bas a he average eghs of he cofece ervas are saer ha he oher hree esaes soees by a we arg I parcuar whe 08 a = 000 he 95% cofece erva average egh for our eho s a 3 es saer ha he hree aerave esaors respecvey Ths s par ue o he use of oe-se cofece ervas sce for a Nauray whe 3 Davs a Resc eas square esaor s ore effce ha a hree eree vaue esaors Whe our esaor w aways perfor sghy beer ha he a esaor Bare s a McCorc [5] esaor a a avaage es wh s versay o perfor we for varous oegave e seres cug bu o resrce o hgher orer auoregressve oes aog wh ARMA oes Tabe reveas ha our esaor for geeray perfors beer ha he hree aerave esaors for whe 0 Ths s parcuary rue whe coparg average cofece erva eghs Ahough a hree esaors a rage e coverge o he rue vaue of he paraeer as es o fy respecvey hs seg hey ay o copee asypocay wh say a cooa eas square esaor e whe Noeheess for sa sape szes our suao suy favors rage over he oher hree esaors The ffcuy for a eas square esae s ha a sa egave bas for he esae of he auocorreao paraeer gves rse o a uch arger posve bas he esae of e Whe he affec s o as grea he posve bas fou our esaor a he ohers for has a sgfca effec o he esae for Fgures -4 show a coparso be wee he probaby ha esaors a rage a LS are wh 00 of he rue auocorreao paraeer vaue respecvey Wh a sape sze of 500 hese fgures poe he sape fraco of esaes whch fe wh a bou of 00 of he rue vaue Goo perforace wh respec o hs easure s refece curves ear o 0 wh shg goo behavor as curves approach 00 Whe 0 he fgures see o show ha our esaor copare o he oher hree Fgure P 00 for RV Copyrgh 0 ScRes

13 A BARTLETT W MCCORMICK 45 Fgure P a 00 for RV Fgure 3 P rage 00 for RV Copyrgh 0 ScRes

14 46 A BARTLETT W MCCORMICK Fgure 4 P LS 00 for RV Fgure 5 Eprca vs heoreca probaby Copyrgh 0 ScRes

15 A BARTLETT W MCCORMICK 47 prouce a hgher fraco of precse esaes especay copare o Davs a Resc esaor Whe he reguar varao e vaue s coser o we see a hgher fraco of he Davs-Resc esaes showg beer accuracy by hs easure The fgures aso cae ha McCorc a Mahew s rage esaor prouce a cosse hgh fraco of precse esaes whe Lasy we perfore a Moe Caro suao o suy he egree of approao for he eprca probaby Pb P a y a Pb a o s - EW g vaues e e y a e EW y respecvey The eprca srbuos were cacuae fro 5000 repcaos of he oegave e seres 0 for a sape sze of 5000 where M 3 EW a M was se o Aoay we resrce The op wo pos Fgure 5 beow shows he perforace whe F a he auocorreao coeffce s 09 for a equa o 08 5 respecvey Observe for 0 7 ha he eprca a probaby b rrors he heoreca probaby que cey The ower ef po Fgure 5 spays he asypoc perforace whe F a he ocao paraeer s for Noce ha he covergece rae of he eprca probaby o he heoreca probaby s ereey sow Ths s o surprsg sce o average our esae fas ore ha 0 fro he rue vaue whe 08 The ower rgh po Fgure 5 spays he asypoc perforace whe F for he o srbu o of Ob - serve ha hs po sofes he asypoc epeece bewee b a a REFERENCES [] P J Brocwe a R A Davs Te Seres: Theory a Mehos Eo Sprger New Yor 987 [] W P McCorc a G Mahew Esao for Noegave Auoregressve Processes wh a Uow Locao Paraeer Joura of Te Seres Aayss Vo 4 No 993 pp 7-9 o:0/ b0030 [3] A E Rafery Esao Efcace Pour u Processus Auoregressf Epoee a Dese Dscoue Pubcaos e Isu e Sasque e Uversé e Pars Vo 5 No 980 pp 65-9 [4] R A Davs a W P McCorc Esao for Frs- Orer Auoregressve Processes wh Posve or Boue Iovaos Sochasc Processes a Ther Appcaos Vo 3 No 989 pp o:006/ (89) [5] A Bare a W P McCorc Esao for No- Negave Te Seres wh Heavy-Ta Iovaos Joura of Te Seres Aayss 0 hp://oebraryweyco/oura/0/%8issn % /earyvew [6] S I Resc Po Processes Reguar Varao a Wea Covergece Avaces Appe Probaby Vo 8 No 986 pp o:0307/4739 [7] Jří Aě No-Negave Auoregressve Processes Joura of Te Seres Aayss Vo 0 No 989 pp - o:0/ b000 [8] Jří Aě No-Negave Lear Processes Appcaos of Maheacs Vo 36 No 99 pp [9] S Daa a W P McCorc Boosrap Iferece for a Frs-Orer Auoregresso wh Posve Iovaos Joura of Aerca Sasca Assocao Vo 90 No 995 pp o:0080/ [0] S I Resc Heavy-Ta Pheoea: Probabsc a Sasca Moeg Sprger New Yor 007 [] R A Davs a S Resc L Theory for Movg Averages of Rao Varabes wh Reguary Varyg Ta Probabes Aas of Probaby Vo 3 No 985 pp o:04/aop/ [] O Kaeberg Rao Measures Aaee-Verag Ber 976 [3] R A Davs a S Resc L Theory for he Sape Covarace a Correao Fucos of Movg Averages Aas of Sascs Vo 4 No 986 pp o:04/aos/ Copyrgh 0 ScRes