STRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING

Transcription

1 Joural of tatstcs: Advaces Theory ad Alcatos Volume 5, Number, 6, Pages 3- Avalable at htt://scetfcadvaces.co. DOI: htt://d.do.org/.864/jsata_7678 TRONG CONITENCY FOR IMPLE LINEAR EV MODEL WITH v/ -MIXING ERROR DI HU ad PINGYAN CHEN Deartmet of Mathematcs Ja Uversty Guagzhou, 563 P. R. Cha e-mal: Abstract I the aer, the smle lear errors--varables (EV) model wth v/ -mg errors s studed. Uder sutable mg coeffcet rate ad momet codtos, the covergece rates of strog cosstecy for the least square estmators of arameters the smle lear EV model are obtaed.. Itroducto ad Ma Result The regresso models are wdely used may area because of ther good tutve ature ad etesve alcabltes. However, the measuremet errors est durg the collecto ad statstcs data. ce the ordary regresso models do ot do well cosderg measuremet errors, the errors--varables (EV) regresso models come to beg. Mathematcs ubject Classfcato: 6F5. Keywords ad hrases: smle lear errors--varables model, least square estmator, strog cosstecy, v/ -mg radom varable. Receved Jue, 6 6 cetfc Advaces Publshers

2 4 DI HU ad PINGYAN CHEN The most basc form of the regresso models s the smle lear EV model: η θ β,,, (.) wth the followg assumtos: () θ, β,,, are ukow costats (arameters); () (, ),, are radom vectors; (3) η,,, are observable. By (.), we have η θ β ( β ),. (.) The cosderg (.) as a ordary lear regresso model of η o wth the errors β, we get the least square (L) estmators βˆ ad θˆ of the ukow arameters β ad θ as βˆ ( )( η η ), ( ) θˆ η βˆ, (.3) where ad η η. Deato [] frst added the measuremet errors to the usual regresso model ad roosed the cocet of EV model. The chage makes EV model more ractcal tha the usual model. Fuller [] summarzed the early results about measuremet errors lear EV model. Mttag [3] studed the estmatos of the ukow arameters smle EV model. Gao ad Lag [4] roved the strog cosstecy ad asymtotc ormalty of geeralzed least square estmators lear EV model. The strog ad weak cosstecy estmators of the arameters varyg-coeffcets structural lear EV model were roved by Ouyag [5]. Lu ad Che [6] roved that the weak ad strog cosstecy are equvalet. Mao et al. [7] geeralzed the results of Lu ad Che [6] ad obtaed the rate of the strog cosstecy.

3 TRONG CONITENCY FOR IMPLE LINEAR 5 All of the above results are uder the codto that errors are deedet detcally dstrbuted radom varables, ad ths codto s very dffcult to meet the real world. o, there are stll a lot of restrctos o ractcalty. I recet years, may authors ut ther eyes to weakeg the assumto of deedece. Fazekas ad Kukush [8] started to add α -mg errors to the olear EV model, however, they cosdered that the L estmators of the ukow arameters were ot cosstecy, ad just roved the cosstecy of aother alteratve estmators. Also uder the assumto of α -mg errors, the cosstecy of the estmators whch were based o the Fourer trasform EV model was studed by Bara [9], wthout cosderg the L estmato. Fa et al. [] gave the strog cosstecy of the L estmators uder the codto that the errors varables are statoary α -mg radom varables, but he dd ot show the covergece rates. Mao et al. [] assumed that the errors varables {(, ), } are egatvely deedet radom sequeces, ad roved the strog cosstecy ad asymtotc ormalty of the L estmators of β ad θ. I the aer, uder the assumtos that the errors varables are v/ -mg, we ot oly rove the strog cosstecy of the L estmators βˆ ad θˆ of the ukow arameters β ad θ, but also obta the covergece rates of β ˆ β ad θˆ θ. Although a sequece of v/ -mg radom varables s also α -mg (see Lu ad L []), but we have more lmt roertes uder obta more eact results EV model. We frstly troduce the cocet of the radom vectors). v/ -mg setu ad hece we ca v/ -mg radom varables (or

4 6 DI HU ad PINGYAN CHEN Defto.. Defe the v/ -mg coeffcet for { X, }, a sequece of radom varables or radom vectors, as v/ ( ) su ( ), ( ) ( ) P AB su m m, P A P A F B F P( A) P( B) B m m where F σ( X : m). The { X, } s sad to be v/ -mg, f v /( ) as. The cocet of v/ -mg s troduced by Blum et al. [3]. For more detal, we refer to the book of Lu ad L []. We ow state the ma results. ome lemmas ad the roofs of the ma results wll be detaled the et secto. Theorem.. Uder model (.), suose that {(, ), } s a sequece of detcally dstrbuted v- / mg radom vectors wth v/ ( ) <. If E E, E <, E <, ad lm, (.4) for some, where ( ) ad. The / ( ˆ β). β (.5) Remark.. It s easy to show that by (.4) sce, thus β ˆ β almost surely by (.5). o, we ot oly obta the strog cosstecy estmators of β, but also obta the covergece rate. Theorem.. Uder the codtos of Theorem., further assume that α lm su <, (.6) for some α (, ). The

5 TRONG CONITENCY FOR IMPLE LINEAR 7 α ( θˆ θ). (.7). Lemmas ad Proofs of Ma Results To rove the ma results, we eed the followg two lemmas. The frst oe s from Lu ad L [] ad the secod oe s due to Yag [4]. Lemma.. Let { X, } be a sequece of detcally dstrbuted r v-mg / radom varables wth v /( ) <. uose that E X < for some r <, the r ( ) ( ( / X ) EX o ). (.) Lemma.. Let, { X, } be a sequece of detcally dstrbuted v-mg / radom varables wth v/ ( ) <, { a,, a array of costats wth su ma a < ad } θ lm su a <, (.) for some θ (, ). The EX ad E X < mly that a X. (.3) Proof of Theorem.. By (.) ad (.3), t s easy to show that βˆ β ( )( η η ) β ( ) ( ) [( ) ( )] [ β( ) ( )] β [( ) ( )] ( )

6 DI HU ad PINGYAN CHEN 8 ( ) ( )( ) ( ) ( ). β β (.4) Hece to rove (.5), t suffces to rove that ( )., (.5) ( )( )., β (.6) ( )., (.7) ( ). (.8) By Lemma.,. ad., E E (.9) Thus by (.4) ad (.9), ( )., ad. The ( ) [( ) ].

7 TRONG CONITENCY FOR IMPLE LINEAR 9 o (.5) ad (.7) hold. et ( ).,, a The a for all ad by Lemma., ( )., β a whch follows (.6). We fally rove (.8). Note that ( ) ( ) ( )( ) ( ). The by the Hölder s equalty, ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ). (.) By (.4) ad (.5), ( ) ( ).,

8 DI HU ad PINGYAN CHEN whch by (.) esures that ( ).,.e., (.8) holds. The roof s comleted by (.5), (.6), (.7), ad (.8). Proof of Theorem.. By (.3), we have θ ˆ ( ˆ ) ( ˆ θ β β β β ) β. (.) By E <, E < ad Lemma., we have for ay α (, ) α α.,. (.) By Theorem., β ˆ β almost surely ad hece by (.), By Theorem. ad (.6), we have α ( β βˆ ). (.3) α α ( ˆ β β ) ( β βˆ ). (.4) Therefore, by (.), (.), (.3), ad (.4), α ( ˆ α θ θ) [( β βˆ ) ( β βˆ ) β ].,.e., (.7) holds. The roof s comleted. Ackowledgemets The research s suorted by the Natoal Natural cece Foudato of Cha (No. 76).

9 TRONG CONITENCY FOR IMPLE LINEAR Refereces [] A. Deato, Pael data from tme seres of cross-sectos, J. Eco. 3() (985), 9-6. [] W. A. Fuller, Measuremet Error Models, New York, 987. [3] H. J. Mttag, Estmatg arameters a smle errors--varables model: A ew aroach base o fte samle dstrbuto theory, tat. Paers 3() (989), [4] Y. F. Gao ad H. Lag, O geeralzed least square estmates EV lear regresso model, Mathematcs Practce ad Theory (4) (996), ( Chese). [5] G. Ouyag, O arameter estmato for lear varyg-coeffcets structural EV models, Acta Math. Al. ca () (5), ( Chese). [6] J. X. Lu ad X. R. Che, Cosstecy of L estmator smle lear EV regresso models, Acta Math. c. er. B, Egl. Ed. 5() (5), [7] Y. Mao, K. Wag ad F. F. Zhao, ome lmt behavors for the L estmator smle lear EV regresso models, tatst. Probab. Letters 8() (), 9-. [8] I. Fazekas ad A. G. Kukush, Asymtotc roertes of a estmator olear fuctoal errors--varables models wth deedet error terms, Com. Math. Al. 34() (997), [9]. Bara, A cosstet estmator for lear models wth deedet observatos, Comm. tats.: Theory Methods 33() (4), [] G. L. Fa, H. Y. Lag, J. F. Wag et al., Asymtotc roertes for L estmators EV regresso model wth deedet errors, Asta Advaces tatstcal Aalyss 94() (), [] Y. Mao, F. F. Zhao, K. Wag et al., Asymtotc ormalty ad strog cosstecy of L estmators the EV regresso model wth NA errors, tat. Paers 54() (3), [] C. Y. Lu ad Z. Y. L, Lmt Theory for Mg Radom Varables, Dordrecht/Bejg: Kluwer Academc Publshers/cece Press, 997. [3] J. R. Blum, D. L. Haso ad L. Koomas, O the strog law of large umbers for a class of stochastc rocesses, Z. Wahrsch. Verw. Gebete (963), -. [4]. C. Yag, Almost sure covergece of weghted sums of mg sequeces, J. ys. c. Math. cs. 5(3) (995), g