A Result of Convergence about Weighted Sum for Exchangeable Random Variable Sequence in the Errors-in-Variables Model

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1 AMSE JOURNALS-AMSE IIETA publcato-17-sere: Advace A; Vol. 54; N ; pp 3-33 Submtted Mar. 31, 17; Reved Ju. 11, 17, Accepted Ju. 18, 17 A Reult of Covergece about Weghted Sum for Exchageable Radom Varable Sequece the Error--Varable Model Me Yag Chogqg College of Electroc Egeerg, Chogqg 41331, Cha (mez1116@163.com) Abtract Th the dcue lear EV (error--varable) regreo model, that, regreo model wth meauremet error. Becaue practce, data are ofte obtaed wth meauremet error, EV model more ft for applcato tha the ordary regreo model. However, t more complcated the tattcal ferece ad aaly, o reearch about th theory very dffcult. Due to the applcato of tattc, whe the weght fucto ue real varable EV model, we exted the cotecy of the weghted um for the depedet radom varable equece ad obta a reult of covergece about the weghted um for the exchageable radom varable equece EV model. Key word Error--varable Model, The Weght Fucto Cota Real Varable, Covergece about the Weghted Sum. 1. Itroducto I the early 193, De Fett put forward the cocept of radom varable exchageablty [1]. The o-called exchageablty of fte equece {X k } k 1 refer to that f the jot dtrbuto of radom varable X1, X,, X uchaged dplacemet, that mea the jot dtrbuto to ay dplacemet of 1,,, o π, Xπ(1), Xπ(),, Xπ() ad o X1, X,, X are detcal. The fte equece {X k } + k 1 of exchageable radom varable exchageable, cae that ay fte ubet thereof exchageable. Theore have proved that a fte equece of exchageable 3

2 radom varable depedet detcally dtrbuted uder the tal σ-algebrac codto, o o woder that t aymptotc mlar to the depedet detcally dtrbuted equece. A the fudametal tructure theorem for fte exchageable radom varable equece, De Fett theorem tate that a fte exchageable radom varable equece depedet ad detcally dtrbuted wth the codto of the tal σ-algebra. So, ome reult about depedet detcally dtrbuted radom varable are mlar to exchageable radom varable. A the fudametal tructure theorem for fte exchageable radom varable equece, De fett theorem ot applcable to fte exchageable radom varable equece, ad t therefore eceary to fd other techologe to olve the approxmate behavor problem of fte exchageable radom varable equece. By ug the revere martgale approach, cholar have gve ome reult. I th paper, we do ome reearche about the mlarty ad dfferece betwee detcally dtrbuted radom varable ad exchageable radom varable equece ad maly dcu the lmt theory of exchageable radom varable. We uppoe X ad Y are radom varable ad (X, Y), =1,,.., are the ample from the paret. Here the ample (X, Y) a fucto of the mportace of x.,,, W x W x X X 1 If the followg codto are atfed:, 1,,, W x ; 1 W x 1. W(x) called the weght fucto The followg queto are propoed the lterature []: Uder what codto, whe, we obta 1 W x Y E Y X x a. (1) Dcuo o th ue rare at home ad abroad. 33

3 Whe we etablh the regreo model, we aume that the depedet varable fxed, ad that the depedet varable affected by radom factor or meauremet error. y, f x; I the above equato, x ad θ are vector of dmeo k, ad y, μ, ad ԑ are vector of dmeo, ad E(ԑ)=. For a log tme, the reearch o EV model ha ot codered the error model electo; other word, the model, the depedet varable ad the depedet varable are really coected by the fucto, but later t wa foud that ecoomc aaly, there ext error. I may cae actual producto, the depedet varable the meaured value ad thu wll be affected by radom factor. So whe the parameter a radom varable, the model ca be expreed a follow: x, y, f ; I the formula E(ԑ)=E(δ)=, ξ, ad θ are repreeted a parameter, we call th model the error--varable model. There are ome lterature about the error--varable model [3-1], Geerally we aume that X = X + ε rght, whe tudyg th model. Here X a varable that caot be drectly oberved whle X a varable that ca be oberved. Uder ormal crcumtace, the relatohp betwee X ad X complcated. For example, X = ψ(x, ε), where ԑ the error of meauremet depedet of (X, Y). ψ repreet a arbtrary kow fucto. I the oe-dmeoal lear tructural relato of the error--varable model y=a+bx, Y=y+ ԑ, X=x+u, f the parameter a,b are bouded cotuou fucto a(t), b(t) where the real varable t (,1)(b(t) ), the the EV model of the oe-dmeoal lear tructure wth varable coeffcet obtaed a follow: y a t b t x Y y, X x u Where x, y are a radom varable, ad (ԑ,u) are meauremet error. 34

4 Suppoe t (,1), ad we wat to etmate the parameter (a(t),b(t)) at t. If we are ot able to oberve tme at t, we jut have to oberve tme the vcty of t. Suppog t1, t,, t, are deg pot [,1], ad atfy t1< t< < t 1, we oberve (y,x) at every pot t, ad the wll get group of obervato of (Y, t, X), =1,,. If we ue thee group of obervato to etmate the parameter (a(t),b(t)) at t, we hould ote that the oberved value of (Y, t, X) at t are ot the ame a thoe at t. The mportace ca be meaured by the weght fucto W(t) of the real varable t, =1,,...,. We frt gve the followg defto: Suppoe (Y, t, X), =1,,,. are the ample take from the paret of (Y, X), that t1, t,, t are deg pot [,1], ad that t a pot wth the terval (,1). W (x) W (t, t 1, t ) the fucto of the real varable t1, t,, t, (=1,,..., ), ad we call t the real varable weght fucto, f t atfe the followg codto: W t, 1,,, ; W t 1 1. We aume the oe-dmeoal probablty dety fucto h (,1/), ad the we obta: ad that the badwdth t t h W t, 1,, t t 1 h W(t) called the kerel weght fucto.. Reult Here the weght fucto W(x) a real varable kerel fucto. W(t) tuded (1). We obta the cocluo o the exchageable radom varable of {Y } =1, ad obta the cotecy betwee the weghted um of =1 W (t )Y ad the weghted um of the equece of exchageable radom varable the EV model. The theorem aume t atfe the followg codto: 35

5 A1 For ay real varable kerel fucto {W (t )} =1, there ext the teger A: max W ( t ) 1 Alog h A Radom varable of Y, Y1, Y,, Y are the exchageable radom varable equece. Cov(Y1,Y)=, ad EY ext, ad there a potve umber D for whch, Var(Y) D, o h (1) If ( ) log hold, the W ( t) Y EY h () If ( ) log hold, the W ( t ) Y EY a.. W ( t) Y EY ad W ( t ) Y EY ad W ( t ) Y EY a.. ca be decrbed a follow: 1 W ( t ) Y EY a.. E Y EY, Var Y EY Var Y D, 1,,,, ; therefore, the theorem ca be chaged a follow: Theorem 1 For the weght fucto {W (t )} =1 of ay real varable uder the followg codto, there a potve umber A: A1 max W ( t ) 1 Alog h A Radom varable of Y, Y1, Y,, Y are exchageable radom varable equece. Cov(Y1,Y)=, ad EY=, ad there a potve umber D, for whch EY Var( Y) D, o 1 h If ( ) log 1 hold, the W t Y EY h () If ( ) log ( ) 1 hold, the roof: (1) becaue E W ( t ) Y 1 eed to prove: W ( t ) Y EY a.., to prove 1 W ( t ) Y EY rght, we oly 36

6 lm Var W ( t) Y 1 I fact, due to the reult of A1, A ad W t 1 1, we obta DAlog Var W ( t ) Y W ( t ) Var Y Dmax W ( t ) h h Becaue of the reult of ( ) log, we kow that lm Var W ( t ) Y rght, 1 ad the 1 W ( t ) Y EY (1) proved. ( ) a.. rght, for ay gve potve umber ԑ (uppoe () To prove W t Y EY ԑ<d/), we ote 1 1, Y Y I Y Y Y I Y The we obta Y = Y (1) + Y (), ad baed o EY=, we obta EY (1) = EY (), o 1 1 Y Y EY Y EY rght. 1 1 W ( t ) Y W ( t ) Y EY W ( t ) Y EY () 37

7 , There are two tep to prove the reult. Frt we eed to prove the reult a follow: whe W ( t ) Y EY a.. (3) We ote Z W t Y EY, exchageable radom varable equece, ad Becaue 1 1 ad b maxw t b 1 Alog h. Y EY Y EY, o we obta , o {Z } =1 the zero mea max Z maxw t Y EY max max 1 1 Y EY b b Db Becaue 1 1 Var Z Var W t Y EY 1 W t Var Y maxw t Var Y b D 1 Var Z 1 So b D rght. Becaue log h ad 1 log are rght whe, for arbtrary ԑ>, whe uffcetly large, log h ad 1 log equalty, we ote B=DA(A+1), the rght. From the reult of Beett expoetal Z exp 1 b D Db exp 1 1 B B B log 38

8 1 Becaue rght, 1 Z alo rght. 1 1 By Borel-Catell lemma, for arbtrary ԑ>, whe uffcetly large, we obta: 1 Z a.. Whe uffcetly large, we obta W t Y EY a that W ( t ) Y EY a.. We prove the reult a follow. Whe uffcetly large, W t Y EY a.. (4) Frtly, Y EY a zero mea exchageable radom varable equece, o 1 Y E Y I Y x I x df x Y df x ad o max E Y max E Y max E Y I Y max 1 I I max Y E Y max Y E Y max Y Becaue 39

9 Var Y E Y E Y E Y Var Y D we have Becaue Var Y E Y D. 1 xdf Y Y x Var Y E Y I Y x I x df x x 1 1 D whe uffcetly large, we obta 1 1 E Y D log (we ue the formula 1 above. Whe uffcetly large, 1 4D log.) Becaue 1 expoetal equalty, we obta 1 E Y D log rght ad due to the reult of beett log Y log exp 1 1 ( 1 ) Becaue 1 rght, we obta Y log 1 1 By Borel-Catell lemma, we obta 33

10 1 Y log a. 1 Becaue E Y D log ad 1 uffcetly large, we have Y log.. 1 a rght, whe 1 3 log W t Y E Y A a. h log Becaue of the reult of h, whe uffcetly large, 1 W t Y E Y a. (4) proved. 1 Whe uffcetly large, becaue we obta W t Y a.. ad (), (3), ad (4) are all rght, 1 W ( t ) Y EY a. The theorem proved. Cocluo I th paper, a the applcato of tattc, whe the weght fucto ue real varable error--varable model, we exted the cotecy of the weghted um for the equece of depedet radom varable, ad obta a reult of covergece about the weghted um for the equece of exchageable radom varable error--varable model. The exchageable radom varable are depedet ad detcally dtrbuted o, t ha ome properte of depedet detcally dtrbuto ad more applcable to ome tattc. The error-- 331

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