Relatnhp between tatn ued b ew Grant Applcatn, and Regren Predctr Develpment f Gnzala wth Suggeted Change t Cmmn tatn Baed n Gnzala and Stanek ntrductn Ed Stanek We lt ntatn ued n tw prncpal dcument, wth a uggeted cmmn ntatn. The purpe f th ltng facltatn f cmparn f the dcument, and checkng reult. Dcument ued n develpng th dcument are: c08ed0v6.dc A vern f the grant prpal t be ubmtted fr revew n 008. Gnzala&Snger&Stanek008v.te Etmatn f Fnte Ppulatn Parameter wth aular nfrmatn and repne errr. Draft manucrpt. Other dcument that were cndered n the develpment are: c08ed04.dc (an update n c07ed66.dc A develpment f a predctr f a meaure f acatn mlar t the develpment f Gnzala ung dfferent ntatn ntrduced b Stanek. c08ed05.dc A develpment f a predctr wth errr n aular varable, and meaurement errr n repne, cmbnng reult f and Gnzala ung dfferent ntatn ntrduced b Stanek. W_errr_n_X_v0-ej.dc An edted vern f Wenjun develpment f a predctr f the repne mean cntrllng fr an aular varable when there endgenu meaurement errr n the aular varable. tatn We ummarze ntatn n a table, gvng epren fr varu uantte ued n the dfferent dcument. C08ed03v3.dc 9/5/008 :06 PM
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v Fnte Ppulatn,...,,...,,..., w (ue when w (nt wted ( (nt wted ( μ (nt wted μ (nt wted w w μ μ w (eample nl w ( ( β μ μ (weghted ( β β β β (weghted μ μ μ (weghted μ μ (weghted,...,,..., ( ( μ ( k μ μ ( ( ( ( ( ( C08ed03v3.dc 9/5/008 :06 PM
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v μ μ ( μ μ μ μ μ ( μ μ μ μ (weghted μ μ μ 0 ( μ z ( 0 z ( z ( ( z ( z ( z ( γ / ρ M M O M ( Σ Σ ( z μ ( z μ Σ ( z μ ( z μ z z Σ ( z z Σ M M O M (weghted C08ed03v3.dc 9/5/008 :06 PM 3
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v Addng Samplng U U u (realzed value f U U (( U U (( U U ( U ( U ( U U U U U U U U U ( U ( U U U U ( U U U U U U U U n U U ( n n π E U U U (weghted U U (weghted U U b Uβ d U (wted, centered uβ (realzed μ + b μ + d (weghted b Uβ E μ d U (wted, centered b b b b d U ( d d d d d U U d U (centered C08ed03v3.dc 9/5/008 :06 PM 4
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v Z Uz Z Uz U( Z vec U ( vec ( U U vec ( U d Z vec U ( vec ( U U vec ( U d X U (nt centered E vec Z μ (wted, nt centered S ( ( ( vec( Z Σ P cv vec ( Z Σ P var S ( vec, varξ ξ (amplng nde S (amplng nde - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - P J P P n, n n n, var P ξ V V, varξ V, V V P n, V P ( n, V V J,, / n ( n P b J ab, a a Z Σ P (wted - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C08ed03v3.dc 9/5/008 :06 PM 5
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v Addng Endgenu Meaurement Errr t R nde fr epectatn wth repect t meaurement errr when weght are eual t r + Er + E (unwted E Er (r E Ek (unwted, r E ( E E E (r ( E ( E 0 0 R r ( E var R r e e e E ( E E E E E (unwted, r E E (unwted, r R var R e r we r we (unwted, r general weght E (wted, r ER( E r 0 R 0 ( E w ( E w var R r e e e e var R E r e E E (wted, r var R e e e e (wted, r (wted, r E ( E r Er E r (wted, r E ( E E E (wted, r Addng Endgenu Meaurement Errr t R nde fr dt. f meaurement errr U??? r,..., R X r + Wr + W (aume and r W Wr ( R W (aume and r ( ( ( r 0 ( W W W W W ( W W W W ( R R R W ( E W E ( W 0 var R r e e e ( W W W W U var e e e C08ed03v3.dc 9/5/008 :06 PM 6
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v Addng Samplng t Endgenu Meaurement Errr t r Ur r UEr Ur UEr E E ( R ( R U + UE + E ( E UE ( E E E R Addng Samplng t Endgenu Meaurement Errr n W UW X U + W Z vec X Z Z+ U E W ( d d d Z ( d U + W U + W k d U + UW μ E X Z vec U ( + U ( E W vec ( ( + ( + U E U W vec( ( + U W d Z vec U ( + U ( E W vec ( ( + ( + U E U W vec( ( + U W d C08ed03v3.dc 9/5/008 :06 PM 7
Prped Cmmn tatn W_errr_n_X_v0 Gnzala008v Addng Egenu Meaurement Errr t % r μ + b + E% r E E % R ( r 0 ( E% var R r ( E % E % E % E % Mdel ntatn X X X X degn matr fr MV mdel fr Z (eample X where X ( p ( ( X X X X ample degn matr fr MV mdel fr Z (eample X n p X X X X remander degn matr fr MV mdel fr Z (eample X n p + vec Z G (wth (te: When Z, X X X degn matr fr Z X X degn matr fr + + X degn matr fr Z Xμ + b X + b+ E μ X b μ + X b var V V, ξ V, V V P n,, ( n, V P, V V J - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Z vec U z ( Z vec U ( b Ub [ ] b U b, d vec Ud d vec[ U d ], Z X μ + b d b d + X z + μ Z + X b d d U d, d ( d d d + X E ξ μ X z r E ξ μ where X n, X n Z + X Z X 0 0 V V, V varξ V Σ P n,, V Σ P n,, V, V, Σ J Z V, V n ( n - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -,, / n ( n C08ed03v3.dc 9/5/008 :06 PM 8
Z X μ + b d + E 0 + X X μ where X n, X n Z X 0 0 V V, VR 0 e 0 V varξ R + V VR VR VR ΣR n, VR ΣR n, and ΣR Z V, V 0 VR 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - z ( vec U E 0 Z Z Z Z vec U ( E 0 + E ξ R μ z r E Z + X ξ Mdel ntatn Z X μ z + b d + E W + X X μ where X n, X n Z X 0 0 e 0 0 V V, VR 0 e 0 V varξ R + V VR VR VR ΣR n, VR ΣR n, and ΣR where 0 e 0 ΣR Z V, V 0 VR 0 ΣR M M O M 0 0 e Z Repreentatn fr develpng predctr. Z Target ntatn c c c c c c c Z Z vec U E W Z Z vec U ( E W + E ξ R μ z r E Z + X ξ c ( c c c ( n n+ n+ T c g c g c Eample: T cz gz + g Z g c g c Eample: c, 0 c, ( C08ed03v3.dc 9/5/008 :06 PM 9
T α α w T ( 0 n g g Z w μ / where Z when C08ed03v3.dc 9/5/008 :06 PM 0
Slutn ntatn Tˆ g ˆ, ( ˆ + g Xα + V V Xα ˆ X V X X V α ˆ + ( k / ( + e T k T% ˆ μ + % + ( ˆ k μ ˆ n n T + γ d n d n n n n n n ˆ μ w% k / w % % w% / ( + n d d n ( d d d, ˆ g ˆ ˆ ˆ ˆ X + VV Z X + g X + V V Z X T α α α α ˆ X V X X V Z α C08ed03v3.dc 9/5/008 :06 PM