Investigation of Partially Conditional RP Model with Response Error. Ed Stanek

Size: px

Start display at page:

Download "Investigation of Partially Conditional RP Model with Response Error. Ed Stanek"

Bertram Barrett
5 years ago
Views:

1 Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a dfferet model s postulated We call ths model a partally codtoal model The results ths documet were developed c06ed06doc The prcpal dfferece s that we mage that whe addg respose error, we cosder the model W Uy uw stead of the model * * W Uy UW A smple example llustrates that the model for μ U β uw s meagful, eve though the model cludes both U ad ts realzato, u For example, let X represet a radom varable wth equally lkely realzatos, x j j, for j,,, ad Z represet a secod radom varable wth equally lkely realzatos, zj j, for j,, Wthout ambguty, we ca express Z X x, x s a realzato of X the model for, the dcator radom varables U correspod to X EXAMPLE OF ALTERATVE PREDCTOR WTH RESPOSE ERROR Settg: Expected Respose: y s s,, Respose: sk ys Wsk k,, r s dex of order of measure 3, u,, dex for possble values of respose error equally lkely Assumptos: EW sk 0 for all s,,, k,, r s var Wsk s for all s,,, k,, r s var Wsk, W s* k* 0 for all s,, ; s*,, ; k,, r s k* k,, r s Parameters: μ ys, βs ys μ, y s μ, e se s s ; s 3 FTE POPULATO otato ad termology C07ed63doc /9/008 6:33 PM

2 Partally Codtoal Radom Permutato Model 7- Expected Respose: y s s,, Respose: sk ys Wsk k,, r s dex of order of measure Assumptos: EW sk 0 for all s,,, k,, r s W var sk se for all s,,, k,, r s var Wsk, W s* k* 0 for all s,, ; s*,, ; k,, r s k* k,, r s Assume r s for all s,, so that k measure of respose Parameters: μ ys, βs ys μ, ys μ, e se s s s Addtoal otato: U U U U, Defe y y y y, sk k k k Uy, R sk k k k W W W W W ad U U U U U u u u u u W uw uw s sk s k Uy uw W yu Wu vec y vec U W vec u 4 DEFE THE EXPADED MODEL Let sk Us ys uswsk s Wsk s Us ys ad W sk uswsk Defe ; C07ed63doc /9/008 6:33 PM

3 Partally Codtoal Radom Permutato Model 7-3 ys U y Uy Uy U s W Wsk u Wk uwk uwk u s Wk Wk Wk W U y u W U y u W U y u W k k k Defe the expaded radom varables as Uy Uy Uy uw k uwk uw k Uy Uy Uy uwk uwk uw k vec vec or Uy Uy Uy uwk uwk uwk W uw k uw k uw k uw k uw k uw k vec W, uwk uwk uwk vec 5 EXPECTED VALUE AD VARACE Deote expected value wth respect to samplg va a subscrpt, ad expected value wth respect to respose error va a subscrpt R E y W 0 so that E R E y R Expected Value: C07ed63doc /9/008 6:33 PM 3

4 Partally Codtoal Radom Permutato Model 7-4 Varace: varr E varr var ER E varr W var ow, uw k uw k uw k uw k uw k uw k varr vec W uwk uwk uwk Du W Du W Du W var R vec D D D D D D D D D D D D D D D D D D Du D D w u D u D D D D D u w u u w u u w u u w u u w u u w u w u u w u u u 0 D u ad 0 0 u We express ths as var R W Dvec u D D w vec u D w e e e Usg ths expresso, or var D D D P Δ vec u w vec u R var P Δ Δ y s P y s, ad s s 6 COLLAPSG TO THE USUAL RADOM VARABLES We collapse the expaded radom varables to a vector of the usual radom varables To do so, we pre-multply by The C07ed63doc /9/008 6:33 PM 4

5 Partally Codtoal Radom Permutato Model 7-5 s s s s s s s sk s sk s sk s s s s s s vec U y U y U y vec u W u W u W k k k vec vec W W W W W ad var k uswsk s Also, we fd that E R vec vec u u D D D P Δ R w ow vec u We defe D u ad Δ As a result, var R u D w u P e ud u w us se s ud u w μ P Ths defto makes use of a assumpto that oly oe elemet of u s s oe, ad all others are zero t would be possble to make other assumptos about the elemets, u s for,, ad s,, The k e P var R e e k 7 PARTTOG TO THE SAMPLE AD REMADER C07ed63doc /9/008 6:33 PM 5

6 Partally Codtoal Radom Permutato Model 7-6 We defe sample ad remader radom varables by parttog usg 0 L L, resultg L 0 L x k k k k ad X The, E R μ ad var X k var R k k k k k R V V, V, V 8 TARGET PARAMETERS, RADOM VARABLES, AD TERMOLOG We assume that there s a terest a target that ca be defed as a lear combato of the expected value over respose error of the collapsed radom varables We use the sample data to estmate/predct the target f the target s a fxed costat, the we use the sample data to estmate the target f the target s a radom varable, the we use the sample data to predct the target We oly cosder ferece for a sgle target ot jot targets We represet collapsed radom varables by E R L ad partto g represetg We defe a target as P g P g g g g g so that 9 DEVELOPG THE BEST LEAR UBASED PREDCTOR BLUP 976: We defe the BLUP of P as P, P satsfes the followg crtera see Royall, Lear the sample: P g Ubased: E 0 R P P Mmum MSE: var R P P a s mmzed C07ed63doc /9/008 6:33 PM 6

7 Partally Codtoal Radom Permutato Model 7-7 order to develop the BLUP of P, we frst preset expressos for P P ad ts varace Frst, ote that P P g a g g V V, We represet var The V, V V V V, var R V V V, V, V, V Recall that V k ad V Let us defe k VR k so that V V V R The 0 VR 0 0 V V, 0 var R 0, V V The Ubased Costrat: We ca expad the ubased costrat: X E P P a g γ, such that X E P P ax g X γ order for ths expresso to equal zero for ay value of γ, the ubased costrat wll be always be satsfed whe ax g X 0 Ths s troduced as a costrat usg Lagraga multplers whe mmzg the MSE Fdg the Mmum MSE: otce that V V V, g a var R P P g a g g V V V, g, or,, V V V g V V, a var R P P a g g a VR g a V, V g C07ed63doc /9/008 6:33 PM 7

8 Partally Codtoal Radom Permutato Model 7-8 Expadg ths expresso, var P P, av a g V g V a g V g g V g R R cludg the costrat va a Lagraga multpler, we seek to fd the value of a that wll mmze f a, λ av a g VR g V, a ax g X λ g Vg gvr g Dfferetatg wth respect to a ad λ, f a, λ Va V Rg V, g Xλ ad a f a, λ Xa Xg λ f a, λ, we set these dervatves to zero smultaeously, To fd the value of a that mmzes ad solve for a The estmatg equatos are gve by f a, λ a V X a VR g V, g 0 f, a λ X 0 λ Xg 0 λ V X a VR g V, g or equvaletly, by The soluto to ths equato s gve X 0 λ Xg by a, V V X X V X X V V g V g V X X V X X g We ca ow express the best lear ubased predctor Recall that P g g, P g a, the best predctor replaces a by â Let us defe R ad we predct P by α X V X X V The a g VR g V, V V X X V X X V α α, α g X X V X X V g VR V X g X V V X As a result, the best lear ubased predctor s gve by P g, R α α α V V X g X V V X We ca express the predctor a slghtly dfferet maer by substtutg VR V V, resultg C07ed63doc /9/008 6:33 PM 8

9 Partally Codtoal Radom Permutato Model 7-9 Example Target:, P g α α α α X V V X g X V V X Collapsg: usual radom varables otce that V k The The R su R ur s R sur Let R The R k Also, let k u ad s The k ur s k fk k k The ur s Also, R s k k fk k k k k k, ad ur k k The these results, V k k kk fk ow α X V X X V ad X The X V X kk fk k k fk k k k fk k k so that fk fk X V X The, k R sur kk Usg C07ed63doc /9/008 6:33 PM 9

10 Partally Codtoal Radom Permutato Model 7-0 α fk X V X X V k kk k fk fk k k fk k fk k fk k k k ext, we smplfy the expresso for V V k kk fk k kk fk k kk k fk kk fk k k kk k fk fk fk k kk k fk fk k kk k fk fk k k k fk or V V k k k Fally, we use these expressos to evaluate fk P, α g α α α X V V X g X V V X whe g e ad g 0 The C07ed63doc /9/008 6:33 PM 0

11 Partally Codtoal Radom Permutato Model 7- α P g α X V V X e k k k k k fk k k k k k e e k k fk k k k 0 k 0 k k k fk k k We smplfy ths further otce that 0 k 0 k k P k k k k k fk k k 0 k 0 k k k k k k k k fk k k k 0 k 0 k k k We re-express ths equato so as to form a devato the secod part As a result, k P k 0 k 0 k k k k k 0 0 k k k k k k k k Thus, whe, P μ k μ μ k, k k e e e e ud u w us se otce that the weghts are gve by w s e e Ths predctor s dfferet from the predctor developed the earler settg whch was gve by * * * P e ad C07ed63doc /9/008 6:33 PM

CHAPTER 4 RADICAL EXPRESSIONS

CHAPTER 4 RADICAL EXPRESSIONS 6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube