Determining the Best Linear Unbiased Predictor of PSU Means with the Data. included with the Random Variables. Ed Stanek
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1 Detemining te Bet Linea Unbiaed Pedicto of PSU ean wit te Data included wit te andom Vaiable Ed Stanek Intoduction We develop te equation fo te bet linea unbiaed pedicto of PSU mean in a two tage andom pemutation upepopulation model. We do o in geneal, and do not implify expeion fo te vaiou matice. Te development follow a cloe a poible te development peented by Bolfaine and Zack (99, p3-5. ee we include te data wit te andom vaiable. Ti development i identical to tat in c00ed5.doc, wit te exception tat te data ae included in te andom vecto, a oppoed to being iolated a a epaate matix. I jump back and fot between auming equal ize PSU, and unequal ize PSU. Te odel Ou tating point i a imple linea model given by y X β + e (uing te B& Z B& Z notation of Bolfaine and Zack, equation (.., p9. Te paamete coepond to te yt t PSU mean given by µ fo,...,. N N Te vecto y in te model i a vecto of andom vaiable. In ealy wok B& Z (999, we ave epaated te andom vaiable tat aie fom te upepopulation c00ed.doc 0/3/00 :33 P
2 model fom te contant te make up te finite population. Uually, te andom vaiable coeponding to te i t elected PSU and te t j elected SSU i epeented conceptually a N (. We ay conceptually ince wen Y U U y ij i jt t t N diffe ( between PSU, te andom vaiable U jt may not be defined fo j > N. To avoid ti complication, we aume equal numbe of SSU pe PSU ee, and et N fo all,...,. Ti combine te andom vaiable and contant. We epeent te vecto of uc quantitie a te N vecto Y, wee Y ( Y Y Y Y Y Y Y Y Y. Ti epeentation paallel tat of te population value, wic we denote a te N vecto y, wee y ( y y y and ( y y y y. Wit ti epeentation, te fit ubcipt in Y ij indexe te election of PSU, wile te fit ubcipt in yt indexe te actual PSU. Altoug it i tempting to tink of te andom vaiable y a being equivalent B& Z to Y, ti epeentation can not be ued and till enable te paamete fo PSU mean to be etimable in te model. Te eaon it doen t wok i tat eac value of N ( i a function of all te paamete, and a a common expected Y U U y ij i jt t t value. In ealy invetigation of ti model, we expanded te epeentation o a to epeent te andom vaiable epaately fo PSU election, Y, wee element of c00ed.doc 0/3/00 :33 P
3 Yae given by N ( Y U U y. Tee andom vaiable wee ummaized in a ij i jt t t vecto tat wa epeented by Y UU y. A illutated in c99ed.doc (p0, wen, N, N, and N, 3 ( UU y ( U U y ( U U y ( 3 U3U y3 ( UU y ( U U y Y ( U U y ( 3 U3U y3 ( U3U y ( U U y ( U U y ( 3 U33U y3 t t t t t t t t t t t t 3 t t t 3 t t t oe geneally, te vecto Ya dimenion, wee te fit pat i U Ut yt U Ut yt U Utyt U Ut yt U Ut yt U Ut yt. t t t t t t ( ( ( ( ( ( given by We fute expand te expeion fo te andom vaiable o a to epeent eac andom vaiable uniquely. Te baic andom vaiable in te expeion fo c00ed.doc 0/3/00 :33 P 3
4 N ( Y U U y ae UU ij i jt t t development. ( i jt t y. Tee andom vaiable fom te bai of ubequent We aange tee andom vaiable in a vecto of dimenion N wic we define a uc tat ( N N N wee te individual vecto ( wee tat t t t t ae of dimenion N and ae given by t ( ( ( ( y UU UU UU uc it t i t i t i Nt ( ( ( ( ( ( ( ( ( y U U U U U U U U U U U U. t t t t Nt t t Nt We can multiply te expanded et of andom vaiable by a matix tat will collape it to fom te vecto Y. Fit, note tat c00ed.doc 0/3/00 :33 P 4
5 t t t t t t t t t I ( U U y ( U U y ( U U y ( U U y ( U U y ( U U y t t t t t t t t t oe geneally, te element in Y can be obtained by te poduct (. I I. oweve, te tem in ti poduct will not be in te ame ode a te element of Y. We can e-aange te tem n te ame ode a Y by pemultiplying by ( 0 0 ( 0 0 I I I ( 0 0 I. e-aanging into a Sampled and emaining Patition We define a pemutation matix K tat will e-aange te andom vaiable in into te ampled and emaining potion. Ti i te ame matix a wa defined in c00ed.doc. Ti matix a dimenion N N. We ue ti matix to c00ed.doc 0/3/00 :33 P 5
6 e-aange te andom vaiable in matix Ki defined a K into a ampled, and a emaining potion. Te I II 0 Im 0 I ( I m ( m I m I I 0 0 I I ( I ( m m K I m I 0 I I 0 K. ( I I m ( m I m I 0 I 0 I ( I I ( m m We illutate te poduct of K wen 3, 3, and m. Ten II 0 Im 0 I ( I m ( m II 0 Im 0 I ( I m ( m II 0 Im 0 3 I ( I m ( m II 0 Im 0 I ( I m ( m K II 0 Im 0 I ( I m ( m II 0 Im 0 3 I ( I m ( m II 0 Im 0 3 I ( I m ( m II 0 Im 0 3 I ( I m ( m II 0 Im 0 33 I ( I m ( m wee c00ed.doc 0/3/00 :33 P 6
7 UUt y t ( UUt yt ( UU3t yt 0 0 ( U U 0 0 UUt yt ( U U y U U I 0 I ( UU3 t y U U t U U ( U3Ut yt ( U3Ut yt ( U3U3 t y t I m t t t I ( I m ( m. ( ( t t ( t t ( t t ( y y y y t t We define K K. Te vecto i of dimenion K I nn. Te vecto i of dimenion ( ( I N n N + I N. Wit SSU pe PSU, and mssu ampled pe PSU, (wit I PSU ampled, dimenion ( Im. Te vecto Detemining te Deign atix X i of dimenion ( Im. i of We pecify te deign matix Xby fit detemining a deign matix fo a imila model expeed in tem of te uual epeentation of te population, y. Te model i a imple PSU mean model. We denote te deign matix fo te non-tocatic model a X In tem of te population, N B& Z y X β + e. Te paamete β ae defined a B& Z yt t te PSU mean given by µ fo,...,. N N c00ed.doc 0/3/00 :33 P 7
8 We would like to pecify a model compaable to te deteminitic population model fo te expanded andom vaiable given by Xβ+ε. Now K and K K I. We fit detemine a model of te fom X β+ε tat i compaable to te model y X β+ e. Ten, ince K B& Z, we detemine X KX. We fit detemine poible value of X in te model X β+ε tat ae conitent wit te deign matix in te model fo y X β+ e wee B& Z X. One B& Z matix tat will atify ti popety i X I [ ]. Te model can be pemuted to te e-aanged model (given by Xβ+ε by multiplying all ide by K. Tu, K KXβ+ K ε, o KX β+ Kε. Now c00ed.doc 0/3/00 :33 P 8
9 KX I II 0 Im 0 I ( I m ( m I m I I 0 0 I I ( I ( m m wic I m I 0 I I 0 ( I I m ( m I m I 0 I 0 I ( I I ( m m [ ] implifie to ( I m Im ( X KX ( I m ( I m ( K X X I m I m. K X X ( I m ( I ( m Note tat tee ae many ote poible definition of X tat would atify ti equiement. Tee ote definition would include weigt uc tat ( w w w w w w w w w w t t t Nt t t Nt t t Nt wit element, w. Ten let wijtuc tat fo all,...,, and t,..., N N t w w w w, and note tat X can be defined a any matix X w uc tat fo all w, w. t N t Pedicto of β Te vecto β i a vecto of PSU mean. Note tat a a eult of te definition of te andom vaiable, fo all and t, i j UU i ( jt. Ten c00ed.doc 0/3/00 :33 P 9
10 ( IN N N β N o wit equal numbe of SSU pe PSU, te expeion implifie to β. Since K K and K K K I, ten K + K. A a eult, β K + K, wic we epeent a βl + L. We implify tee expeion uing K I II 0 Im 0 I ( I m ( m I m I I 0 0 I I ( I ( m m K K. Ten I m I 0 I I 0 ( I I m ( m I m I 0 I 0 I ( I I ( m m Im L K, wile I( m ( I m ( I ( m L K. It i inteeting to note tat β i non-tocatic, but te vecto and ae tocatic. Ti elationip old due to te containt implied on te andom vaiable a a eult of te ypegeometic upepopulation model. Bolfaine and Zack (p c00ed.doc 0/3/00 :33 P 0
11 decibe a imila elationip tat may exit fo cetain paamete. oweve, te elationip tey decibe i wit epect to a ealized population vecto, y, not te andom vaiable and. Te epeentation we ue doe not equie te upepopulation to be ealized. Fo all ealization, te expeion will old. Ti vecto of paamete can alo be epeented (a in Bolfaine and Zack (p a βθ + θ.tu, te paamete of inteet can be expeed a te um of two tem, wee te fit tem i a linea function of te andom vaiable econd tem i a linea function of te andom vaiable, wile te. Upon ealizing te ample, we will obeve te fit tem. Te econd tem will not be obeved, and ence we need to pedict it. We etimate β by etimating te un-obeved andom vaiable, βθ ˆ conide etimato tat ae a linea function of te ealized andom vaiable (linea + θ ˆ. We etimato, uc tat ˆ L θ (imila to Bolfaine and Zack, p4, equation... Te matix L i of dimenion ( Im I nn o equivalently (wit equal SSU,. Ten Bolfaine and Zack (p4, eq.. tate tat te etimato will be model unbiaed if and only if L X L X. Ti make ene ince ( L L X X β. β o equivalently, E ξξ ( L L ( L L c00ed.doc 0/3/00 :33 P
12 Taking te expected value, E ξξ ( L L ( L L X β. Ti implie X tat te etimato will be unbiaed if and only if L X L X o X L X L. An Expeion fo te SE We want to detemine Ltat will minimize te mean quaed eo (SE. Note tat ince we ae etimating a vecto of paamete, te SE will be a SE matix. Bolfaine and Zack (p7 define te genealized SE a GSE ( β β ˆ ( β β ˆ β β β β, and tate tat te bet pedicto of a vecto will E ξξ minimize ti quantity. Anote citeia tat i often ued i te expected quaed eo. Te expected quaed eo i given by E ( ˆ ( ˆ ( ˆ y Xβ y Xβ E ( ˆ ξξ Xβ Xβ+ε Xβ Xβ+ε β β β β+ε β β+ε. Tee two citeia do ξξ not appea to be te ame. Wit multi-vaiate data, tee may be diffeent citeia fo optimization. We fit fom an expeion fo te SE. To do o, we follow te idea of Lemma... (p4 in Bolfaine and Zack. We ave detemined tat te etimato (given by βθ ˆ + θ ˆ L + L will diffe fom te paamete (given by βθ + θ L + L by an amount given by L L. Tu, te SE i. Ti i given by defined a vaξξ ( L L ( L L ( L L L. Expeion ave been vaξξ va ξξ L c00ed.doc 0/3/00 :33 P
13 deived (ee c00ed3.doc [witout including te data] fo va ξξ ( wee we note tat ( K K va va ξξ ξξ o va K va ( K ξξ ξξ. Te matice and contain te ame andom vaiable, but ave diffeent dimenion. Te matix alo diffe fom ince include te data. Te matix a dimenion N N, wile te matix a dimenion N. Fo ti eaon, expeion fo te vaiance of V V vaξξ ae vey imila. Uing ti V V patition of te vaiance, an expeion fo te SE i given by ( L L ( L L va V V L ξξ. Ti expeion agee V V L wit Coollay.. (p4 of Bolfaine and Zack. Te SE can be expanded o a to be L L L VL L VL LVL + LVL. Te given a vaξξ ( i dimenion of vaξξ ( L L. Deivation of te Bet Linea Unbiaed Pedicto We deive te bet linea unbiaed pedicto following Teoem.. in Bolfaine and Zack (p5. Fit, note tat te matix L i of dimenion ( Im ], wile te matix L i of dimenion I nn [o c00ed.doc 0/3/00 :33 P 3
14 I N n N + I N ( ( [o ( mi citeia fo ou etimato te genealized SE, given by ]. We ue a te ( ( + GSE va ξξ L L L VL L VL L V L L V L tat ote citeia, uc a minimizing te tace, ae poible.] We account fo te unbiaed containt, ( mi. [Note L Im I in te minimization by adding a tem wic coepond to te metod of Lagangian multiplie. Poceeding along te line of Bolfaine and Zack, it i clea tat in tei deivation, te SE i a cala. In te poblem tat i conideed ee, te SE i a matix, and we fom a cala expeion by conideing te GSE. In ode to evaluate te deivative, and et te Lagangian multiplie coectly, we expand te GSE and containt o a to expe tem a a um of cala expeion. Ti involve expanding te SE uing vec opeato, uc tat ( ( ξξ ( GSE va ξξ vec va L L L L. Tee will be equation in vecvaξξ ( L L. Let u define te ow of ( L L uc tat ( L L L L L L o L L c00ed.doc 0/3/00 :33 P 4
15 L L L L L L L L. Ten vec va ξξ ( L L ( L L L vaξξ L L va ξξ L L va ξξ L L vaξξ L L L vaξξ L L L vaξξ L L L va ξξ L L L va ξξ L L L va ξξ L ( L L ( L L ( L L ( L ( L ( L ( L ( L will epeent an vecto of equation woe um we wi to minimize. Ti can be epeented L GSE. L L va ξξ L equivalently a ( c00ed.doc 0/3/00 :33 P 5
16 Te containt tat apply to tee equation ae given by X L X L o ( ( X L L L X L L L. Uing tee definition and expanion, we define te Lagangian tat i to be minimized a (, ;,..., L L λ ( va L L + λ ( X L X L L λ ξξ λ wic in L tun i given by ( L, λ ;,..., ( L VL L VL L VL + L VL + λ ( X L X L L wee we note tat cala, we can wite it equivalently a. λ i of dimenion. Since eac tem in ti ummation i a ( L, λ ;,..., ( L VL L VL L VL + L VL + λ ( X L X L L We fute expand te fit tem by epaating tem in te double ummation wen fom toe wee. Tu, ( L, λ ;,..., ( L VL L VL + L VL + ( L VL L VL L VL + L VL + λ ( XL X L L. We diffeentiate wit epect to L and λ fo,...,, and ten et te eulting deivative to zeo. Te olution (wic we ditingui by te upecipt i te bet linea unbiaed pedicto. Aide on a Geneal Solution c00ed.doc 0/3/00 :33 P 6
17 aville (p460, Teoem 9.. conide a poblem vey imila to te poblem tat we face. To ee te imilaity, let u define ( V vaξξ. Alo, let A L L. Ten te SE i given by AVA. Let te containt be defined uc tat XA D. Tee tem can be detemined. Let λbe a matix of abitay contant, uc tat te quantity to be minimized can be expeed a AVA λ ( D XA tat aville doen t tate te poblem like ti, ince wat i actually meant by. [Note minimizing a matix i not clea.] Nevetele, uppoe we diffeentiate ti expeion wit epect to A. Te eult (wic we et equal to zeo (need cecking will be VA+ Xλ0. Te containt can be expeed a XA D. A a eult, te two equation ae given by V X A 0. Te olution of ti equation i addeed X 0λ D in Teoem 9.. (p460 of aville. Evaluation of Deivative We fit evaluate te deivative wit epect to L fo,...,. We make ue of tandad matix diffeentiation eult (ee aville, p95 uc tat ( xax x ( A+ A x, ( Ax x Aand ( Ax x A. Te deivative L L ( wit epect to L fo,..., will eult in te equation L L ( ( ( VL VL + VL VL + X λ o ( L ( VL VL + X λ and te deivative L c00ed.doc 0/3/00 :33 P 7 L λ ( wit epect to
18 L λ fo,..., will eult in te equation X L X L. Setting tee λ ( equation equal to zeo, te bet linea unbiaed pedicto i te olution to te equation: ( VL VL 0 + Xλ fo,..., and X L X L 0 fo,...,. VL VL X λ 0 Now ince ( +, VL VL X λ + and ence L V VL V Xλ +. Ten X L X V VL X V Xλ +, and a a eult of te econd equation, X L + X V VL X V X λ. Let u define a genealized invee ( X V X of ( X V X. Ten ( ( λ X V X X L + X V X X V VL. We ubtitute ti value into te fit et of equation uc tat ( ( L V VL V X X V X X L + X V X X V VL Note tat te vecto L i of dimenion ( mi.. c00ed.doc 0/3/00 :33 P 8
19 Te olution i fo te model X ε β+ X ε wee KX K X X KX X and, X and Im X ( I m ( I m ( ( I m. Alo, ε Ke ε Ke ε Ke [ee NOTE]. Alo, βl + L uc tat Im L K, wile I( m ( I m ( I ( m L K. Now te etimato i of te fom βθ ˆ + θ ˆ L + L and we ave own tat ( ( L V VL V X X V X X L + X V X X V VL Finally, ( matix. mi I k C wee k mi and C i a column of an identity. c00ed.doc 0/3/00 :33 P 9
20 efeence Bolfaine,. and Zack, S. (99. Pediction Teoy fo Finite Population, Spinge- Velag, New Yok. c00ed.doc 0/3/00 :33 P 0
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