Model-based Variance Estimation for Systematic Sampling

Transcription

1 ASA Sectio o Survey Researc Metods Model-based Variace Estimatio for Systematic Samplig Xiaoxi Li ad J D Opsomer Ceter for Survey Statistics ad Metodology, ad Departmet of Statistics, Iowa State Uiversity, Ames, IA 500 Abstract: Systematic samplig is a frequetly used samplig metod i surveys, because of its ease of implemetatio ad its desig efficiecy A importat drawback of systematic samplig, owever, is tat o direct estimator of te desig variace is available We describe a ew estimator of te model-based expectatio of te desig variace, uder a oparametric model for te populatio Te oparametric model is sufficietly flexible tat it ca be expected to old at least approximately for may practical situatios We prove te cosistecy of te estimator for bot te aticipated variace ad te desig variace uder te oparametric model, ad illustrate its practical properties troug a simulatio experimet KEY WORDS: aticipated variace, oparametric model, local polyomial regressio Itroductio Systematic samplig is widely used i surveys of fiite populatios due to its appealig simplicity ad efficiecy Te metod was first studied by Madow ad Madow 944, i wic te expressio of desig-based variace for te sample mea was developed However, it is impossible to derive a ubiased desig-based estimator for tis variace, because systematic samplig is equivalet to cluster samplig wit oly oe cluster selected Iaca 982 Some less-ta-perfect approaces for dealig wit tis problem exist i literature Oe is to use biased variace estimators, ad aoter oe, for example, is to use a auxiliary simple radom sample For te former approac, Särdal et al 992 remarked tat te estimator due to Yates ad Grudy 953 ad Se 953 will overestimate te variace A more compreesive review ca be foud i Wolter 985, were eigt biased variace estimators were described ad guidelies for coosig amog tem were give For te latter approac, Ziger 980 pursued a approac, defied as partially systematic samplig, tat gives a ubiased variace estimator, by mixig systematic ad simple radom samples togeter 204 Sedecor Hall, Iowa State Uiversity, Ames, IA 500; lixiaoxi@iastateedu Te above variace estimatio metods are coditioal o te desig I oter words, tey are desig-based i te way tat we treat te fiite populatio as fixed Tere also exist some model-based variace estimators were te populatios are cosidered radom realizatios from a superpopulatio model For te case of a liear superpopulatio, Motaari ad Bartolucci 998 proposed a modelbased variace estimator, wic is approximately ubiased for te aticipated variace, ie te expectatio of te desig-based variace for te sample mea uder te superpopulatio model However, it may lack accuracy ad efficiecy due to a iger cotributio of te bias if te systematic compoet of te superpopulatio is sigificatly differet from liear A ew class of ubiased estimators tat icludes some simple oparametric estimators was proposed by Bartolucci ad Motaari 2005 ad was sow to be ubiased uder liear superpopulatio models I tis article, we propose a model-based oparametric variace estimator based o local polyomial regressio Te systematic samplig framework is briefly described i Sectio 2, ad Sectio 3 reviews te model-based variace results uder te liear superpopulatio model I Sectio 4, we study te properties of te proposed local polyomial variace estimator uder te oparametric superpopulatio model Simulatio results are preseted i Sectio 5 ad coclusios are draw i Sectio 6 2 Systematic samplig Suppose te study variable Y for a fiite populatio is Y, Y 2,, Y N Te te populatio mea is Ȳ N = N N Y j, j= wic is estimated by te sample mea Let deote te sample size ad k = N/ deote te samplig iterval For simplicity, we assume k to be a iteger So te bt systematic sample b =,, k cosists of te observatios wit te followig labels b, b + k,, b + k 3307

2 ASA Sectio o Survey Researc Metods Te bt sample mea is Ȳ Sb = j= Y Sb j ad te desig-based variace for sample mea ȲS, deoted by Var p ȲS, is Var p ȲS = k k ȲS b ȲN 2 b= 3 Variace estimatio uder liear models I te model-based cotext, te populatio is cosidered beig draw from a superpopulatio model Let Y j R j =, 2, be a set of idepedet ad idetically distributed radom variables Let X j R d j =, 2, be vectors of auxiliary variables, wic we cosider as fixed Suppose te liear superpopulatio model, deoted by L, is Y = Xβ + ε, 2 were E L ε = 0 ad Var L ε = σl 2 Ω, wit E L ad Var L deotig te expectatio ad variace uder te model L, respectively For simplicity, we assume Ω to be diagoal ie Ω = diag{ω, ω 2,, ω N } I model 2, Y = Y, Y 2,, Y N T, β = β 0, β,, β d T, ad X = X X d X N X Nd We ca rewrite te desig variace as Var p ȲS = k 2 YT NY, 3 were N = M T HM, wit M = T I k ad H = I k k k T k Here is te Kroecker product ad r is a colum vector of s of legt r Te model aticipated variace is E L [Var p ȲS] = k 2 βt X T NXβ + k 2 trnωσ2 L 4 Bartolucci ad Motaari 2005 discuss a ubiased estimator for E L [Var p ȲS], defied as ˆV L ȲS = βt ˆ k 2 b X T NX ˆβ b tˆσ Lb 2 + k 2 trnωˆσ2 Lb, 5 were ˆβ b is te ordiary least square OLS estimator for β for te bt sample ad ˆσ Lb 2 is a model ubiased estimator for σl 2, wic is defied as ˆσ Lb 2 = Y b Xˆβ b T Ω b Y b Xˆβ b, rakx were Ω b is a sub matrix of Ω correspodig to te bt sample I equatio 5, tˆσ Lb 2 is a bias correctio term wit t = k k 2 b= trpt b XT NXP b Ω b ad a coice of P b is P b = X T b Ω b X b X T b Ω b We assume tat X T b Ω b X b exists Li 2005 sows tat, uder a asymptotic framework i wic N ad a set of assumptios, ad Var p ȲS E L [Var p ȲS] = O p N /2, 6 ˆV L ȲS E L [Var p ȲS] = O p /2 7 ad ece by 6 ad 7, ˆV L ȲS Var p ȲS = O p /2 8 Equatio 7 suggests tat ˆV L ȲS is a cosistet estimator for E L [Var p ȲS] Equatio 6 idicates tat te coditioal variace Var p ȲS coverges to E L [Var p ȲS] i probability So ˆV L ȲS ca be used as a cosistet predictor for Var p ȲS, as 8 suggests Details o tese results are i Li Variace estimators uder oparametric models Parametric metod are appropriate we we ca correctly specify te superpopulatio model However, if te superpopulatio model is icorrectly specified, parametric metod may result i biased or iefficiet estimatio We propose a cosistet variace estimator uder oparametric models We study te case were d = Let X = X, X 2,, X N Te oparametric superpopulatio model, deoted by NP, is Y = m + ε, 9 were E NP Y j X j = x j = mx j ad Var NP ε = σnp 2 Ω Let m be a cotiuous ad bouded fuctio, ad defie m = mx, mx 2,, mx N We assume tat te s are bouded ad positive, were j =,, N 3308

3 ASA Sectio o Survey Researc Metods Uder model 9, te expected value of Var p ȲS is E NP Var p ȲS = k 2 mt Nm + k 2 trnωσ2 NP 0 As a estimator for E NP Var p ȲS, we propose ˆV NP ȲS = k 2 ˆmT b N ˆm b + k 2 trnωˆσ2 NP b Here ˆσ NP 2 b is defied as ˆσ NP 2 b = Y b ˆm b T Ω b Y b ˆm b, 2 ad ˆm b = ˆmx,, ˆmx were ˆmx j is te local polyomial regressio estimator ˆmx j = e T X T bjw bj X bj X T bjw bj Y b, were e is te q + vector avig i te first etry ad all oter etries 0, ad x x j x x j q X bj =, 3 x x j x x j q { W bj = diag K xi x j, } i =,, 4 were deotes te badwidt, q deotes te degree of local polyomial regressio ad K xi x j is te kerel fuctio We refer to Wad ad Joes 995 for more iformatio o te local polyomial regressio estimator Note tat we are ot icludig te bias correctio term tˆσ Lb 2 used i equatio because tat term is asymptotically egligible To study te covergece properties of ˆV NP ȲS, we make te followig assumptios A Te errors ε j s are idepedet wit mea zero, variace σnp 2 ad compact support, uiformly for all N A 2 For eac N, we cosider te x j s as fixed wit respect to te superpopulatio model N P Te x j s are idepedet ad idetically distributed wit F x = x ftdt, were f is a desity fuctio wit compact support [a x, b x ] ad fx > 0 for all x [a x, b x ] A 3 As N, N p 0,, 0 ad N 2 /log log N A 4 Te q + t derivative of te fuctio m exists ad is bouded o [a x, b x ] Teorem Uder assumptio A - A4, ad Var p ȲS E NP Var p ȲS = O p N /2, 5 ˆV NP ȲS E NP Var p ȲS ˆV NP ȲS Var p ȲS q+ + O p, 6 q+ + O p 7 Teorem sows tat ˆV NP ȲS is a cosistet estimator for E NP Var p ȲS ad a cosistet predictor for Var p ȲS uder te oparametric model NP We provide a outlie of proof for 6 i Appedix Te proof for 5 ad 7 ca be foud i Li Simulatio Study To furter ivestigate te statistical properties of te above variace estimators ad predictors, we perform a simulatio study For simplicity, we assume tat te errors are idepedetly ad ormally distributed wit omogeeous variaces Two superpopulatio models are examied: te liear model were j =,, N ad ε j quadratic model y j = 5 + 2x j + ε j, 8 N0, σ 2, ad te y j = 5 + 2x j 2x 2 j + ε j, 9 were j =,, N ad ε j N0, σ2 2 Let R 2 ad R2 2 deote te coefficiet of determiatio for model 8 ad 9, respectively Te coefficiet of determiatio, also kow as R-square, is te fractio of variatio i te respose tat is explaied by te model So bigger R-square meas bigger predictive power of te model We ivestigated two levels of σ 2 ad two levels of σ2, 2 wic correspodigly determied two levels of R 2 ad two levels of R2 2 Specifically, we ave four differet cases: R 2 075; 2R 2 025; 3R ; 4R We compare te performace of ˆVL ȲS ad ˆV NP ȲS wit te variace estimator for simple radom samplig SI desig, ie ˆV SI ȲS = f S2 Y S 3309

4 ASA Sectio o Survey Researc Metods were f = /N ad SY 2 S = S Y k ȲS 2 We geerate populatios of size N = 2, 000, ad we coose tree differet systematic sample sizes = 500, 00 ad 0, wit correspodig samplig itervals k = 4, 20 ad 200, respectively To draw a systematic sample, we first sort te data by x, from te smallest to te largest, te we radomly coose a observatio from te first k observatios, say te bt oe Te our sample cosists of te observatios wit te followig subscripts: b, b + k,, b + k For eac sample, we calculate te correspodig Var p ȲS, E L [Var p ȲS], ˆV L ȲS, E NP [Var p ȲS] ad ˆV NP ȲS as defied i 3, 4, 5, 0 ad respectively For ˆV NP ȲS, we calculate it usig two badwidt values: = 050 ad = 025 Eac simulatio settig is repeated B = 0, 000 times For eac of ˆV L ȲS, ˆVNP ȲS ad ˆV SI ȲS, we calculate te relative bias RB, mea squared error MSE ad mea squared predictio error MSPE, wic are defied as follows: RB = EˆVȲS E L [Var p ȲS], E L [Var p ȲS] MSE = EˆVȲS E L [Var p ȲS] 2, ad MSPE = EˆVȲS Var p ȲS 2 were ˆVȲS deotes oe of ˆV L ȲS, ˆV NP ȲS ad ˆV SI ȲS For all four cases, ˆVL ȲS assumes liear tred for te superpopulatio model So it is expected tat for case 3 ad 4, wic geerate populatios from quadratic model 9, ˆV L ȲS will ave poor variace estimatio results Table reports te relative bias for ˆV L ȲS, ˆV NP ȲS evaluated at two badwidt values ad ˆV SI ȲS We ca see tat for case 3 ad 4, ˆV L ȲS is a sigificatly biased estimator Ad its bias is of similar magitude to ˆV SI ȲS i correspodig cases For case ad 2, i wic te populatios ave liear treds, te relative biases of ˆV L ȲS are geerally small, ad ted to get smaller we sample size gets larger Te relative biases of ˆV SI ȲS i case ad 2 are similar to tose i case 3 ad 4, respectively, ad beave similarly poorly We we use te model based variace estimator ˆV L ȲS, assumig te wrog model ca be a serious problem Te results i Table also suggest tat ˆV NP ȲS performs better ta ˆV L ȲS i almost all cases, especially we te superpopulatio model is quadratic Tis is because ˆV NP ȲS does ot require a liear specificatio of te model, as it oly requires smootess of te superpopulatio mea fuctio We also see tat its relative biases are geerally small, except for case 3 ad = 0, were te relative bias values are -249% ad -899% for = 050 ad = 025, respectively Tis is mostly likely due to te extremely small sample size for local polyomial regressio Also as far as te relative bias is cocered, tere seems to be o differece betwee tese two badwidt values Table 2 reports te ratios of MSE ad MSPE betwee ˆV L ȲS ad ˆV SI ȲS, ad te ratios of MSE ad MSPE betwee ˆV NP ȲS ad ˆV SI ȲS, evaluated at two differet badwidt values MSE measures te variability of a estimator ad MSPE measures te variability of a predictor Smaller MSE ad MSPE are desired We see tat we te superpopulatio model is liear, ie case ad 2, ˆVL ȲS performs better ta ˆV SI ȲS, because tose ratios are less ta oe Ad we te liear superpopulatio is more precise, ie R 2 075, te advatage of usig ˆV L ȲS is more obvious However, we it comes to case 3 ad 4, were we icorrectly assume te superpopulatio model, ˆV SI ȲS performs sligtly better ta ˆV L ȲS For ˆV NP ȲS, te ratios are less ta oe i all categories, suggestig tat ˆV NP ȲS is a less variable estimator ad a less variable predictor ta ˆV SI ȲS 6 Coclusios From te above study, we coclude tat ˆV L ȲS ad ˆV NP ȲS are cosistet estimators for E L [Var p ȲS] ad cosistet predictors for Var p ȲS, uder te assumed models 2 ad 9, respectively Te liear estimator ˆV L ȲS oly performs well if te model is correctly specified Te oparametric model 9 is less restrictive ta te liear regressio model 2, ad te correspodig variace estimator ˆV NP ȲS performs well i almost all cases So i practice, we recommed te oparametric estimator ˆV NP ȲS 7 Appedix Outlie of proof of equatio 6: ˆV NP ȲS E NP Var p ȲS = k 2 ˆmT b N ˆm b m T Nm + k 2 trnωˆσ2 NP b σnp We ca write as = k 2 ˆm b m T N ˆm b m + k 2 mt N ˆm b m 330

5 ASA Sectio o Survey Researc Metods Liear Quadratic Relative Bias % : R : R : R : R ˆV L ȲS = = = ˆV NP ȲS, = 05 = = = ˆV NP ȲS, = 025 = = = ˆV SI ȲS = = = Table : Simulated relative bias for ˆV L ȲS, ˆVNP ȲS at two badwidt values ad ˆV SI ȲS for four populatios ad tree systematic sample sizes i percet Liear Quadratic : R : R : R : R MSEˆV L ȲS = MSEˆV SI ȲS = = MSPE ˆV L ȲS = MSPE ˆV SI ȲS = = MSEˆV NP ȲS, = 050 = MSEˆV SI ȲS = = MSPE ˆV NP ȲS, = 050 = MSPE ˆV SI ȲS = = MSEˆV NP ȲS, = 025 = MSEˆV SI ȲS = = MSPE ˆV NP ȲS, = 025 = MSPE ˆV SI ȲS = = Table 2: Ratios of MSE ad MSPE betwee ˆV L ȲS ad ˆV SI ȲS, betwee ˆV NP ȲS ad ˆV SI ȲS at two badwidt values 33

6 ASA Sectio o Survey Researc Metods + k 2 ˆm b m T Nm A + 2B Note tat m T N ˆm b m = ˆm b m T Nm because tey are bot scalars By te defiitio of matrix N, we ca write A as were A = k 2 ˆm b m T N ˆm b m = k ˆmx j mx j k b= 2 ˆmx j mx j N = k j U k {a + a2 + a3}, b= a = 2 [ ˆmx j mx j ] 2, a2 = N 2 [ ˆmx j mx j ] 2, j U ad a3 = 2 ˆmx j mx j N ˆmx j mx j Note tat j U ˆmx j mx j = s bj Y b mx j = s bj m b + ε b mx j = b b x j + s bj ε b Here s bj is te smooter matrix ad s bj = e T X T bj W bjx bj X T bj W bj, were X bj ad W bj are defied as i 3 ad 4, respectively For simplicity, we will use K ij to deote K xi x j i future otatio Now expad te pareteses i a, we ave Ea = 2 b 2 bx j + 2 E s bj ε b ε T b s T bj j sb + 2 b b x j b b x l l s b,j l + 2 E j sb l s b,j l s bj ε b ε T b s T bl 2 Te rigt-ad side of 2 cotais four terms We will calculate tem oe by oe i First let us ivestigate 2 b 2 b x j We will use a tecique similar to Ruppert ad Wad 994 Let m b = mx, mx 2,, mx By Taylor s teorem, m b = X bj mxj D m x j + R m x j, were D m x j = m x j, 2 m x j,, q! mq x j ad R m x j is a vector of Taylor series remaider terms So b b x j = s bj m b mx j = s bj R m x j were = e T X T bjw bj X bj X T bjw bj q+! mq+ x j x x j q+ q+! mq+ x j x x j q+ z j z q+j = e T z q+j z q+q+j t kj = t j t q+j e T Z j T, z stj = i= K ij x i x j s+t 2, i= K ij x i x j k+q mq+ x ji, q +! wit s, t =, 2, q +, k =, 2, q + ad x ji is some poit betwee x i ad x j Uder assumptios A2 ad A3, by Lemma 2 ii of Breidt ad Opsomer 2000, for a certai poit x j, tere are at least q + poits i te iterval [x j, x j + ] So Z j is ivertible Lemma Assume tat te kerel fuctio K ij is bouded above, te K ij x i x j r = O r, i= were r = 0,, 2, 332

7 ASA Sectio o Survey Researc Metods Te proof of Lemma is provided by Li 2005 Tus, suppose assumptio A4 olds, by Lemma, we ave z stj = O s+t 2 ad t kj = O k+q So b b x j = e T O O q O q O 2q O q+ O 2q+ Note tat te order of a matrix is te same as its iverse So b b x j = O q+ + + O 3q+ = O q+, 22 ad tus 2 b 2 x j = 2 O 2q+2 2q+2 = O 23 ii Secodly, let us compute E 2 j s s b bjε b ε T b st bj i 2 2 E s bj ε b ε T b s T bj = 2 s bj Ω b s T bj j sb = 2 e T X T bjw bj X bj X T bjw bj Ω b W T bjx bj X T bjw bj X bj e 2 e T Z j C j Z j e, were Ω b is te variace-covariace matrix of model 9 ad Ω b = diagω, ω 2,, ω, ad c j c q+j C j = c q+j c q+q+j wit c stj = 2 2 Kijω 2 i x i x j s+t 2 i= s+t 3 = O by Lemma Li 2005 sows tat e T Z j C j Z j e = O, ad tus 2 E s bj ε b ε T b s T bj = O 2 24 j sb iii Tirdly we will calculate 2 l s b,j l b bx j b b x l i 2 Usig te result i 22, we get 2 l s b,j l is E 2 sows tat 2 E j sb b b x j b b x l = O 2q+2 25 iv Te last term o te rigt-ad side of 2 j sb l s b,j l s bjε b ε T b st bl, ad Li 2005 l s b,j l s bj ε b ε T b s T bl = O 26 Assumptio A3 implies tat, ad by 23, 24, 25 ad 26, Ea = O 2q+2 + O 27 Similarly, we ca calculate Ea2 ad Ea3 Uder A3,, so Ea2 = O 2q+2 + O 28 N ad Ea3 = O 2q+2 + O 29 Also ote tat A > 0, so A = A Tus, by 27, 28 ad 29, E A = EA = k wic implies k {Ea + Ea2 + Ea3} b= = O 2q+2 + O A = O p 2q+2 + O p, Next, usig a similar tecique to tat of A, B q+ + O p 333

8 ASA Sectio o Survey Researc Metods Tus, = A + 2B q+ + O p 30 Now let us calculate = k trnωˆσ 2 2 NP b σnp 2 i 20, were ˆσ2 NP b is defied as i 2 So ˆσ NP 2 b σnp 2 = Y j mx j 2 σnp j= ˆmx j mx j 2 j= j= Li 2005 sows tat ad Y j mx j ˆmx j mx j Y j mx j 2 σnp 2 = O p j= 2 ˆmx j mx j 2 j= j=, 3 2q+2 + O p, 32 Y j mx j ˆmx j mx j q+ + O p 33 Sice k 2 trnω = O, ad by 3, 32 ad 33, we ave k 2 trnωˆσ2 NP b σ 2 NP Terefore by 30 ad 34, q+ + O p 34 Refereces Bartolucci, F ad G Motaari 2005 A ew class of ubiased estimators of te variace of te systematic sample mea Joural of Statistical Plaig ad Iferece To appear Breidt, F J ad J D Opsomer 2000 Local polyomial regresssio estimators i survey samplig Te Aals of Statistics 28 4, Iaca, R 982 Systematic samplig: A critical review Iteratioal Statistical Review 50, Li, X 2005 Model-based variace estimatio for systematic samplig P D tesis Iowa State Uiversity Draft Madow, W G ad L G Madow 944 O te teory of systematic samplig, I Aals of Matematical Statistics 5, 24 Motaari, G ad F Bartolucci 998 O estimatig te variace of te systematic sample mea Joural of te Italia Statistical Society 7, Ruppert, D ad M P Wad 994 Multivariate locally weigted least squares regressio Te Aals of Statistics 22, Särdal, C-E, B Swesso, ad J H Wretma 992 Model assisted survey samplig Spriger-Verlag Ic Se, A R 953 O te estimate of te variace i samplig wit varyig probabilities Joural of te Idia Society of Agricultural Statistics 5, 9 27 Wad, M-P ad M-C Joes 995 Kerel smootig Capma & Hall Ltd Wolter, K M 985 Itroductio to variace estimatio Spriger-Verlag Ic Yates, F ad P M Grudy 953 Selectio witout replacemet from witi strata wit probability proportioal to size Joural of te Royal Statistical Society Series B 5, Ziger, A 980 Variace estimatio i partially systematic samplig Joural of te America Statistical Associatio 75, ˆVȲS E NP Var p ȲS q+ + O p 334