Comparing Four Bootstrap Methods for Stratified Three-Stage Sampling

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1 Joural of Official Statistics, Vol. 26, No. 1, 2010, pp Comparig Four Bootstrap Methods for Stratified Three-Stage Samplig Hiroshi Saigo 1 For a stratified three-stage samplig desig with simple radom samplig without replacemet at each stage, oly the Beroulli bootstrap is curretly available as a bootstrap for desig-based iferece uder arbitrary samplig fractios. This article exteds three other methods (the mirror-match bootstrap, the rescalig bootstrap, ad the without-replacemet bootstrap) to the desig ad coducts simulatio study that estimates variaces ad costructs coverage itervals for a populatio total ad selected quatiles. The without-replacemet bootstrap proves the least biased of the four methods whe estimatig the variaces of quatiles. Otherwise, the methods are comparable. Key words: Multistage samplig; high samplig fractios; resamplig methods; quatile estimatio. 1. Itroductio For most stratified multi-stage samplig desigs, ubiased variace estimators of statistics expressed by liear fuctios of the observatios are available. However, for oliear statistics ad fuctioals, closed-form variace formulas are ofte uavailable. Cosequetly, bootstrap methods ca serve for variace estimatio of such statistics uder stratified multi-stage samplig desigs. Whe the samplig fractios at the first stage are small, bootstrap methods for stratified multi-stage samplig desigs are simplified because without-replacemet samplig ca be approximated by with-replacemet samplig (Shao ad Tu 1995, p. 235). But whe the samplig fractios at the first stage are ot egligible, bootstrap methods for cosistet variace estimatio become complicated ad few have bee developed. For istace, for a stratified three-stage with simple radom samplig without replacemet at each stage (ST SI 3 ) with arbitrary samplig fractios, o bootstrap procedure is available except the Beroulli Bootstrap (BBE) proposed by Fuaoka, Saigo, Sitter, ad Toida (2006) for the 1997 Japaese Natioal Survey of Prices (NSP). A resamplig method for quatile estimatio is particularly importat for the NSP because to aalyze price formatios for major cosumers goods, comprehesive quatile estimates are preseted i the NSP report. I 1997, ST SI 3 was coducted i the NSP. 1 Faculty of Political Sciece ad Ecoomics, Waseda Uiversity, Tokyo , Japa. saigo@waseda.jp Ackowledgmets: This work is supported by a grat from the Japa Society for the Promotio of Sciece. The author thaks Dr. Rady R. Sitter for suggestig the theme of this article. The author also thaks a associate editor ad aoymous referees for valuable commets that substatially improved the article. q Statistics Swede

2 194 Joural of Official Statistics First, 3,233 muicipalities (the primary samplig uits) were stratified ito 537 strata accordig to prefectures, ecoomic spheres, ad populatio sizes. At the first stage, the simple radom samplig without replacemet was coducted. Sice price formatios were locally correlated, large first-stage samplig fractios were adopted accordig to populatio size: 1/1, 2/3, 1/3, 1/5, ad 1/15. At the secod stage, all large-scale outlets were eumerated, while for small scale outlets, sampled muicipalities were divided ito survey areas (the secodary samplig uits) each cosistig of about 100 outlets. Systematic samplig was used to choose survey areas. The secod-stage samplig fractios were betwee 0.1 ad 1.0. Fially, i each selected survey area, about 40 small outlets were selected via ordered systematic samplig. Fuaoka et al. (2006) regarded the systematic samplig at the secod ad third stages as simple radom samplig without replacemet ad applied the BBE for quatile estimatio. However, Fuaoka et al. (2006) studied oly the BBE. A comparative study of several resamplig methods is ecessary. Although thorough simulatio studies have bee coducted about the bootstrap for samplig from a fiite populatio (e.g., Kovar, Rao, ad Wu 1988; Sitter 1992a, 1992b), oe of them has compared the methods uder ST SI 3. The study is particularly importat ot oly for the NSP but also the Family Icome Expediture Survey i Japa, where the ST SI 3 desig is employed. I this article, we exted the followig bootstrap approaches to ST SI 3 ad compare them with the BBE for variace estimatio as well as iterval estimatio through a simulatio study usig pseudo-populatios: the mirror-match bootstrap (BMM) by Sitter (1992a) which covers ST SI 2 ; the rescalig bootstrap (BRS) origially proposed by Rao ad Wu (1988) for ST SI 2 by rescalig the study variable ad the modified by Rao, Wu, ad Yue (1992) for ST SI with replacemet by rescalig the samplig weights to hadle quatile estimatio; ad the without-replacemet bootstrap (BWO) origially argued by Gross (1980) for ST SI ad the exteded by Sitter (1992b) to ST SI 2. The withreplacemet bootstrap (BWR) is icluded as a special case of the BMM (Sitter 1992a). A theoretical compariso of the four methods is beyod the scope of this article. This article is orgaized as follows. Sectio 2 describes ST SI 3. Sectio 3 presets the BBE ad a extesio of the BMM, the BRS, ad the BWO to ST SI 3. I Sectio 4, we coduct a simulatio study to compare the four bootstrap methods uder ST SI 3. Cocludig remarks are made i Sectio Stratified Three-Stage Desig I this sectio, we describe ST SI 3, a stratified three-stage desig with simple radom samplig without replacemet at each stage (see Särdal, Swesso, ad Wretma 1992, pp ). Suppose that a populatio is stratified ito H strata, labeled as h ðh ¼ 1; 2; :::;HÞ. Stratum h has N h primary samplig uits (PSUs) i it, labeled as i [ P h1 ¼ {1; 2; :::;N h }. Primary samplig uit i [ P h1 has M hi secodary samplig uits (SSUs) i it, labeled as j [ P h2i ¼ {1; 2; :::;M hi }. Secodary samplig uit j [ P h2i has L hij ultimate samplig uits (USUs) i it, labeled as k [ P h3ij ¼ {1; 2; :::;L hij }. Ultimate samplig uit k [ P h3ij has the characteristic(s) of iterest y hijk. The populatio total is give by Y:::: ¼ P H P h¼1 Pi[P h1 Pj[P h2i k[p h3ij y hijk.

3 Saigo: Comparig Four Bootstrap Methods 195 Uder ST SI 3, samplig is carried out idepedetly i differet strata. I stratum h, we take a simple radom sample without replacemet (SI) of size h from P h1 ad deote the set of the sampled labels by S h1. The, for i [ S h1, we take a SI of size m hi from P h2i ad deote the set of the sampled labels by S h2i. Fially, for j [ S h2i, we take a SI of size l hij from P h3ij ad deote the set of the sampled labels by S h3ij. A ubiased estimator of the populatio total is give by ^Y:::: ¼ XH X X X h¼1 i[s h1 j[s h2i k[s h3ij w hijk y hijk ð1þ where w hijk ¼ f 21 h1 f h2i 21 f 21 h3ij with f h1 ¼ h =N h, f h2i ¼ m hi =M hi, ad f h3ij ¼ l hij =L hij. A ubiased variace estimator for ^Y:::: is give by vð ^Y::::Þ ¼ XH h¼1 ( N 2 h ð1 2 f h1þ S 2 h1 þ Xh h i¼1 N h M 2 hi ð1 2 f h2iþ S 2 h2i h m hi þ X h X m hi N h M hi L 2 hij ð1 2 f ) h3ijþ S 2 h3ij i¼1 j¼1 h m hi l hij where S 2 h1 ¼ð h 2 1Þ 21 P 2, 2 i[s h1 ^Y hi :: 2 ^Y h ::: Sh2i ¼ðm hi 2 1Þ 21 P 2, ^Y j[sh2ij hij : 2 ^Y hi :: ad S 2 h3ij ¼ðl hji 2 1Þ 21 P 2 i[s h3ijk y hijk 2 y hij : with ^Y hi : ¼ P P j[s h2i k[s h3ij f 21 h2i f 21 h3ij y hijk, ^Y h ::: ¼ 21 h Pj[S h1 ^Y hi ::; ^Y hij : ¼ P k[s h3ij f 21 h3ij y hijk, ^Y hi :: ¼ m 21 P hi j[s h2i ^Y hij :; ad y hij : ¼ l 21 P ijk k[s h3ij y hijk. To estimate the distributio fuctio FðxÞ ¼ P H P h¼1 Pi[P h1 Pj[P h2i k[p h3ij Ið y hijk # xþ= P H P h¼1 Pi[P h1 j[p h2i L hij, where I( ) is the idicator fuctio, a ubiased poit estimator is give by ^FðxÞ ¼ XH X X h¼1 i[s h1 j[s h2i k[s h3ij X whijk I ð y hijk # xþ.x H h¼1 X X X whijk i[s h1 j[s h2i k[s h3ij A closed-form variace formula is provided by replacig y hijk with Ið y hijk # xþ i vð ^Y::::Þ. For estimatig quatile F 21 ð pþ ¼ if {x : FðxÞ $ p} for p [ ð0; 1Þ, the direct iversio estimator ^F 21 ð pþ ¼ if {x : ^FðxÞ $ p} is available. However, o closed-form variace formula is available for ^F 21 ð pþ. Although the Woodruff method ca hadle variace estimatio (Woodruff 1952; see also Shao ad Tu 1995, p. 238), the bootstrap is a reasoable choice because it ca accommodate osmoothed statistics other tha quatiles as well. 3. Bootstrap Methods The bootstrap methods cosidered here ca be described as follows. Suppose a estimator of the parameter u ca be writte as ^u ¼ tðw hijk ; y hijk ; h ¼ 1; 2; :::;H; i [ S 1h ; j [ S h2i ; k [ S h3ij Þ, where the sample weight w hijk is give i (1). The, through the bootstrap method employed, obtai a bootstrap sample S h1 ; h ¼ 1; 2; :::;H; S h2i ; i [ S h1 ; S h3ij ; j [ S h2i ad calculate ^u ¼ tðw hijk ; y hijk; h ¼ 1; 2; :::;H; i [ S 1h ; j [ S h2i ; k [ S h3ij Þ. The methods below are differet i values of w hijk ad creatio

4 196 Joural of Official Statistics of a bootstrap sample {S h1 ; h ¼ 1; 2; :::;H; S h2i ; i [ S h1 ; S h3ij ; j [ S h2i }. But all of them satisfy the coditio that for ^Y :::: ¼ P H P P P h¼1 i[s j[s k[s w hijk y hijk, h1 h2i h3ij E ð ^Y ::::Þ ¼ ^Y:::: ad V ð ^Y ::::Þ ¼vð ^Y::::Þ, where E ð Þ ad V ð Þ are the expectatio ad variace uder repeated bootstrap resamplig, respectively. The proofs of the results for the four methods are give i a separate appedix available from the author upo request. For variace estimatio, oe ca perform a Mote Carlo simulatio, i.e., repeat the above resamplig ad calculatio of ^u a large umber of times B ad obtai a estimate by v boot ð ^uþ ¼ðB21Þ P B b¼1 ^u b 2 ^u 2, _ where ^u b is the value of ^u i the bth bootstrap sample ad ^u _ ¼ B 21P B ^u b¼1 b Beroulli Bootstrap I the BBE proposed by Fuaoka et al. (2006), a bootstrap sample is costructed through radom replacemet of the sampled uits. The procedure is performed idepedetly for h ¼ 1; 2; :::;H. Step 1. Choose ( h 2 1) labels by simple radom samplig with replacemet from S h1. Deote the cadidate set by C h1. For each i [ S h1, we: (a) keep it i the bootstrap 21ð1 sample with probability p h1 ¼ 1 2 ð1=2þ h 2 f 1h Þ; or (b) replace it with oe radomly selected from C h1. If (a) is the case, go to Step 2. Step 2. For i kept at Step 1, choose (m hi 2 1) labels by simple radom samplig with replacemet from S h2i. Deote the cadidate set by C h2i. For each j [ S h2i, we: (c) keep it i the bootstrap sample with probability p h2i ¼ 1 2 ð1=2þp 21 h f 1h ð1 2 m 21 hi Þ 21 ð1 2 f 2hi Þ; or (d) replace it with oe radomly selected from C h2i. If (c) is the case, go to Step 3. Step 3. For j kept at Step 2, choose (l hij 2 1) labels by simple radom samplig with replacemet from S h3ij. Deote the cadidate set by C h3ij. For each k [ S h3ij, we: (e) keep it i the bootstrap sample with probability p h3ij ¼ 1 2 ð1=2þp 21 h f 1hp 21 h2i f 2hi ð1 2 l 21 hij Þ21 ð1 2 f 3hij Þ; or (f) replace it with oe radomly selected from C h3ij. Deote the oresultat bootstrap sample by S h1 ; h ¼ 1; 2; :::;H; S h2i ; i [ S h1 ; S h3ij ; j [ S h2i ad let w hijk ¼ w hijk. Creatig the cadidate sets is ecessary to make the procedure feasible for ay h, m hi, l hij $ 2 (Fuaoka et al. 2006). Obviously, the BBE retais the origial sample sizes ad the origial sample weights. This is desirable i dealig with radomly imputed survey data (Saigo, Shao, ad Sitter 2001) Mirror-Match Bootstrap The BMM proposed by Sitter (1992a) ca be exteded to ST SI 3 The procedure is performed idepedetly for h ¼ 1; 2; :::;H. as follows. Step 0. Choose 0 h ; m0 h ; ad l0 hij such that 1 # 0 h # 0 h =ð2 2 f 1hÞ, 1 # m 0 hi # m hi={1 þð12f h2i Þð h 2 0 h Þ=ðN h 2 h Þ}, ad 1 # l 0 hij # l hij={1 þð12f h3ij Þ ðm hi 2 m 0 hi Þ=ðM hi 2 m hi Þ}. Let f h1 ¼ 0 h = h, f h2i ¼ m0 hi =m hi, f h3ij ¼ l0 hij =l hij, k h1 ¼ { h ð12f h1 Þ}={0 h ð12f h1þ}, k h2i ¼ {N h ð12f h1 Þ}={ h ð12f h1 Þ} {m hið12f h2i Þ}=

5 Saigo: Comparig Four Bootstrap Methods 197 {m 0 h ð12f h2iþ}, ad k h3ij ¼{M hi ð12f h2i Þ}={m hi ð12f h2i Þ} {l hijð12f h3ij Þ}={l0 hij ð12f h3ijþ}. If desirable, we may radomize k h1, k h2i, ad k h3ij to realize E ðf h1 Þ¼f h1, E ðf h2i Þ¼f h2i, ad E ðf h3ij Þ¼f h3ij. See the commet made i the secod paragraph below Step 3. Step 1. If k h1 is a iteger, ~k h1 ¼ k h1. If ot, let ~k h1 ¼ bk h1 c with probability p h1 ¼ k 21 h1 2 dk h1e 21 = bk h1 c 21 2 dk h1 e 21 or ~k h1 ¼ dk h1 e otherwise. Repeat idepedetly ~k h1 times simple radom samplig without replacemet of size 0 h from S h1. Deote the subsampled ~ h ¼ 0 ~ h k h1 labels by S h1. Step 2. For each i [ S h1 : if K h2i is a iteger, ~k h2i ¼ k h2i ; if ot, let ~k h2i ¼ bk h2i c with probability p h2i ¼ k 21 h2i 2 dk h2ie 21 = bk h2i c 21 2 dk h2i e 21 or ~k h2i ¼ dk h2i e otherwise. Repeat idepedetly ~k h2i times simple radom samplig without replacemet of size m 0 hi from S h2i. Deote the subsampled ~m hi ¼ m0 ~ hi k h2i labels by S h2i. Step 3. For each j [ S h2i :ifk h3ij is a iteger, ~k h3ij ¼ k h3ij ; if ot, let ~k h3ij ¼ bk h3ij c with probability p h3ij ¼ k 21 h3ij 2 dk h3ije 21 = bk h3ij c 21 2 dk h3ij e 21 or ~k h3ij ¼ dk h3ij e otherwise. Repeat idepedetly ~k h3ij times simple radom samplig without replacemet of size l 0 hij from S h3ij. Deote the subsampled ~l hij ¼ l0 ~ hij k h3ij labels by S h3ij. The bootstrap sample weight is give by w hijk ¼ N h=~ h Mhi = ~m hi Lhij =~l hij. To coduct a Mote Carlo simulatio, repeat Steps 1 3 a large umber of times B. I the separate appedix, it is show that h =ð2 2 f h Þ $ 1, m hi ={1 þð12f h2i Þð h 2 0 h Þ=ðN h 2 h Þ} $ 1 ad l hij ={1 þð12f h3ij Þðm hi 2 m 0 hi Þ=ðM hi 2 m hi Þ} $ 1 for 2 # h, N h ; 2 # m hi, M hi ; ad 2 # l hij, L hij. Thus, the method is always feasible. Note that by lettig 0 h ¼ m0 hi ¼ l0 hij ¼ 1, the with-replacemet bootstrap (BWR) follows from the BMM. If 1 # h f h1 ; 1 # m hi f h2i ; ad 1 # l hij f h3ij ; radomizig 0 h ; m0 hi ; ad l0 hij may yield E f h1 ¼ f h1, E f h2i ¼ f h2i,ade f h3ij ¼ f h3ij. Specifically, replace the first setece i step 0 by Let ~ 0 h ¼ b h f h1 c with probability d h f h1 e 2 h f h1 or ~ 0 h ¼ d hf h1 e otherwise; let ~m 0 hi ¼ bm hi f h2i c with probability dm hi f h2i e 2 m hi f h2i or ~m 0 hi ¼ dm hi f h2i e otherwise; ad let ~l 0 h ¼ bl hij f h3ij c with probability dl hij f h3ij e 2 l hij f h3ij or ~l 0 hij ¼ dl hij f h3ij e otherwise. The replace 0 h ; m0 hi ; ad l0 hij with ~0 h ; ~m0 hi ; ad ~l 0 hij i the secod setece i Step 0. Uder ST SI, the coditios E f h1 ¼ f h1 ðh ¼ 1; 2; :::;HÞ esure third-order momet matchig (Sitter 1992a). Although o theoretical explaatio is available for a multistage desig, we may pursue the coditios E f h1 ¼ f h1 ; E f h2i ¼ f h2i, ad E f h3ij ¼ f h3ij that possibly improve the BMM s performace. To implemet a Mote Carlo simulatio, repeat Steps 0 3 a large umber of times B Rescalig Bootstrap Rao ad Wu (1988) proposed a BRS method that rescales residuals to provide cosistet variace estimatio for a parameter defied as a smooth fuctio of the populatio meas. This approach, however, caot hadle variace estimatio for sample quatiles. Rao, Wu, ad Yue (1992) studied a modified BRS method that rescales sample weights to accommodate quatile estimatio uder stratified multistage samplig where the first stage samplig fractios are egligible. Here, we preset a weight-rescalig BRS for ST SI 3 with ay samplig fractios. The procedure is coducted idepedetly for h ¼ 1; 2; :::;H.

6 198 Joural of Official Statistics Step 0. Choose positive itegers h ; m hi ; ad l hij for i [ S h1; j [ S h2i. To avoid egative weights, choose h # ð1 2 f h1þ 21 ð h 2 1Þ, m hi # hð h 2 1Þ 21 ð1 2 f h1 Þ f 21 h1 ðm hi 2 1Þ ð1 2 f h2i Þ 21, ad l hij # m hi ðm hi 2 1Þ 21 ð1 2 f h2i Þ f 21 h2i ðl hij 2 1Þð1 2 f h3ij Þ 21 if possible. For example, if f h1 # 1=2 ad f h2i # 1=2 for all h ad i, we may choose h ¼ h 2 1; m hi ¼ m hi 2 1; ad l hij ¼ l hij 2 1. Step 1. Choose h labels radomly with replacemet from S h1. Let ~ hi be the umber of times label i [ S h1 is selected. We may equivaletly carry out this step by lettig ~ h1 ; ~m h2 ;:::;~ h h, MN h ; 1= h; 1= h ;:::;1= h, where MN stads for the multiomial distributio. Step 2. If ~ hi $ 1 for i [ S h1, choose ~ h m hi labels radomly with replacemet from S h2i. Let ~m hij be the umber of times label j [ S h2i is selected. Equivaletly, let ~m hi1 ; ~m hi2 ;:::;~m him hi, MN hi m hi ; 1=m hi; 1=m hi ;:::;1=m hi, If ~ hi ¼ 0, let ~m hij ¼ 0 for j [ S h2i. Step 3. If ~m hij $ 1, choose ~m ~ hij l hij labels radomly with replacemet from S h3ij. Let ~l hijk be the umber of times label k [ S h3ij is selected. Equivaletly, let ~ l hij1 ; ~l hij2 ;:::; ~l hijl hij, MN ~m hi l hij ; 1=l hij; 1=l hij ;:::;1=l hij. If ~m hij ¼ 0, let ~l hijk ¼ 0 for k [ S h3ij: The bootstrap sample is the same as the origial sample, but the bootstrap sample weights are give by w hijk ¼w hijk 1þa hi ~ hi 2 h = h þbhij ~m hij 2 ~ hi m hi =m hi þg hijk ~l hijk 2 ~m ~ o hij l hij =l hij ; where p ffiffiffiffiffiffiffiffiffiffiffiffiffi. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a hi ¼ h 12f h1 h ð h 21Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b hij ¼ h f h1 = hm hi 12f h2i m hi ðm hi 21Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g hijk ¼ h f h1 = h m hi f h2i =m hil hij 12f h3ij l hij ðl hij 21Þ: Rao ad Wu (1988) studied the choice of the resample sizes for ST SI with replacemet whe rescalig the study variable y to match the third order momets ad to capture the secod term of Edgeworth expasios with kow strata variaces. Note, however, that o theory has bee developed for multistage desigs. For variace estimatio, repeat Steps 1 3 for a large umber of times B ad perform a Mote Carlo simulatio Without-Replacemet Bootstrap A exteded BWO method for a stratified two-stage desig (Sitter 1992b) ca be exteded further to ST SI 3 as follows. The procedure is coducted idepedetly i h ¼ 1; 2; :::;H. Step 0. We eed the followig iteger radom variables to create a pseudo-populatio by copyig the origial sampled uits.

7 Saigo: Comparig Four Bootstrap Methods 199 For PSUs Compute the followig costats: v h ¼ h h þ N 21 h ; vh ¼ dv h e; ad _v h ¼ bv h c; k h1 ¼ N h 22 h v h; k h1 ¼ dk h1 e; ad _k h1 ¼ bk h1 c; ad a h1 ¼ k h1 {1 2 _v h =ð h k h1 Þ}={ _v h ð h k h1 2 1Þ}; ad _a h1 ¼ _k h1 {1 2 v h =ð h_k h1 Þ}={v h ð h_k h1 2 1Þ}: Defie ð~ h ; ~k h1 Þ as : Pr ~ h ; o ~k h1 ¼ _v h ; k h1 ¼ p h1 ad Pr ~ h ; o ~k h1 ¼ vh ; _k h1 ¼ 1 2 p h1 ; where p h1 ¼½ð12f h1 Þ={ h ð h 2 1Þ} 2 _a h1 Š={a h1 2 _a h1 Þ For SSUs Compute the followig costats: E ð~ h ;~ k h1 Þ ~ 21 h ¼ ph1_v 21 þð12p h1 Þv 21 h h ; o m hi ¼ m hi N h E ð~ h ; k ~ h1 Þ ~ 21 h 1 2 m 21 hi þ M 21 hi ; m hi ¼ dm hie; ad _ m hi ¼ bm hi c; k h2i ¼ M hi m 22 hi m hi ; k h2i ¼ dk h2i e; ad _k h2i ¼ bk h2i c; ad o. o a h2i ¼ k h2i 1 2 _ m hi =ðm hi k h2i Þ _m hi ðm hi k h2i 2 1Þ ; ad o. o _a h2i ¼ _k h2i 1 2 m hi =ðm hi_k h2i Þ m hi ðm hi_k h2i 2 1Þ Defie ~m hi ; ~k h2i as: Pr ~m hi ; o ~k h2i ¼ð_ m hi ; k h2i Þ ¼ p h2i ad Pr ~m hi ; o ~k h2i ¼ðmhi ; _k h2i Þ ¼ 1 2 p h2i ; where p h2i ¼ ð1 2 f h2i Þ= N h E ~ 21 ~ h ;~ k hi h mhi ðm hi 2 1Þo 2 _a =ða h2i 2 h2i _a h2i Þ For USUs Compute the followig costats: E ~m ð hi ; k ~ h2i Þ ~m h 21 ¼ ph2i _ m21 hi þð12p h2i Þ m 21 ; hi

8 200 Joural of Official Statistics l hij ¼ l hij N h E ~ ð h ;~ k h1 Þ ~21 h Mh E ~m ð hi ;~ k h2ij Þ ~m 21 hi 1 2 l 21 hij l hij ¼ dl hij e; ad _l hij ¼ bl hij c; k h3ij ¼ L hij l 22 hij l hij; k h3ij ¼ dk h3ij e; ad _k h3ij ¼ bk h3ij c; ad a h3ij ¼ k h3ij 1 2 _l hij =l hij k h3ij Þ = _l hij l hij k h3ij 2 1 ; ad o _a h3ij ¼ _k h3ij 1 2 l hij =l hij_k h3ij Þ = l hij l hij_k h3ij 2 1 þ L 21 hij Defie ~l hij ; ~k h3ij as : o o Pr ~l hij ; k h3ij ¼ _l hij ; k h3ij ¼ p h3ij ad Pr ~l hij ; ~k h3ij ¼ l hij ; _k h3ij ¼ 1 2 p h3ij ; where p h3ij ¼ ð1 2 f h3ij Þ= N h E 21 ~ h ;~ k h1 h M hi E 21 ~m hi ;~ ~m k h2i hi l hij l hij _a h3ij =ða h3ij 2 _a h3ij Þ: ; Step 1. Geerate ~ h ; ~k h1. Copy Sh1 ~k h1 times to create P h1 of size h ~k h1. Step 2. For each i [ P h1 ; geerate ð ~m hi ; ~k h2iþ. Copy S h2i ði [ P h1 Þ ~k h2i times to create P h2i of size m hi k ~ h2i. Step 3. For each j [ P h2i ði [ P h1þ, geerate ~l hij ; ~k h3ij. Copy S h3ij j [ P h2i ; i [ P h1 k h3ij times to create P h3ij of size l hijk h3ij. Step 4. Coduct ST SI 3 from the pseudo-populatio to obtai a bootstrap sample: first, take a SI of size ~ h from P h1 to get S h1 ; the for i [ S h1, take a SI of size ~m hi from P h2i to get S h2i ad fially for j [ S h2i i [ S h1, take a SI of size ~l hij from P h2ij to get S h3ij. The bootstrap samplig weights are give by w hijk ¼ N h=~ h Mhi = ~m hi It is show i the separate appedix that p h1 ; p h2i ; p h3ij [ ½0; 1Š ad that v h ; k h1 ; m hi ; k h2i ; l hij ; ad k h3ij are all positive itegers for h,m hi,l hij $ 2. For variace estimatio, repeat Steps 1 4 for a large umber of times B. For efficiet computatios, we may avoid uecessary radom umber geeratio by replacig Steps 1 4 with the followig steps: Lhij =~l hij. Step 1 0. Geerate ~ h ; ~k h1. Copy Sh1 ; ~k h1 times to create P h1 of size h k ~ h1. Sample S h1 of size ~ h from P h1 without replacemet. Step 2 0. For each i [ S h1, geerate ~m hi ; k h2i. Copy S h2i i [ P ~ h1 k h2i times to create P h2i of m hi k ~ h2i. Sample S h2i of size ~m hi from P h2i without replacemet. Step 3 0. For each j [ S h2i i [ S h1, geerate ~l hij ; k h3ij. Copy S h3ij j [ S h2i ; i [ S h1 k ~ h3ij times to create P h3ij of size l hijk h3ij. Sample S h3ij of size ~l hij from P h3ij. This procedure is similar to the BMM.

9 Saigo: Comparig Four Bootstrap Methods Numerical Illustratio To illustrate, let us cosider a sigle-stratum ðh ¼ 1Þ populatio which has N ¼ 10 PSUs each composed of M i ¼ 20 SSUs each with L ij ¼ 30 USUs i it. The sample sizes used are ¼ 4; m i ¼ 8; ad l ij ¼ 3. Sice H ¼ 1, we suppress subscript h i what follows. Creatio of a bootstrap sample i the four methods is carried out as follows. I the BBE: Step 1. Choose 3 PSUs as a cadidate set through simple radom samplig with replacemet from the 4 PSUs i the sample. For each of the 4 PSUs, keep it with probability 0.60 or replace it with oe radomly selected from the 3 PSUs i the cadidate set. For the PSUs kept i this step, go to the ext step. Step 2. Choose 7 SSUs as a cadidate set through simple radom samplig with replacemet from the 8 SSUs i the sample. For each of the 8 SSUs, keep it with probability or replace it with oe radomly selected from the 7 PSUs i the cadidate set. For the SSUs kept i this step, go to the ext step. Step 3. Choose 2 USUs as a cadidate set through simple radom samplig with replacemet from the 3 USUs i the sample. For each of the 3 USUs, keep it with probability or replace it with oe radomly selected from the 2 USUs i the cadidate set. I the BMM: Step 0. Let 0 ¼ 2; m 0 i ¼ 3; ad l0 ij ¼ 1, say. Step 1. Let ~k 1 ¼ 1 with probability 0.20 or ¼ 2 otherwise. Geerate ~k 1 ad repeat ~k 1 times simple radom samplig without replacemet of size 2 from the 4 PSUs. Step 2. Let ~k 2i ¼ 8 with probability 0.64 or ¼ 9 otherwise. For each of the PSUs take i Step 1, geerate ~k 2i ad repeat ~k 2i times simple radom samplig without replacemet of size 3 from the 8 SSUs. Step 3. Let ~k 3ij ¼ 5 with probability 0.63 or ¼ 6 otherwise. For each of the SSUs take i Step 2, geerate ~k 3ij ad repeat ~k 3ij times simple radom samplig without replacemet of size 1 from the 3 USUs (This is equivalet to simple radom samplig with replacemet of size ~k 3ij ). I the BRS: Step 0. Let ¼ 3, m i ¼ 7; ad l ij ¼ 2, say. Step 1. Choose 3 PSUs from 4 PSUs via simple radom samplig with replacemet. Let ~ i be the umber of times that PSU i is selected. Step 2. For i such that ~ i $ 1, choose 7~ i SSUs from 8 SSUs i PSU i via simple radom samplig with replacemet. Let ~m ij be the umber of times that SSU j i PSU i is selected. For i such that ~ i ¼ 0, let ~m ij ¼ 0; Step 3. For ij such that ~m ij $ 1, choose 2 ~m ij USUs from 2 USUs i SSU j i PSU i via simple radom samplig with replacemet. Let ~l ijk be the umber of times that USU k i SSU j i PSU i is selected. For ij such that ~m ij ¼ 0, let ~l ijk ¼ 0. Use a i ¼ 1:03; b ij ¼ 0:65; ad g ijk ¼ 0:70 for weight rescalig.

10 202 Joural of Official Statistics I the BWO: Step 0. Defie the followig iteger radom variables: ð~ ; ~k 1 Þ¼ð3; 3Þ with probability 0.44 or ¼ (4,2) otherwise; ð ~m i ; ~k 2i Þ¼ð20; 7Þ with probability 0.47 or ¼ (21, 6) otherwise; ad ð~l ij ; ~k 3ij Þ¼ð5; 19Þ with probability 0.29 or ¼ (6, 18) otherwise. Step 1. Geerate ð~ ; ~k 1 Þ ad copy 4 PSUs ~k 1 times. Step 2. For each i i the copied PSUs i Step 1, geerate ~m i ; ~k 2i ad copy 6 SUSs ~k 2i times. Step 3. For each j i the copied SSUs i PSU i i Step 2, geerate USUs ~k 3ij times. Step 4. Mimic the origial samplig uder the created pseudo-populatio with realized sample sizes ~ ; ~m i ; ad ~l ij. 4. Simulatio Study ~l ij ; ~k 3ij ad copy 2 We compared the four bootstrap methods through a limited simulatio study. To focus o desig-based properties, we geerated a populatio ad fixed it uder repeated samplig. We created two populatios. I Populatio I, stratificatio was less effective ad the itra-cluster correlatio was low. O the other had, i Populatio II, stratificatio was more effective ad the itra-cluster correlatio was high. Both had H ¼ 4 strata. Each stratum had N h ¼ 10 primary samplig uits, each of which had M hi ¼ 20 secodary samplig uits, each havig L hij ¼ 30 ultimate samplig uits. We chose the small umber of strata to obtai a clear picture of the pseudo-populatios uder study. The study variables y hijk were geerated as follows. First, we geerated m hi ¼ m h þ s h u hi, where m h ad p s h are listed i Table 1 ad u hi, iidnð0; 1Þ. Secod, we geerated m hij ¼ m hi þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1 2 r 2 Þ=r 2 s h u hij, where u hij, iidnð0; 1Þ. Fially, we geerated y hijk ¼ m hij þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2 r 3 Þ=r 3 s h u hijk, where u hijk, iidnð0; 1Þ. We set ðr 2 ; r 3 Þ¼ð0:2; 0:3Þ for Populatio I (low itra-cluster correlatio) ad ðr 2 ; r 3 Þ¼ð0:5; 0:5Þ for Populatio II (high itra-cluster correlatio). Figure 1 shows the histograms of y hijk i Stratum h ðh ¼ 1; 2; 3; 4Þ i Populatio I, with five vertical lies showig the populatio quatiles for p ¼ 0:10; 0:25; 0:50; 0:75; 0:90. Sice stratificatio is weak i Populatio I, the characteristic y hijk i a stratum overlaps those i the other strata. O the other had, stratificatio i Populatio II is effective. I particular, the first quartile F 21 ð0:25þ, the media F 21 ð0:5þ, ad the third quartile F 21 ð0:75þ are located at the stratum boudary betwee h ¼ 1 ad h ¼ 2; h ¼ 2 ad h ¼ 3; ad h ¼ 3 ad h ¼ 4, respectively (see Figure 2). Table 1. Parameter Values to Create a Populatio Populatio I Populatio II h m h s h m h s h

11 Saigo: Comparig Four Bootstrap Methods 203 Fig. 1. Histograms of y hijk i Strata h ðh ¼ 1; 2; 3; 4Þ i Populatio I. Note: The five vertical lies show F 21 ð pþ with p ¼ 0:1; 0:25; 0:50; 0:75; 0:9 Sice high samplig fractios were the cocer, all the first stage samplig fractios were oegligible. Table 2 shows the first- ad secod-stage samplig fractios i the simulatio. The third-stage samplig fractios were set to be f h3ij ¼ 0:1. I the BMM, 0 h ; m0 h ; ad l0 hij were radomized to get E f h1 ¼ f h1 ; E f h2i ¼ f h2i, ad E ¼ f h3ij whe possible (see Sectio 3.2) or 0 h ¼ 1; m0 hi ¼ 1 ad l0 hij ¼ 1 f h3ij Fig. 2. Histograms of y hijk i Strata h ðh ¼ 1; 2; 3; 4Þ i Populatio II. Note: The five vertical lies show F 21 ð pþ with p ¼ 0:1; 0:25; 0:50; 0:75; 0:9

12 204 Joural of Official Statistics Table 2. The samplig Fractios i the Simulatio Study h f h1 f h2i f h3ij otherwise. I the BRS, h ¼ h 2 1; m hi ¼ m hi 2 1; ad l hij ¼ l hij 2 1, which assures that w hijk. 0. The parameters of iterest u were the populatio total Y:::: ad quatiles F 21 ð pþðp ¼ 0:1; 0:25; 0:5; 0:75; 0:9Þ, ad their poit estimators were give by ^Y... ad ^F 21 ð pþ i Sectio 2. I each simulatio ru, we geerated B ¼ 1; 000 bootstrap samples to estimate variace ad made a bootstrap histogram. Variace was estimated by v boot ð ^uþ i Sectio 3. To evaluate the empirical coverage, the 0.1 ad 0.9 poits of the bootstrap histogram were employed to calculate the lower- ad upper-tail errors defied below (see Shao ad Tu 1995, pp , for usig the bootstrap histogram to costruct a cofidece iterval). The relative bias (%Bias) ad the istability (the coefficiet of variatio, %Istb) i variace estimatio, ad the lower- ad upper-tail errors (%L ad %U) were evaluated through S ¼ 1; 000 simulatio rus while the variace Vð ^uþ was estimated by 10,000 iteratios: %Bias ¼ ( ) S 21XS v ðsþ boot ð ^uþ 2 Vð ^uþ Vð ^uþ 100; s¼1 %Istb ¼ ( ) 1=2 S 21XS v ðsþ boot ð ^uþ 2 Vð ^uþ 2 Vð ^uþ 100; s¼1 %L ¼ S 21 #{0, the 0:1 poit of bootstrap histgram of ^u b } 100; %U ¼ S 21 #{0. the 0:9 poit of bootstrap histgram of ^u b } 100; The empirical coverage rate was computed as (100 2 %L 2 %U) percet. Tables 3 ad 4 show the results for the four bootstrap methods for Populatios I ad II, respectively. We observe from Tables 3 ad 4 the followig poits. 1. I estimatig variaces for the estimated populatio total i both Populatios I ad II, the performace measures for the four methods are almost idetical. This was expected because the four methods satisfy E ^Y :::: ¼ ^Y::: ad V ^Y :::: ¼ v ^Y:::, so the differeces i %Bias reflect oly Mote Carlo errors. 2. I estimatig variaces for the five estimated quatiles i Populatio I, the four methods perform similarly although the BWO is the least biased ad the most stable. 3. I estimatio for Populatio II, the four methods show remarkable differeces: the bias i variace estimatio by the BRS ca be serious; the istability of the BBE teds to be greater tha that of the BMM ad the BWO; ad the BMM ad the BWO perform similarly, although the latter is margially less biased ad more stable.

13 Saigo: Comparig Four Bootstrap Methods 205 Table 3. Variace Estimatio ad Tail Errors for Populatio I %Bias %Istb %L %U %Bias %Istb %L %U ^Y: :: ^F 21 ð0:10þ BBE BMM BRS BWO ^F 21 ð0:25þ ^F 21 ð0:50þ BBE BMM BRS BWO ^F 21 ð0:75þ ^F 21 ð0:90þ BBE BMM BRS BWO %Bias ¼ S 21P o. S s¼1 vðsþ boot ð uþ ^ 2 VðuÞ ^ VðuÞ ^ 100; %Istb ¼ S 21P S s¼1 vðsþ boot ð uþ ^ 2 VðuÞ ^ 2 o 1=2=Vð uþ ^ 100; %L ¼ S 21 #{u, the 0:10 lower trail of u ^ b } 100; %U ¼ S 21 #{u, the 0:10 upper trail of u ^ b } 100. Table 4. Variace Estimatio ad Tail Errors for Populatio II %Bias %Istb %L %U %Bias %Istb %L %U ^Y: :: ^F 21 ð0:10þ BBE BMM BRS BWO ^F 21 ð0:25þ ^F 21 ð0:50þ BBE BMM BRS BWO ^F 21 ð0:75þ ^F 21 ð0:90þ BBE BMM BRS BWO %Bias ¼ S 21P o. S s¼1 vðsþ boot ð uþ ^ 2 VðuÞ ^ VðuÞ ^ 100; %Istb ¼ S 21P S s¼1 vðsþ boot ð uþ ^ 2 VðuÞ ^ 2 o 1=2. VðuÞ ^ 100; %L ¼ S 21 #{u, the 0:10 lower trail of u ^ b } 100; %U ¼ S 21 #{u. the 0:10 upper trail of u ^ b } 100.

14 206 Joural of Official Statistics 4. Variace of sample quatiles is usually overestimated by the bootstrap methods. However, the BRS seriously uderestimates variace of ^F 21 ð0:25þ; ^F 21 ð0:50þ; ad ^F 21 ð0:75þ i Populatio II. A possible explaatio for this is as follows. To fix the idea, defie ~W hijk such that ^Y :::: ¼ P H P P h¼1 Pi[S h1 j[s h2i k[s h3ij ~W hijk y hijk. That is, ~W hijk are the sum of bootstrap weights associated with USU k i SSU j i PSU i i Stratum h i the origial sample. I the BRS, ^W hijk ¼ w hijk. 0 for all the uits i the origial sample. Namely, the order statistics i bootstrap samples do ot chage at all. By cotrast, i the BBE, the BMM, ad the BWO, ~W hijk ca be zero. I Populatio II, y hijk aroud F 21 ð pþ; where p ¼ 0:25; 0:5; ad 0:75, are scarce, ad so are the sampled y hijk aroud the poits. Possibly, the bootstrap pseudo-estimates, ^F 21 ð pþ; where p ¼ 0:25; 0:5, ad 0.75, i the BRS vary too smoothly because of the fixed order statistics while those i the BBE, the BMM, ad the BWO fluctuate largely sice some ~W hijk are zero. 5. Overall poor performaces of iterval estimatio are probably due to a small populatio size with a small sample size. This is particularly true for the media i Populatio II, where y values aroud the quatile are scarce. To test this poit, we coducted a simulatio usig the parameter values for Populatio II with doubled N h ¼ 20 ad M hi ¼ 40 to fid the actual tail error rates closer to the omial rates. The result showed that the lower ad the upper-tail error rates were, respectively, about 8% ad 24%. We ca, perhaps, improve the coverage rates through a more sophisticated bootstrap cofidece iterval. But that is beyod the scope of this article ad we do ot ited to pursue it here. 5. Coclusio I this article, we first exteded the three bootstrap methods to a stratified three-stage desig with simple radom samplig without replacemet at each stage: the mirror-match bootstrap (BMM), the rescalig bootstrap (BRS), ad the without-replacemet bootstrap (BWO). The, we coducted a simulatio study to examie the three methods as well as the Beroulli bootstrap (BBE). The simulatio showed that (1) the four methods perform similarly for estimatig the variace of the estimated populatio total; (2) the four methods perform differetly for quatile estimatio whe stratificatio is effective; ad (3) overall, the BWO was the least biased while bias i variace estimatio by the BRS was sometimes remarkably large. The last observatio supports the ituitio that methods which better mimic the origial samplig will perform better as the estimators ad the samplig desig become more complex (Sitter 1992b, p. 153). Theoretical research is beyod the scope of this article ad will be a future topic. 6. Refereces Fuaoka, F., Saigo, H., Sitter, R.R., ad Toida, T. (2006). Beroulli Bootstrap for Stratified Multistage Samplig. Survey Methodology, 32, Gross, S. (1980). Media Estimatio i Sample Surveys. Proceedigs of the America Statistical Associatio, Sectio o Survey Research Methods,

15 Saigo: Comparig Four Bootstrap Methods 207 Kovar, J.G., Rao, J.N.K., ad Wu, C.F.J. (1988). Bootstrap ad Other Methods to Measure Errors i Survey Estimates. The Caadia Joural of Statistics, 16, Rao, J.N.K. ad Wu, C.F.J. (1988). Resamplig Iferece with Complex Survey Data. Joural of the America Statistical Associatio, 83, Rao, J.N.K., Wu, C.F.J., ad Yue, K. (1992). Some Recet Work o Resamplig Methods for Complex Surveys. Survey Methodology, 18, Saigo, H., Shao, J., ad Sitter, R.R. (2001). A Repeated Half-Sample Bootstrap ad Balaced Repeated Replicatios for Radomly Imputed Data. Survey Methodology, 27, Särdal, C.-E., Swesso, B., ad Wretma, J. (1992). Model Assisted Survey Samplig. New York: Spriger-Verlag. Shao, J. ad Tu, D. (1995). The Jackkife ad Bootstrap. New York: Spriger-Verlag. Sitter, R.R. (1992a). A Resamplig Procedure for Complex Survey Data. Joural of the America Statistical Associatio, 87, Sitter, R.R. (1992b). Comparig Three Bootstrap Methods for Survey Data. The Caadia Joural of Statistics, 20, Woodruff, R.S. (1952). Cofidece Itervals for Media ad Other Positio Measures. Joural of the America Statistical Associatio, 47, Received April 2008 Revised September 2009

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