EFFICIENCY OF MODEL-ASSISTED REGRESSION ESTIMATORS IN SAMPLE SURVEYS

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1 Statistica Sinica , doi:ttp://dx.doi.org/ /ss EFFICIENCY OF MODEL-ASSISTED REGRESSION ESTIMATORS IN SAMPLE SURVEYS Jun Sao 1,2 and Seng Wang 3 1 East Cina Normal University, 2 University of Wisconsin-Madison and 3 Matematica Policy Researc Abstract: Model-assisted regression estimators are popular in sample surveys for making use of auxiliary information and improving te Horvitz-Tompson estimators of population totals. In te presence of strata and unequal probability sampling, owever, tere are several ways to form model-assisted regression estimators: regression witin eac stratum or regression by combining all strata, and a separate ratio adjustment for population size, or a combined ratio adjustment, or no adjustment. In te literature, tere is no compreensive teoretical comparison of tese regression estimators. We compare te asymptotic efficiencies of six model-assisted regression estimators under two asymptotic settings. Wen tere are a fixed number of strata wit large stratum sample sizes, our result sows tat one of te six regression estimators is a clear winner in terms of asymptotic efficiency. Wen tere are a large number of strata wit small stratum sample sizes, owever, te story is different. Some comparisons in special cases are also made. Some simulation results are presented to examine finite sample performances of regression estimators and teir variance estimators. Key words and prases: Asymptotic efficiency, bootstrap, combined regression estimators, separate regression estimators, unequal probability witout replacement sampling, variance estimation. 1. Introduction Auxiliary information from sources suc as administrative records is often available in sample surveys. Model-assisted regression estimators are te most popular estimators tat utilize auxiliary information to gain efficiency in estimating population totals. Altoug regression estimators are constructed using a regression model between te variables of interest and some covariates, tey are consistent and asymptotically normal under te traditional design-based framework in wic te randomness of survey estimators is from repeated sampling. Tus tey are robust against violation of te assumed regression model. But, wen te regression model is correct, tey are more efficient tan te estimators tat do not use te auxiliary information. Let U denote te finite population of interest wit N units stratified into H strata, U = U 1 U H, were U contains N units and N 1 + +

2 396 JUN SHAO AND SHENG WANG N H = N. For unit i U, let y i be a nonconstant variable of interest and x i be a nonconstant auxiliary variable associated wit y i. Altoug we consider a univariate x i trougout, all results can be generalized to te case of multivariate x i in a straigtforward manner. Consider te estimation of te population total Y = y i based on a sample S = S 1 S H from U, were eac S is a sample from U according to some probability sampling plan and S 1,..., S H are independently selected. Let n be te size of S and n = n n H. For i S, y i is observed. Te Horvitz-Tompson estimator of te total Y is Ŷ = Ŷ, Ŷ = i S w i y i, were wi 1 is te first-order inclusion probability of unit i in S, a known quantity from te sampling design. Under te traditional design-based framework, y i s are fixed values and te only randomness is from te repeated selection of S from U. Let E s be te expectation wit respect to repeated sampling. Ten, it is well known tat E s Ŷ = Y. For te auxiliary variable, te value of x i is observed for i S, and te value of X = x i is known e.g., from administrative records for eac. To utilize te auxiliary variable x i, te model-assisted approac e.g., Särndal, Swensson, and Wretman 1992 can be adopted to form regression estimators. In te presence of strata, owever, tere are two ways to apply te regression: regression witin eac stratum or regression by combining all strata. Also, one can apply a separate ratio adjustment for population size, or a combined ratio adjustment, or no adjustment. Tis leads to six regression estimators of Y : Ŷ C0 = [Ŷ + ˆβX ˆX ] = Ŷ + ˆβX ˆX, 1.1 Ŷ S0 = [Ŷ + ˆβ X ˆX ] = Ŷ + ˆβ X ˆX, 1.2 Ŷ CC = [ N Ŷ + ˆβ X N ˆX ] = NŶ + ˆβ X N ˆX, 1.3 Ŷ SC = [ N Ŷ + ˆβ X N ˆX ] = NŶ + ˆβ X N ˆX, 1.4 Ŷ CS = [ N Ŷ + ˆβ X N ˆX ] = N Ŷ + ˆβ X N ˆX, 1.5 Ŷ SS = [ N Ŷ + ˆβ X N ˆX ], 1.6

3 MODEL-ASSISTED REGRESSION ESTIMATORS 397 were X = x i, ˆX = i S w ix i, ˆX = i S w i x i, = i S w i, = i S w i, i S ˆβ = w i y i x i ˆX / i S w i x i ˆX / is te slope estimator assuming a single regression line over all strata, and i S ˆβ = w i y i x i ˆX / i S w i x i ˆX / is te slope estimator for te regression line witin stratum. Te first subscript u in Ŷuv given by indicates weter a combined regression line over all strata is assumed u = C or a separate regression line for eac stratum is considered u = S. Te second subscript v in Ŷuv indicates weter no ratio adjustment for population size is applied v = 0, a combined over strata ratio adjustment N/ is applied v = C, or a separate ratio adjustment N / witin eac stratum is applied v = S. Some of te estimators in are te same or similar to tose proposed by previous researcers e.g., Isaki and Fuller 1982, Wrigt 1983, Montanari 1987, Särndal, Swensson, and Wretman 1992 and Fuller 2009 and are frequently used in practice; te oters are included for comparison. Wen combined regression is applied, knowing X, instead of eac X, is enoug for estimators ŶC0, ŶCC, and ŶCS. Intuitively, regression by combining strata is motivated by te tinking tat it produces more efficient estimators wen all regression models across strata are te same wic is not always true as we sow in Section 2. Altoug some discussions on te relative efficiencies among te six estimators in can be found in te literature e.g., Cocran 1977, Särndal, Swensson, and Wretman 1992 and Fuller 2009, a compreensive teoretical comparison of tese estimators is not available. Te purpose of tis paper is to study te asymptotic relative efficiencies among tese six estimators under te model-assisted approac tat imposes a regression model between y i and x i in eac stratum. We consider two types of sampling designs, one wit a fixed number of strata and a large number of sampled units witin eac stratum and te oter one wit a large number of strata and a few sampled units witin eac stratum. Under te first type of design, we are able to draw some definite conclusions on te relative performance of estimators provided te regression models are correct. Note tat if we do not use any model, all six estimators are consistent and asymptotically normal under te design-based framework, but teir asymptotic relative efficiencies cannot be precisely assessed unless some special designs are considered, for example, stratified simple random sampling. An example is given in Section 2.

4 398 JUN SHAO AND SHENG WANG Under te model-assisted framework, some asymptotic results for two types of designs are given in Sections 2 and 3, respectively. Some simulation results are presented in Section 4. Te last section contains some discussion. All tecnical proofs are given in te Appendix. 2. Asymptotic Results for Large n s and Fixed H To consider asymptotics, te population U is viewed as a member of a sequence of populations {U k, k = 1, 2,...}, were te number of units in U k increases to infinity as k. All quantities, suc as y i, N, etc., depend on te index k but k is omitted for simplicity. All limiting processes are wit respect to k. In tis section, we assume tat, in eac stratum, te population size N and te sample size n as k. Te number of strata, H, is fixed. Te sampling design wit a fixed H and large n s is commonly used in many surveys, for example, business surveys. We can establis te asymptotic normality of estimators in wen te y i s and x i s are fixed values and te randomness is from te sample selection of S, = 1,..., H. A simple example, owever, indicates tat, under te design-based approac, we cannot generally tell wic estimator in is te most efficient one unless a special design is considered. Consider tat H = 1, eac i U as probability p i to be selected, and sampling is wit replacement. We compare te estimators Ŷ1 = i S y i/np i and Ŷ2 = NŶ1/, = i S 1/np i, tat are special cases of estimators in 1.1 and 1.2, respectively. Suppose tere exists a positive constant M suc tat M 1 Np i M for all i U and N 1 y2 i M. Ten, by te Central Limit Teorem, nŷ1 Y /N d d v 1 N0, 1, were denotes convergence in distribution and v 1 = 1 2 yi N 2 p i Y. p i Applying te delta metod, we obtain tat nŷ2 Y /N v 2 d N0, 1, were v 2 = v 1 2Y 1 yi N 3 p i N Y + Y p i p i N 4 p i N. p i Ten, Ŷ2 is asymptotically more efficient tan Ŷ1 if and only if Ȳ 2 N 2 1 < 2Ȳ p i N 2 y i p i Ȳ 2, 2.1

5 MODEL-ASSISTED REGRESSION ESTIMATORS 399 were Ȳ = Y/N. Under te design-based framework, owever, y i s are fixed values and we cannot tell weter 2.1 olds except for a few special cases. Tis is similar to te discussion in Section 6.6 of Cocran If we assume te y i s are iid wit mean µ y, ten 2.1 olds for sufficiently large N unless µ y = 0 or p i = N 1 simple random sampling. Tis can be seen as follows. Suppose µ y = 0 and p i is not a constant. By te Law of Large Numbers and te condition M 1 Np i M for all i U, y i Ȳ µ y and p i N 2 µ y p i N 2 0 a.s.. Tis togeter wit Jensen s inequality > N p i N 1 p i = N implies tat 2.1 olds for large enoug N. We consider te model-assisted approac tat views y i, x i, i U, as random vectors following a model. To utilize te auxiliary variable x i, we consider te regression model y i = α + β x i +, i U, = 1,..., H, 2.2 were α and β are, respectively, te unknown intercept and slope for te regression witin stratum U,, i U, are iid wit mean 0 and an unknown variance, x i, i U, are iid wit an unknown variance, and s are independent of x i s. Te estimator ˆβ in 1.8 is te least squares estimator of β, = 1,..., H. If we combine all regression lines into a single line, ten te estimator ˆβ in 1.7 is te least squares estimator of te common slope. Altoug te combined regression is wrong wen β s are unequal, estimators in are still consistent and asymptotically normal since tey are model-assisted estimators. Tis is even true wen model 2.2 is incorrect. A teorem establises te asymptotic normality of estimators , wit respect to te probability under regression model 2.2 and repeated sampling. Let P s, E s, and V s be te probability, expectation, and variance wit respect to repeated sampling, and let P m, E m, and V m be te probability, expectation, and variance wit respect to model 2.2. We require some conditions. C1 For any, n /N 0 and tere exists φ suc tat N /N φ. C2 For any, σx 2 = V mx i <, σ 2 = V m <, i U, and tere exists δ > 0 suc tat as k, E m R w i 1x i µ x 2+δ + E m x i µ x 2+δ i S /S 0, [ ]1+δ/2 V m R i S w i x i x i

6 400 JUN SHAO AND SHENG WANG E m R w i 1 2+δ + E m 2+δ i S /S 0, [ ]1+δ/2 V m R i S w i were R = 1, /N, or /N, and µ x = E m x i for i U. C3 Tere exists M > 1 suc tat for any, N /n M < w i < N M/n. Teorem 1. If 2.2 and C1 C3 old, ten Ŷ uv Y N ϕ uv d N0, 1, were u = C or S, v = 0, C, or S, [ ] ϕ SS = N 2 σ2 w i N, ϕ CS = ϕ SS + 1 N 2 σ2 x β β 2 w i N, ϕ SC = ϕ SS + V s α N N, ϕ CC = ϕ CS + V s [α + µ x β β] N N, ϕ S0 = ϕ SS + 1 and ϕ C0 = ϕ CS + 1 N 2 V s V N 2 s α N, [ ] α + µ x β β N, β = N β σx 2 N σx 2. Te proof is given in te Appendix. Te condition n /N 0 is not needed wen sampling is wit replacement or simple random sampling witout replacement. Te following are some conclusions for te relative efficiencies among estimators in For two estimators Ŷu and Ŷv, Ŷu Ŷv denotes tat Ŷu is asymptotically as efficient as Ŷv, Ŷ u Ŷv denotes tat Ŷu is asymptotically more efficient tan Ŷv, and Ŷu = Ŷv denotes tat Ŷu and Ŷv are asymptotically equivalent. 1. Asymptotically, no oter estimator is more efficient tan ŶSS, even wen regression lines across strata are te same. Some oter estimators may be asymptotically as efficient as ŶSS under some special situations.

7 MODEL-ASSISTED REGRESSION ESTIMATORS Wen α = α 0 and β = β for all, ŶCC = ŶCS = ŶSC = ŶSS ŶC0 = Ŷ S0. Wen α = 0 and β = β for all, all estimators in are asymptotically equivalent. 3. Wen β = β for all, ŶC0 = ŶS0, ŶCC = ŶSC, ŶCS = ŶSS, ŶSS ŶS0, and Ŷ SS ŶSC. Wen = N, ŶS0 = ŶSC. No general conclusion can be made about ŶS0 and ŶSC. 4. Wen tere exist 1 and 2 suc tat β 1 β 2, Ŷ S0 ŶC0, Ŷ SC ŶCC and ŶSS ŶCS. Tus, applying te same ratio, te estimators using separate lines are more efficient tan tose using a single line. 5. Wen α = α 0 for all, Ŷ SC = Ŷ SS ŶS0. Wen α = 0 for all, Ŷ SC = Ŷ SS = Ŷ S0. 6. Wen tere exist 1 and 2 suc tat α 1 α 2, ŶSS ŶSC and ŶSS ŶS0 unless 1 = N 1 and 2 = N 2. Tere is a definite conclusion. Asymptotically, ŶSS is te winner among all estimators in even wen te regression lines across strata are te same. Tis penomenon as been noticed by previous researcers see, e.g., Fuller One explanation is tat, wen all n s are large, te information about te equality of regression lines does not elp to improve model-assisted estimators of te forms tat are robust against model violation. Some model-based estimators may be more efficient tan ŶSS wen all regression lines are te same, but tey are not robust against model violation. Furtermore, te situation is quite different if some n s are small, as te results in te next section indicate. Anoter point is tat te result in Teorem 1 still olds if ˆβ in 1.8 is replaced by a different consistent estimator of β. Fuller 2009 indicates tat, given a sampling design, an optimal slope estimator can be derived e.g., in Fuller 2009 tat is asymptotically more efficient tan ˆβ in 1.8 in terms of estimating β. For te estimation of Y, owever, te efficiency of ˆβ is a second-order asymptotic effect. It does not improve te asymptotic efficiency of Ŷ SS. 3. Asymptotic Results for Large H and Small n s To increase efficiency, many surveys involve a large number of strata. To reduce te cost, a few units are sampled from eac stratum. See, for example, Krewski and Rao We consider a different asymptotic setting: all n s are bounded by a fixed constant and H. For estimators based on separate regressions, Ŷ SS, Ŷ SC, and ŶS0, eac n as to be sufficiently large so tat a regression witin stratum can be fitted. Since model 2.2 involves a two term regression function, we require n > 2 for

8 402 JUN SHAO AND SHENG WANG Ŷ SS, ŶSC, and ŶS0. If x i is multivariate wit dimension p, ten n > p + 1 for eac is required. Wen n is small, it is not possible to obtain a consistent estimator for a parameter related only wit stratum U. However, quantities suc as totals over all strata can be consistently estimated. Consider ŶSS as an example. Let E m x be te expectation under 2.2, conditional on te x i s. Ten E m x y i = α + β x i, i U, and [ N E m x Ŷ + E m x ˆβ X N ˆX ] E m x ŶSS = = [ N α + N β ˆX + β X N ˆX ] = = N α + β X E m x Y = E m x Y. Tus, ŶSS is unbiased wit respect to model 2.2. Since ŶSS is a sum over of independent random variables aving finite means and variances under some conditions, we can apply Liapounov s Central Limit Teorem to establis te asymptotic normality of ŶSS. Tis is te approac considered by Krewski and Rao We need some conditions. D1 Tere exists M 1 suc tat α + β µ x < M 1 for any. D2 Tere exists M 2 suc tat 1/M 2 < σ 2 x = V mx i < M 2 and 1/M 2 < σ 2 = V m < M 2 for any, and tere exists δ > 0 suc tat, as H, H R E m Ŷ + Q X R ˆX E m [R Ŷ + Q X R ˆX ] =1 { H =1 2+δ } 0, 2+δ V m [R Ŷ + Q X R ˆX ] were R = 1, /N, or /N, and Q = ˆβ or ˆβ. D3 As H, β i S w i x i ˆX / 2 i S w i x i ˆX / p β, 2 E s { n N H N } [α + β βµ x ] N 0. =1

9 MODEL-ASSISTED REGRESSION ESTIMATORS 403 Teorem 2. If model 2.2, C3, and D1 D3 old, ten were u = C or S, v = 0, C, or S, and Ŷ uv Y N ϕ uv d N0, 1, ϕ C0 = 1 N 2 V s Λ + 1 [ N 2 Γ ϕ S0 = 1 [ N 2 E se m ϕ CC = V s σ 2 i S Λ ϕ SC = 1 [ N 2 E se m ϕ CS = 1 N 2 E s ϕ SS = 1 N 2 [ + 1 i S σ 2 N 2 Γ 2 [ σ 2 E se m ζ i +w i 1 [ N 2 Γ ζ i +w i 1 i S ] w i N, 2 ] + 1 N 2 σ 2 N n + 1 N 2 V s α, ] w i N, 2 ] + 1 N 2 σ 2 N n +V s ], wi 2 N i S ψ i + N w i 1 2 ] + 1 N 2 σ 2 N n, wit Λ = α + β βµ x, Γ = σ 2 + β β 2 σ 2 x, x = ˆX /, ζ i = α, x i x w i ˆX x i x 2 X, and ψ i = x i x w i N ˆX w i x i x 2 X. w i i S i S Te proof is given in te Appendix. Note tat i S ψ i = 0 and i S ζ i = 0 for any. If i S x i x w 2 i = 0 for any, wic occurs for example wen w i is a constant witin stratum, ten ϕ SS = 1 { N 2 σ 2 E } se m ψi 2 N β β 2 σ 2 2 x E s w i S 2 i 2 N +ϕ CS. i S 3.1 As for te relative efficiencies among estimators in , we ave te following discussion. 1. Wit simple random sampling SRS witin eac stratum, = N, = N, Ŷ C0 = ŶCC = ŶCS, and ŶS0 = ŶSC = ŶSS. Tus, we compare ŶCS and ŶSS. Under SRS witin eac stratum, i S ψ i w i = 0 for any. If β = β for

10 404 JUN SHAO AND SHENG WANG any, ten it follows from 3.1 tat ϕ SS = 1 { N 2 σ 2 E se m i S ψ 2 i } + ϕ CS > ϕ CS 3.2 and, ence, ŶCS ŶSS. However, it also follows from 3.1 tat, wen β β 2 increases to, te ratio ϕ SS /ϕ CS increases to infinity, so ŶSS ŶCS if β β 2 is sufficiently large for some. 2. Under a general sampling design, tere is no definite conclusion about te relative efficiency among ŶC0, Ŷ CC, and ŶCS, because ϕ C0, ϕ CC, and ϕ CS involve expectations and variances wit respect to sampling. For example, tere is no definite conclusion on wic of 1 N 2 V s Λ and V s Λ is larger. Similarly, we are not able to draw a definite conclusion about te relative efficiency among ŶS0, ŶSC, and ŶSS. 3. If β = β and i S ψ i w i = 0 for any, ten ŶCS ŶSS. Tis is because 3.1 olds and, if β = β, ten 3.2 also olds. Similarly, if i S ζ i w i = 0 for any, ten ŶCC ŶSC and ŶC0 ŶS0. If we furter assume tat α = α 0 for any, ten ŶCC ŶC0 and ŶSC ŶS0. Wen α = 0 for any, ŶCC = ŶC0 and ŶSC = ŶS0. 4. If te β s are not all te same, β β 2 increases to for some, ϕ C0, ϕ CC, and ϕ CS all diverge to infinity. On te oter and, none of ϕ C0, ϕ CC, and ϕ CS depends on β β 2. Tis means tat ŶS0 ŶC0, ŶSC ŶCC, and Ŷ SS ŶCS wen β s are different and te differences are large enoug. Te result is quite different from tat in Section 2 were all n s are large and H is fixed. In Section 2 we found a clear winner in terms of asymptotic efficiency, Ŷ SS, regardless of te scenario, wereas for small n s, tere is no winner even if we use SRS witin eac stratum. Te asymptotic relative efficiency between Ŷ CS and ŶSS depends on ow te β s differ. 4. Empirical Results In tis section, we present some empirical results using simulated data sets. We also discuss variance estimation and examine related confidence intervals Simulation results for large n s and fixed H We considered a population wit H = 4 strata. In stratum, we generated independent x i Γϑ, θ, i = 1,..., N = 2000, were Γϑ, θ is te gamma

11 MODEL-ASSISTED REGRESSION ESTIMATORS 405 distribution wit sape parameter ϑ and scale parameter θ, y i = α + β x i + and z i = 30 + x i + ν i, were s and ν i s are independent and normally distributed wit mean 0 and variance 50, and tey are independent of x i s. Values from different strata were independently generated. Te values of te parameters of te gamma distribution in eac stratum were as follows. ϑ 1 θ 1 ϑ 2 θ 2 ϑ 3 θ 3 ϑ 4 θ For eac, we adopted te Rao-Hartley-Cocran RHC sampling sceme to obtain a sample of size n = 200 witout replacement from U using z i s as te weigts see, e.g., Sampford Divide U into n subgroups U 1,..., U n, were eac group consists of k = N /n units. 2. Select unit j from eac subgroup U q using {z i : i U q } as weigts. Under tis sampling sceme, wj 1 = z j / q z i. 3. Samples across strata are independently sampled. Table 1 reports te value of α, β and, based on 2000 simulations, te biases and standard deviations SD of six estimators, two estimated SD, ŜD S obtained by substituting asymptotic variance formulas derived in Section 3 and ŜD B based on te bootstrap variance estimator described in Bickel and Freedman 1984, Sao and Tu 1995, and Antal and Tillé 2011 wit B = 300, togeter wit te coverage probability, CP u, of te approximate 95% confidence interval for te population total Y : [estimated total 1.96ŜD u, estimated total+1.96ŝd u], were u = S or B. Te following observations summarize te results in Table All estimators ave negligible biases. 2. Te simulation results support te asymptotic teory in Teorem 1. Wen β s and α s are different, ŶSS as te best performance. Wen α s and/or β s are te same, oter estimators are comparable wit ŶSS. If ŶSS is not te best, ten te difference between te SD of ŶSS and te SD of te best estimator is negligible. 3. Substitution and bootstrap variance estimators perform reasonably well. Wen te estimator of SD performs well, so does te 95% approximate confidence interval. In a few cases te substitution estimator ŜD S overestimates substantially, wic results in a too large CP. Te bootstrap estimator ŜD B is more stable, but always overestimates in our simulation.

12 406 JUN SHAO AND SHENG WANG Table 1. Simulation Results for Six Estimators Large n s and Fixed H. α β Y Ŷ C0 Ŷ S0 Ŷ CC Ŷ SC Ŷ CS Ŷ SS Bias SD ŜD S ŜD B CP S 93.7% 93.2% 93.4% 93.3% 93.5% 93.1% CP B 94.9% 94.8% 94.8% 94.6% 94.9% 94.6% Bias SD ŜD S ŜD B CP S 97.1% 95.1% 97.0% 94.9% 95.6% 95.1% CP B 96.2% 96.1% 96.4% 96.1% 96.4% 96.1% Bias SD ŜD S ŜD B CP S 95.0% 95.0% 94.8% 95.0% 94.5% 94.8% CP B 95.2% 95.4% 95.6% 96.1% 96.0% 95.7% Bias SD ŜD S ŜD B CP S 97.0% 95.0% 99.9% 94.3% 96.1% 93.9% CP B 96.8% 95.2% 97.2% 94.7% 96.4% 95.1% 4.2. Simulation results for large H and small n s We considered a population wit H = 600 strata. In stratum, we independently generated x i Γϑ, θ, i = 1,..., N = 60, were ϑ and θ were independently generated from te uniform U6, 8. In stratum, y i s and z i s were generated as in Section 4.2. Values from different strata were independently generated. In stratum, a sample of size n = 4 was taken witout replacement witin U, using SRS or RHC as described in Section 4.1. Samples across strata were independently sampled. Te regression parameters were cosen as follows: α, β = a 1, b 1, = 1,..., H/2, and α, β = a 2, b 2, = H/2 + 1,..., H, were values of a k and b k, k = 1, 2, are sown in Table 2. Table 2 reports te same quantities as tose in Table 1 for SRS or RHC, except tat tere is only one estimated SD and ence one CP for eac estimator. For ŶC0, ŶCC, and ŶCS, te estimated SD, ŜD, is based on te bootstrap wit B = 300 and bootstrap sample size n 1 see, for example, McCarty and Snowden 1985, for te reason of using n 1. Tis bootstrap metod, owever,

13 MODEL-ASSISTED REGRESSION ESTIMATORS 407 Table 2. Simulation Results for Six Estimators Large H and Small n s. k a k b k Y Sampling Ŷ C0 Ŷ S0 Ŷ CC Ŷ SC Ŷ CS Ŷ SS SRS Bias SD ŜD CP 95.6% 95.8% RHC Bias SD ŜD CP 95.2% 95.0% 94.8% 95.4% 95.0% 94.4% SRS Bias SD ŜD CP 96.8% 94.8% RHC Bias SD ŜD CP 94.8% 95.6% 95.2% 97.2% 95.4% 94.6% SRS Bias SD ŜD CP 95.2% 96.2% RHC Bias SD ŜD CP 95.4% 94.8% 94.6% 96.8% 94.0% 95.2% SRS Bias SD ŜD CP 96% 95.1% RHC Bias SD ŜD CP 96.8% 95.0% 96.8% 96.3% 96.5% 94.2% does not work for ŶS0, Ŷ SC, and ŶSS, because it is igly possible tat te bootstrap sample in stratum contains only one or two points y i, x i from S wen n is small, wic prevents us from regression fitting. Tus, we consider substitution estimators based on te result in Teorem 2: ˆϕ S0 = 1 N 2 i S ˆσ 2 ζ i+w i N 2 ˆσ 2 N n + 1 N 2 [ ˆα N ] 2,

14 408 JUN SHAO AND SHENG WANG ˆϕ SC = 1 N 2 i S ˆσ 2 ζ i+w i [ N 2 ˆσ 2 N n + ˆα N ] 2, N ˆϕ SS = 1 N 2 ˆσ 2 ψ i + N 2 1 w i 1 + N 2 ˆσ 2 N n, i S were ζ i and ψ i are given in Teorem 2. Under SRS, ŶC0 = ŶCC = ŶCS and ŶS0 = ŶSC = ŶSS. Tus, te results are sown for ŶCS and ŶSS only. Te following is a summary of te results in Table Te biases of all estimators are negligible, altoug tey are relatively large compared wit tose in Table In terms of SD, ŶSS is no longer always te best. Wen regression lines are all te same, te estimators using combined regression are muc more efficient tan tose using separate regression lines, wic is quite different from te situation wit large n s. On te oter and, wen te slopes of regression lines are different, te estimators using separate regression lines are more efficient tan tose using combined regression. Under RHC, among te tree estimators using separate regression lines, ŶSS is no longer te best, ŶSC is. 3. Te substitution variance estimators for ŶS0, ŶSC, and ŶSS and te bootstrap variance estimator for ŶC0, ŶCC, and ŶCS perform well and result in good CP of confidence intervals. 5. Discussion We study te asymptotic efficiencies of estimators in two different asymptotic settings under stratified sampling wit H strata and model 2.2. In te case were H is fixed and all stratum sample sizes tend to infinity, te estimator ŶSS in 1.6 is asymptotically te most efficient estimator, regardless of weter regression models in different strata are te same or not. Wen all stratum sample sizes are small and H tends to infinity, owever, no general conclusion can be made. Wen H is fixed and all stratum sample sizes are large, it is not difficult to derive design-based consistent variance estimators for all six estimators in In fact, it can be sown tat te bootstrap variance estimators used in Section 4.1 are design-based consistent. For te case were all stratum sample sizes are small and H is large, it can still be sown tat te bootstrap variance estimators for ŶC0, Ŷ CC and ŶCS described in Section 4.2 are designbased consistent.

15 MODEL-ASSISTED REGRESSION ESTIMATORS 409 Tere are estimators of Y oter tan tose in For example, an anonymous referee commented tat we could consider regression adjustments instead of sample size adjustments, wic lead to an estimator of te form H [N Ŷ ˆβ X ˆX ˆα N ] =1 wit some estimators ˆα and ˆβ. Asymptotic results can be similarly derived. Acknowledgements We would like to tank tree referees and an associate editor for providing elpful comments and suggestions. Te researc was partially supported by an NSF grant. Appendix Proof of Teorem 1. Let P, E, and V be te te probability, expectation and variance under bot model 2.2 and design. By te fact tat y i = α +β x i +, i S and te definition of ŶSS, we ave Ŷ SS Y N = First, notice tat [ β ˆβ ˆX X + w i N 1 i S N [ ˆX E X ] [ ˆX = E s E m X ] [ = 0 and E w i N N 1 i S By C2 and Liapounov s Central Limit Teorem, ] N N. A.1 ] = 0. N were 1 ψ1 [ ˆX X N ] d N0, 1, ˆX ψ 1 = V X N ˆX = V s E m X ˆX + E s V m X N N ˆX = E s V m X N [ = σ 2 x E s i S wi 1 N N /S 2 ]

16 410 JUN SHAO AND SHENG WANG = σ 2 x [E s w2 i i S 2 1 N ], and te tird equality follows since E m ˆX / X /N = 0. By C3, we ave i S wi 2/ 2 < 1/n M. It is easy to sow tat N / p 1. By te Dominated Convergence Teorem and Liapounov s Central Limit Teorem, we ave 1 [ ˆX X ] d N0, 1, A.2 ϕ1 N were ϕ 1 = σ2 ] x [E N 2 s wi 2 N i S and σ 2 x = E[x i], i U. Similarly, we ave 1 [ ϕ2 1 i S w i = σ2 x N 2 w i N, N ] d N0, 1, were ϕ 2 = σ 2 w i N /N 2. Terefore, by A.1, Ŷ SS Y N ϕ SS = ϕ2 [ ϕ 1 β ˆβ ˆX X + 1 w i ϕss ϕ 2 ϕ1 N ϕ2 i S were ϕ SS = ϕ 2N 2 /N 2 = σ x /σ, A.2, and ˆβ β p 0, we obtain Ŷ SS Y N ϕ SS = ϕ2 N ] N N, [ σ2 w i N ] /N 2. Using ϕ 1 /ϕ 2 = [ 1 ϕss ϕ2 1 Terefore, by Slutsky s Teorem, i S w i Ŷ SS Y N ϕ SS d N0, 1. From te conditions, tere exists b suc tat N i S w i x i ˆX / 2 i S w i x i ˆX / 2 p b. ] N N + o p1. According to te definition of ˆβ, we ave ˆβ p β, were β = b β. Similarly, we can obtain Ŷ CS Y N ϕ CS d N0, 1,

17 MODEL-ASSISTED REGRESSION ESTIMATORS 411 were ϕ CS = τ N 2 /N 2 and τ = β β 2 ϕ 1 + ϕ 2. Terefore, ϕ CS = 1 N 2 σx 2 β β 2 w i N + ϕ SS. For ŶSC, we ave Ŷ SC Y N = α N + N ˆX β ˆβ X + N w i ϵ i. N i S Similar to te previous proof, by te fact tat ˆβ p β, te asymptotic distribution of ŶSC Y /N is as same as T, were T = α N + N 1 i S w i 1 N. Terefore, under C1 C3, T E[T ] ϕsc d N0, 1, were E[T ] = E s [ α / N /N] and ϕ SC = ϕ SS +V s [ α / N /N]. Under C3, N /M < < N M, N 2/Mn < i S wi 2 < MN2 /n. Terefore N/M < < NM and N 2/n M < i S wi 2 < M N 2/n. Ten N α 1 < α M N A.3 Since n /N 0 ten, for any fixed ϵ 0 > 0, wen n is large enoug, σ 2 N n M 1 > ϵ 0, Ten E[T ] ϕsc ϕ SC ϕ SS > 1 N N 2 σ 2 2 n M N > 0. [ E s α N / N ] 1 1 [ E s α N ] 1 0, ϵ 0 N σ2 N /n M 1

18 412 JUN SHAO AND SHENG WANG were te last inequality olds from /N p 1, A.3, and te Dominated Convergence Teorem. Terefore, T/ d ϕ SC N0, 1, wic implies Ŷ SC Y N ϕ SC d N0, 1. Similarly, we can get te asymptotic properties of ŶS0, ŶCC, and ŶC0. Proof of Teorem 2. By te Law of Large Numbers, we ave ˆβ β. Ten, similar to te proof of Teorem 1, we ave Ŷ CS Y = [ β N ˆβ ˆX X + w i ] N N 1 N N. A.4 i S By D2, it is easy to sow tat ŶCS Y /N as te same distribution as T 2, were T 2 = [ ˆX β β X + w i ] N N 1 N N. i S For T 2, E[T 2 ] = E s E m [T 2 ] = 0. Terefore, by Liapounov s Central Limit Teorem, we ave T 2 ϕcs d N0, 1, were [ 1 N ϕ CS = E s N 2 β β 2 σx ] σ2 w 2 i 2 N. i S For ŶCC, by Liapounov s Central Limit Teorem, we ave ŶCC Y /N E s [α + β βµ x / ] N /N d ϕ N0, 1, CC were [ ϕ CC = V s α + β βµ x N ] + N [ ] σ 2 + β β 2 σx 2 i S E wi 2 s 2 i S w i + N 2 N N 2. It can be sown tat n /N 1 = O p 1 and N 2/ nn 2 = o1. Based on tese facts, and similar to te proof of Teorem 1, we get ŶCC Y /N ] E s [α + β βµ x N N d N0, 1. ϕcc

19 MODEL-ASSISTED REGRESSION ESTIMATORS 413 Similar to te proof of Teorem 1, by D1 and D3, ] E s [α + β βµ x N N 0. ϕcc Terefore, For ŶSS, we ave were Ŷ SS Y N Terefore, were = = ϕ SS = Ŷ CC Y N ϕ CC d N0, 1. [ β ˆβ ˆX X [ i S ψ i = N ψ i + w i 1 N + w i 1 i S /S x i x w i ˆX i S x i x 2 X. w i N Ŷ SS Y N ϕ SS d N0, 1, N 1 N ] N N, N 2 [ N 2 σ2 E se m σ 2 ψ i + w i 1 2 N n ] + i S N N 2. ] N N, Similarly, we can get te asymptotic properties of ŶC0, ŶSC, and ŶS0. References Antal, E. and Tillé, Y A direct bootstrap metod for complex sampling designs from a finite population. J. Amer. Statist. Assoc. 106, Bickel, P. J. and Freedman, D. A Asymptotic normality and te bootstrap in stratified sampling. Ann. Statist. 12, Cocran, W. G Sampling Tecniques. Tird edition. Wiley, New York. Fuller, W. A Sampling Statistics. Wiley, Hoboken, New Jersey. Isaki, C. T. and Fuller, W. A Survey design under te regression superpopulation model. J. Amer. Statist. Assoc. 77, Krewski, D. and Rao, J. N. K Inference from stratified samples: properties of te linearization, jackknife and balanced repeated replication metods. Ann. Statist. 9,

20 414 JUN SHAO AND SHENG WANG McCarty, P. J. and Snowden, C. B Te bootstrap and finite population sampling. Vital and Healt Statistics, 2-95, Public Healt Service Publication U.S. Government Printing Office, Wasington, D.C. Montanari, G. E Post-sampling efficient Q-R prediction in large-sample surveys. Internat. Statist. Rev. 55, Sampford, M. R On sampling witout replacement wit unequal probability of selections. Bimetrika 54, Särndal, C. E., Swensson, B. and Wretman, J Model Assisted Survey Sampling. Springer- Verlag, New York. Sao, J. and Tu, D Te Jackknife and Bootstrap. Springer-Verlag, New York. Wrigt, R. L Finite population sampling wit multivariate auxiliary information. J. Amer. Statist. Assoc. 78, Scool of Finance and Statistics, East Cina Normal University, Sangai , Cina. Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, U.S.A. Matematica Policy Researc, Princeton, NJ 08540, U.S.A. Received February 2012; accepted December 2012