A comparison of stratified simple random sampling and sampling with probability proportional to size

Transcription

1 A comparison of stratified simple random sampling and sampling with probability proportional to size Edgar Bueno Dan Hedlin Per Gösta Andersson Department of Statistics Stockholm University

2 Introduction Objective: To find an efficient strategy (in terms of variance) for estimating the total of a study variable, y. y is known to be right-skewed. One quantitative auxiliary variable, x, is available. We will work under the model assisted approach.

3 General Regression Estimator ˆt GREG = U ŷ k + s e ks π k with ŷ k = x ˆB k and e ks = y k ŷ k, where ( ˆB = s x k x k a k π k ) s x k y k a k π k. AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k.

4 General Regression Estimator, Case 1 Let x k = 0 for all k U, we have ˆB = ( s x k x k a k π k ) s x k y k a k π k = 0 Then ŷ k = x k ˆB = 0 and e ks = y k ŷ k = y k 0 = y k. The GREG-estimator becomes ˆt GREG = U ŷ k + s e ks π k = U 0 + s y k π k = ˆt π The HT-estimator can be seen as the case where no auxiliary information is used into the GREG-estimator.

5 General Regression Estimator, Case 2 Let a k = c j and x k = (x 1k, x 2k,, x Jk ) with x jk defined as { 1 if k U j x jk = 0 if not where the U j (j = 1,, J) form a partition of U. The post-stratified estimator is obtained when this type of auxiliary information is used in the GREG-estimator. The residuals become E k = y k ȳ U j (k U j ).

6 General Regression Estimator, Case 3 Let a k = c and x k = (1, z k ), with z k = f (x k ) and f known. The regression estimator is obtained when this x k is used in the GREG-estimator. The residuals become E k = y k + B 2 t z N t y N B 2z k with B 2 = Nt zy t z t y Nt z 2 t 2 z where t y = U y k, t z = U z k, t z 2 = U z2 k and t zy = U z ky k.

7 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; 2 3

8 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; 2 π k E k ; 3

9 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; 2 π k E k ; 3

10 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; 2 π k E k ; 3 π k E k together with π kl = π k π l if k U + and l U ;

11 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; 2 π k E k ; 3 π k E k together with π kl = π k π l if k U + and l U ;

12 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; 2 π k E k ; 3 π k E k together with π kl = π k π l if k U + and l U ; Although not leading to a zero-variance, we can consider 2 π k E k.

13 General Regression Estimator AV p (ˆt GREG ) = 1 2 U U ( Ek kl E ) 2 l π k π l with E k = y k x k B where B = ( U ) x k x k x k y k a k U a k. The following are sufficient conditions for a zero-variance: 1 E k = 0 for all k U; Estimator 2 π k E k ; 3 π k E k together with π kl = π k π l if k U + and l U ; Although not leading to a zero-variance, we can consider 2 π k E k. Design

14 Super-population model The statistician is willing to admit that the following model adequately describes the relation between y and x. The values of y are realizations of the model ξ 0 Y k = δ 0 + δ 1 x δ 2 k + ɛ k E ξ0 (ɛ k ) = 0 V ξ0 (ɛ k ) = δ 3 x 2δ 4 k E ξ0 (ɛ k ɛ l ) = 0 (k l) where moments are taken with respect to the model ξ 0 and δ i are constant parameters. δ 0 + δ 1 x δ 2 k will be called trend and δ 3 x 2δ 4 k will be called spread.

15 Super-population model δ 0 + δ 1 x δ 2 k δ 3 x 2δ 4 k

16 Super-population model δ 0 + δ 1 x δ 2 k δ 3 x 2δ 4 k

17 Strategy πps reg Y k = δ 0 + δ 1 x δ 2 k + ɛ k with V ξ0 (ɛ k ) = δ 3 x 2δ 4 k If ξ 0 holds and δ 2 and δ 4 are known, it is natural to use x k = (1, x δ 2 k ) in the GREG-estimator. And a proxy for E k is Ẽ k = δ 1/2 3 x δ 4 k. This suggest the strategy πps reg with π k = n xδ 4 k tx δ4 sometimes referred as optimal., which is

18 Strategy πps reg

19 Strategy πps reg

20 Research questions Our hypothesis is that, as it strongly relies on the model, the strategy above is not robust. We will compare πps reg with other four strategies. 1 When ξ 0 holds and δ 2 and δ 4 are known, is, in fact, πps reg the best strategy? 2 How does πps reg behave with respect to other strategies in terms of finite population characteristics? 3 When ξ 0 does not hold, is πps reg the best strategy? 4 How does πps reg behave with respect to other strategies in terms of finite population characteristics?

21 Strategy STSI reg We use again x k = (1, x δ 2 k ) in the GREG-estimator. The proxies Ẽ k x δ 4 k are now partitioned, creating H strata. A Simple Random Sample of elements is selected in each stratum. This strategy, STSI reg, is often called model-based stratification. The stratum boundaries are obtained using the cum f rule on x δ 4 k ; The sample is allocated using Neyman allocation, i.e. n h = n N hs x δ 4,Uh j N j S x δ 4,Uj

22 Strategy STSI reg

23 Strategy STSI reg

24 Strategy STSI HT We use x k = 0 in the GREG-estimator (i.e. the HT-estimator). The population is stratified with respect to x δ 2 k and a Simple Random Sample of elements is selected in each stratum. This strategy, STSI HT, uses the auxiliary information only at the design stage. It will be considered as a benchmark. The stratum boundaries are obtained using the cum f rule on x δ 2 k ; The sample is allocated using Neyman allocation, i.e. n h = n N hs x δ 2,Uh j N j S x δ 2,Uj

25 Strategy STSI HT

26 Strategy STSI HT

27 Strategy πps pos Let s assume that ξ 0 holds and δ 2 and δ 4 are known, but still we plan to use the post-stratified estimator. As the estimator must explain the trend, the population is post-stratified with respect to x δ 2 k in the same way as in STSI HT. ( ) A proxy for E k is Ẽk = δ 1/ N j x 2δ t x 2δ 4,U 4 k + j = δ 1/2 Nj 2 3 v k. The design is a πps with π k Ẽ k.

28 Strategy πps pos

29 Strategy πps pos

30 Strategy πps pos

31 Strategy πps pos

32 Strategy STSI pos We decide to use the post-stratified estimator again in the same way as above. The proxies Ẽ k v k are now partitioned, creating H strata. A Simple Random Sample of elements is selected in each stratum: The stratum boundaries are obtained using the cum f rule on v k ; The sample is allocated using Neymal allocation, i.e. n h = n N hs v,uh j N j S v,uj

33 Strategy STSI pos

34 Strategy STSI pos

35 Strategy STSI pos

36 Strategies Estimator Design HT Pos Reg STSI πps 3 5

37 Simulation study under the correct model A finite population of size N was generated as follows. The ( auxiliary ) variable, x, is obtained as N realizations from a Γ 4, 12γ 2 plus one unit, where γ is the skewness. γ 2 The study variable is generated as ( ) Y k = δ 0 + δ 1 x δ 2 k + ɛ k with ɛ k N 0, δ 3 x 2δ 4 k For each strategy, the variance of sampling n elements is computed. The procedure is repeated R = 5000 times. The number of strata/post-strata, H, is the same for every strategy.

38 The simulation study N = 5000 n = 500 γ = 3, 12 H = 5 δ 0 = 0 δ 1 = 1 δ 2 = 3 4, 4 4, 5 4 δ 3 two levels in order to obtain ρ(x, Y ) = 0.65, 0.95 δ 4 = 2 4, 3 4, 4 4

39 Results γ = 3, δ 2 = 1, δ 4 = 0.5, ρ = 0.95

40 Results γ = 3, δ 2 = 1, δ 4 = 0.5, ρ = 0.95

41 Results γ = 3, δ 2 = 1, δ 4 = 0.5, ρ = 0.95

42 Results γ = 3, δ 2 = 1, δ 4 = 0.5, ρ = 0.95

43 Results γ = 3, δ 2 = 1, δ 4 = 0.5, ρ = 0.95

44 Results γ = 3, δ 2 = 1, δ 4 = 0.5, ρ = 0.95

45 Results γ = 3, δ 2 = 0.75, δ 4 = 0.5, ρ = 0.65

46 Results γ = 12, δ 2 = 0.75, δ 4 = 0.75, ρ = 0.95

47 Results γ = 12, δ 2 = 0.75, δ 4 = 1.00, ρ = 0.95

48 Results γ = 12, δ 2 = 1.25, δ 4 = 1.00, ρ = 0.65

49 Research questions Our hypothesis is that, as it strongly relies on the model, the strategy above is not robust. We will compare πps reg with other four strategies. 1 When ξ 0 holds and δ 2 and δ 4 are known, is, in fact, πps reg the best strategy? 2 How does πps reg behave with respect to other strategies in terms of finite population characteristics? 3 When ξ 0 does not hold, is πps reg the best strategy? 4 How does πps reg behave with respect to other strategies in terms of finite population characteristics?

50 Research questions Our hypothesis is that, as it strongly relies on the model, the strategy above is not robust. We will compare πps reg with other four strategies. 1 When ξ 0 holds and δ 2 and δ 4 are known, is, in fact, πps reg the best strategy? Not always! 2 How does πps reg behave with respect to other strategies in terms of finite population characteristics? 3 When ξ 0 does not hold, is πps reg the best strategy? 4 How does πps reg behave with respect to other strategies in terms of finite population characteristics?

51 Research questions Our hypothesis is that, as it strongly relies on the model, the strategy above is not robust. We will compare πps reg with other four strategies. 1 When ξ 0 holds and δ 2 and δ 4 are known, is, in fact, πps reg the best strategy? Not always! 2 How does πps reg behave with respect to other strategies in terms of finite population characteristics? 3 When ξ 0 does not hold, is πps reg the best strategy? Not always! 4 How does πps reg behave with respect to other strategies in terms of finite population characteristics?

52 Bibliography I Brewer, K.R.W. (1963). A Model of Systematic Sampling with Unequal Probabilities. Australian Journal of Statistics, 5, Brewer, K.R.W. (2002). Combined Survey Sampling Inference: Weighing Basu s Elephants. London: Arnold. Cassel, C.M., Särndal, C. E. and Wretman, J. (1977). Foundations of Inference in Survey Sampling. New York: Wiley. Dalenius, T. and Hodges, J.L. (1959) Minimum variance stratification. Journal of the American Statistical Association, 54,

53 Bibliography II Godambe, V.P. (1955). A unified theory of sampling from finite populations. Journal of the Royal Statistical Society, Series B 17, Hanif, M. and Brewer K. R. W. (1980). Sampling with Unequal Probabilities without Replacement: A Review. International Statistical Review 48, Holmberg, A. and Swensson, B. (2001). On Pareto πps Sampling: Reflections on Unequal Probability Sampling Strategies. Theory of Stochastic Processes, 7(23), Isaki, C.T. and Fuller, W.A. (1982) Survey design under the regression superpopulation model. Journal of the American Statistical Association 77,

54 Bibliography III Kozak, M. and Wieczorkowski, R. (2005). πps Sampling versus Stratified Sampling? Comparison of Efficiency in Agricultural Surveys. Statistics in Transition, 7, Lanke, J. (1973). On UMV-estimators in Survey Sampling. Metrika 20, Rosén, B. (1997). On sampling with probability proportional to size. Journal of statistical planning and inference 62, Rosén, B. (2000a). Generalized Regression Estimation and Pareto πps. R&D Report 2000:5. Statistics Sweden.

55 Bibliography IV Rosén, B. (2000b). On inclusion probabilities for order πps sampling. Journal of statistical planning and inference 90, Särndal, C.E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. Springer. Tillé, Y. (2006). Sampling algorithms. Springer. Wright, R.L. (1983). Finite Population Sampling with Multivariate Auxiliary Information. Journal of the American Statistical Association, 78,

56 Thanks for your attention!