A two-phase sampling scheme and πps designs

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1 p. 1/2 A two-phase sampling scheme and πps designs Thomas Laitila 1 and Jens Olofsson 2 thomas.laitila@esi.oru.se and jens.olofsson@oru.se 1 Statistics Sweden and Örebro University, Sweden 2 Örebro University, Sweden

2 p. 2/2 The problem Assume the population total on the variable y is of interest, i.e. t y = U y k k =1, 2,...,N

3 p. 2/2 The problem Assume the population total on the variable y is of interest, i.e. t y = U y k k =1, 2,...,N An estimator of the population total, t y, is given by ˆt y = s w ky k where w k is the design weight. If w k =1/π k, then ˆt y is unbiased, otherwise it is, more or less, biased.

4 p. 3/2 The problem The precision of the survey, i.e the precision of ˆt y could be increased by using a design with unequal inclusion probabilities compared to using a design were π k = π for all k U.

5 p. 3/2 The problem The precision of the survey, i.e the precision of ˆt y could be increased by using a design with unequal inclusion probabilities compared to using a design were π k = π for all k U. A simple family of designs with unequal first-order inclusion probabilities is the family of PO designs.

6 p. 3/2 The problem The precision of the survey, i.e the precision of ˆt y could be increased by using a design with unequal inclusion probabilities compared to using a design were π k = π for all k U. A simple family of designs with unequal first-order inclusion probabilities is the family of PO designs. An optimal choice of π k is π k y k

7 p. 4/2 The problem If a fixed-size design and w k =1/π k, the variance of ˆt y is V(ˆt π )= 1 2 U (π kl π k π l ) ( yk y ) 2 l π k π l

8 p. 4/2 The problem If a fixed-size design and w k =1/π k, the variance of ˆt y is V(ˆt π )= 1 2 U (π kl π k π l ) ( yk y ) 2 l π k π l It is easily seen that the variance is zero if π k y k (2)

9 p. 4/2 The problem If a fixed-size design and w k =1/π k, the variance of ˆt y is V(ˆt π )= 1 2 U (π kl π k π l ) ( yk y ) 2 l π k π l It is easily seen that the variance is zero if π k y k (3) If the design is not a fixed-size design as the PO design with π k y k, the variation of ˆt y would solely be due to the variation in the sample size.

10 p. 5/2 The problem Letting π k y k is impossible in practise.

11 p. 5/2 The problem Letting π k y k is impossible in practise. In some cases, however, there exits some positive variable x that co-varies positively with y and whose value is known for all k U.

12 p. 5/2 The problem Letting π k y k is impossible in practise. In some cases, however, there exits some positive variable x that co-varies positively with y and whose value is known for all k U. Using auxiliary information the πps rule is defined as π k x k

13 p. 5/2 The problem Letting π k y k is impossible in practise. In some cases, however, there exits some positive variable x that co-varies positively with y and whose value is known for all k U. Using auxiliary information the πps rule is defined as π k x k With-out replacement (wor) designs complying with the πps rule are said to be πps designs.

14 p. 6/2 The problem In literature different fixed-size designs that comply more or less with the πps rule have been proposed.

15 p. 6/2 The problem In literature different fixed-size designs that comply more or less with the πps rule have been proposed. Most of the designs are not easy to implement in practise.

16 p. 6/2 The problem In literature different fixed-size designs that comply more or less with the πps rule have been proposed. Most of the designs are not easy to implement in practise. Instead of rejective designs or designs that utilize order sampling we suggest a scheme that is based on a two-phase sampling design.

17 Two-phase sampling design p. 7/2

18 Two-phase sampling design p. 7/2

19 p. 8/2 Proposed scheme Let n be the pre-determined sample size and assume target first-order inclusion probabilities to be proportional to a size variable known for all k U, i.e. λ k x k.

20 p. 8/2 Proposed scheme Let n be the pre-determined sample size and assume target first-order inclusion probabilities to be proportional to a size variable known for all k U, i.e. λ k x k. Proposed scheme:

21 p. 8/2 Proposed scheme Let n be the pre-determined sample size and assume target first-order inclusion probabilities to be proportional to a size variable known for all k U, i.e. λ k x k. Proposed scheme: 1. Draw a sample using a PO design, where π ak x k, with expected sample size m n.

22 p. 8/2 Proposed scheme Let n be the pre-determined sample size and assume target first-order inclusion probabilities to be proportional to a size variable known for all k U, i.e. λ k x k. Proposed scheme: 1. Draw a sample using a PO design, where π ak x k, with expected sample size m n. 2. If the the size of the sampled set, s a is smaller than the pre-determined sample size, i.e. n sa <n, repeat step 1. If not, proceed to the next step.

23 p. 8/2 Proposed scheme Let n be the pre-determined sample size and assume target first-order inclusion probabilities to be proportional to a size variable known for all k U, i.e. λ k x k. Proposed scheme: 1. Draw a sample using a PO design, where π ak x k, with expected sample size m n. 2. If the the size of the sampled set, s a is smaller than the pre-determined sample size, i.e. n sa <n, repeat step 1. If not, proceed to the next step. 3. From the sampled set, s a, draw a sample of size n using a SI design.

24 p. 9/2 2Pπps design The proposed scheme correponds to a sampling design, here called the 2Pπps design, with first- and second order inclusion probabilities given by

25 p. 9/2 2Pπps design The proposed scheme correponds to a sampling design, here called the 2Pπps design, with first- and second order inclusion probabilities given by ( ) n π k = π ak E pa k s a,n sa n n sa

26 p. 9/2 2Pπps design The proposed scheme correponds to a sampling design, here called the 2Pπps design, with first- and second order inclusion probabilities given by ( ) n π k = π ak E pa k s a,n sa n n sa ( ) n(n 1) π kl = π akl E pa n sa (n sa 1) k&l s a,n sa n

27 p. 10/2 Estimation Using the proposed scheme and implied design population parameters could be estimated using standard two-phase theory. An unbiased estimator of t y is given by ˆt π = s y k π = s y k π ak π k sa

28 p. 10/2 Estimation Using the proposed scheme and implied design population parameters could be estimated using standard two-phase theory. An unbiased estimator of t y is given by ˆt π = s y k π = s y k π ak π k sa An alternative is to regard the sample as a true πps and use ˆt 2Pπps = s y k λ k as an estimator of y y. This will result in some bias, since π k λ k.

29 p. 11/2 Simulation The proposed design has been evaluated by means of a Monte Carlo simulation with replicates.

30 p. 11/2 Simulation The proposed design has been evaluated by means of a Monte Carlo simulation with replicates. The MU284 population was used and it was multiplied with 10 to the power of z, where z = {0, 1, 2, 3}, in order to also have larger populations.

31 p. 12/2 Simulation U i N m n U U , 50 U , 50, 500 U , 50, 500, 5000 Table 1: Descriptives on the populations and simulation setup

32 p. 13/2 Simulation For comparision, the Pareto πps design was included with as an estimator of t y. ˆt PAR,λ = s y k/λ k

33 p. 13/2 Simulation For comparision, the Pareto πps design was included with as an estimator of t y. ˆt PAR,λ = s y k/λ k The other estimators are ˆt 2Pπps,λ and ˆt π.

34 p. 14/2 Simulation The relative empirical bias is definied as REB = Ê(θ) θ θ 100 where Ê(θ) is MC estimated.

35 Results ˆt PAR,λ ˆt 2Pπps,λ ˆt 2P,π U n =5 U U U U n =50 U U U U n = 5000 U Table 2: Relative empirical bias for ˆt PAR,λ, ˆt 2Pπps,λ and ˆt π p. 15/2

36 Table 3: Empirical design effect of Pareto πps, 2Pπps and 2P PO,SI p. 16/2 Results ˆt PAR,λ ˆt 2Pπps,λ ˆt 2P,π U n =5 U U U U n =50 U U U U n = 5000 U

37 Table 4: Empirical design effect of Pareto πps, 2Pπps and 2P PO,SI p. 16/2 Results ˆt PAR,λ ˆt 2Pπps,λ ˆt 2P,π U n =5 U U U U n =50 U U U U n = 5000 U

38 Table 5: Empirical design effect of Pareto πps, 2Pπps and 2P PO,SI p. 16/2 Results ˆt PAR,λ ˆt 2Pπps,λ ˆt 2P,π U n =5 U U U U n =50 U U U U n = 5000 U

39 p. 17/2 Variances A variance estimator for the 2Pπps design has not yet been derived.

40 p. 17/2 Variances A variance estimator for the 2Pπps design has not yet been derived. An option is to use the variance estimator derived for the Pareto πps design, i.e. ˆV(ˆt PAR,λ )= n n 1 ( (1 λ k ) s ( yk λ k s y ) ) 2 k(1 λ k )/λ k s (1 λ k)

41 p. 18/2 Results ˆt PAR,λ ˆt 2Pπps,λ ˆt 2P,π U n =5 U U U n = 5000 U Table 6: Relative empirical bias for ˆV(ˆt PAR,λ ), ˆV(ˆt 2Pπps,λ )andˆv(ˆt π )

42 p. 19/2 Conluding remarks A two-phase fixed-size sampling scheme with unequal first-order inclusion probabilities has been proposed to generate a sample where the first-order inclusion probabilities comply with the πps rule.

43 p. 19/2 Conluding remarks A two-phase fixed-size sampling scheme with unequal first-order inclusion probabilities has been proposed to generate a sample where the first-order inclusion probabilities comply with the πps rule. The proposed algorithm facilitates unbiased estimation by using the standard two-phase theory and it is more efficient, in terms of precision than using a SI design coupled with the π-estimator.

44 p. 19/2 Conluding remarks A two-phase fixed-size sampling scheme with unequal first-order inclusion probabilities has been proposed to generate a sample where the first-order inclusion probabilities comply with the πps rule. The proposed algorithm facilitates unbiased estimation by using the standard two-phase theory and it is more efficient, in terms of precision than using a SI design coupled with the π-estimator. Viewing the sample as a true πps sample, using the 2Pπps design, works well with respect to empirical bias and precision.

45 p. 20/2 Concluding remarks When using the 2Pπps design, the empirical bias of the point estimator is on a par with the empirical bias using standard two-phase estimation.

46 p. 20/2 Concluding remarks When using the 2Pπps design, the empirical bias of the point estimator is on a par with the empirical bias using standard two-phase estimation. In terms of precision, the precision is on a par with using the Pareto πps design.

47 p. 20/2 Concluding remarks When using the 2Pπps design, the empirical bias of the point estimator is on a par with the empirical bias using standard two-phase estimation. In terms of precision, the precision is on a par with using the Pareto πps design. The proposed scheme is easier theoretically to grasp than many of the other fixed-size schemes suggested for a πps design and it is also easy to use since the designs used are implementet in most statistical softwares.

48 First-order inclusion probabilities p. 21/2

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