1. On coordination of stratified Pareto ps and simple random samples
Annika Lindblom
Alex Teterukovsky
Statistics Sweden

2. The paper focuses on:
Presentation of the sampling designs in the SAMU: stratified SRS and Pareto ps
Sample co-ordination, in particular the implementation of Pareto ps design
Overlap between ps and SRS samples:
- theoretical findings for SRS
- empirical findings based on surveys in practice

3. The SAMU
A system for co-ordination of frame populations and samples from the Business Register at Statistics Sweden since 1972
Three main objectives:
- obtain comparable statistics
- ensure high precision in estimates of change
over time
- spread the response burden
NR behöver jämförbar statistik samma ram population ”frozen” BR

4. Inclusion probabilities
Frame population divided into H disjoint strata Uh, h = 1,..H, where Uh contains Nh units. A sample of fixed size nh from each Uh is to be drawn. Inclusion probabilities are:
Stratified SRS:
same for all units within a stratum
Alla företag är lika inom ett stratum i OSU
(Pareto) ps:
unique for each unit, xk size measure for unit k

5. Permanent random numbers
To each unit k in the Business Register a permanent random number uk uniformly distributed over the interval (0,1), is attached
For SRS we choose the starting point and the direction, and sample the necessary number of units:
Different blocks

6. Pareto ps sampling procedure
Compute the desired inclusion probabilities within each stratum
If k>1 then unit is sampled with probability 1
For other units calculate the ranking variable:
The sample consists of the units with the nh smallest q-values within stratum h
Lambda nästan lika pi

7. Pareto and starting points
Random number transformation is necessary.
The objective of the transformation is to select the nh units with the smallest q-values within a stratum h independently of what starting point S is chosen.
Transform uk into zk as follows.
Sampling direction right:
Sampling direction left:
Rotating the circle
Same distribution

8. Coordination and overlap
Theoretical. SRS/SRS.
Empirical. SRS/SRS.
Empirical. Pareto/Pareto.
Empirical. SRS/Pareto. Same surveys.
Empirical. SRS/Pareto. Different surveys.
Empirical. SRS/SRS and Pareto/Pareto over time.

9. Theoretical overlap. SRS/SRS.
Coordinate 2 equal SRS samples (h is stratum):
sample sizes nh
frame population Nh
completely enumerated units mh
What is the expected overlap for all a’s and b’s?
GEOMETRY!

10. Theoretical overlap. SRS/SRS.
Type
Condition
Expected overlap Ia
b-a<Rh<1-(b-a)
nh –(b-a)(Nh-mh)
IIa
1-(b-a)<Rh<b-a
nh + (b-a-1)(Nh-mh)
IIIa
max(b-a, 1-(b-a))<Rh
2nh - Nh
IVa
Rh<min(b-a,1-(b-a)) mh
Rh=

11. Theoretical overlap. SRS/SRS.
Type
Condition
Expected overlap Ib
0.5(b-a)<Rh<b-a
2nh –(b-a)Nh-(1-b+a)mh
IIb
b-a < Rh < 0.5(1+(b-a))
mh+(b-a)(Nh-mh)
IIIb
max(b-a, 0.5(1+(b-a)))<Rh
2nh - Nh
IVb
Rh<0.5(b-a) mh

12. Empirical overlap. SRS/SRS.
Wages and salaries, private sector (stratified after #employees) N=
SRS: completely enumerated
(23% of the sample)
n = 7 497
SAMU block
Actual
Expected
Start
Direction
Units %
0.0
Right
2 258 30
1.0
Left
3 111 41
3 103
0.2
2 613 35
2 616
0.7
2 956 39
2 927
Turnover in retail trade (stratified after the turnover) N=
SRS: 246 completely enumerated
(10 % of the sample)
n = 2 405
SAMU block
Actual
Expected
Start
Direction
Units %
0.0
Right
444 18
1.0
Left
657 27
639
0.2
549 23
521 22
0.7
611 25
606

13. Empirical overlap. Pareto/Pareto.
Wages and salaries, private sector
(stratified after #employees) N=
Pareto: completely enumerated
SRS: completely enumerated
n = 7 497
SAMU block
Pareto
SRS
Start
Direction
Units %
0.0
Right
2 265 30
2 258
1.0
Left
3 005 40
3 111 41
0.2
2 594 35
2 613
0.7
2 814 38
2 956 39
2 299 31
Turnover in retail trade
(stratified after the turnover) N=
Pareto: 287 completely enumerated
SRS: 246 completely enumerated
n = 2 405
SAMU block
Pareto
SRS
Start
Direction
Units %
0.0
Right
466 19
444 18
1.0
Left
705 29
657 27
0.2
566 24
549 23
0.7
613 25
611
461

14. Empirical overlap. SRS/Pareto. Same surveys.
Use of Information and Communication in Enterprises
(stratified after #employees)
N= 29 124
SRS: completely enumerated
Pareto: 1 116 completely enumerated
n = 4 355
SAMU block
Overlap SRS/Pareto
Start
Direction
Units %
0.0
Right
3 593 83
0.2
3 595
0.7
Left
3 605
3 601
1.0
3 559 82
Stock of goods in the Wholesale and Retail Trade
(optimally stratified after the turnover)
N= 9 322
SRS: 382 completely enumerated
Pareto: 402 completely enumerated
n = 1 753
SAMU block
Overlap SRS/Pareto
Start
Direction
Units %
0.0
Right
1 418 81
0.2
1 406 80
0.7
Left
1 397
1 383 79
1.0
1 405

15. Empirical overlap. SRS/Pareto. Different surveys.
Overlap between two different surveys
(Inf and Sto) placed in different blocks
N (Inf) = 29 124
N (Sto) = 9 322
Completely enumerated
SRS(Inf): 1 214
Pareto(Inf) : 1 116
SRS(Sto): 382
Pareto(Sto): 402
n (Inf) = 4 355
n (Sto) = 1 753
SAMU block
Pareto
SRS
Survey
Start
Direction
Units %
Inf
0.0
Right
241 14
200 11
Sto
1.0
Left
359 21
348 20
0.2
263 15
215 12
0.7
275 16
244
235 13
188
Overlap between two different surveys
(Inf and Sto) placed in different blocks
N (Inf) = 29 124
N (Sto) = 9 322
Completely enumerated
SRS(Inf): 1 214
Pareto(Inf) : 1 116
SRS(Sto): 382
Pareto(Sto): 402
n (Inf) = 4 355
n (Sto) = 1 753
SAMU block
Pareto
SRS
Survey
Start
Direction
Units %
Sto
0.0
Right
222 13
177 10
Inf
1.0
Left
250 14
202 12
0.2
290 17
284 16
0.7
262 15
214
240
196 11
Rader, kolumner, tabeller (nämna)

16. Empirical. Overlap over time.
Use of Information and Communication in Enterprises
(stratified after #employees)
SAMU block
Pareto
SRS
Start
Direction
Units %
0.0
Right
3 560 82
3 428 79
0.2
3 564
3 432
0.7
Left
3 584
3 483 80
3 590
3 455
1.0
3 553
3 411 78
Stock of goods in the Wholesale and Retail Trade
(optimally stratified after the turnover)
SAMU block
Pareto
SRS
Start
Direction
Units %
0.0
Right
1 352 77
1 305 74
0.2
1 322 75
1 288
0.7
Left
1 337 76
1 313
1 342
1 327
1.0
1 316
1 310
Fasta gränser i INF, optimala gränser i STO och SRS blir mer lika Pareto (hopp in hopp ut effekten)