Partial Ranked Set Sampling Design

1. Partial Ranked SetSampling Design By Abdul Haq Ph.D. Student, Department of Mathematics and Statistics, University of Canterbury, Christchurch, NZ.

2. Outline • Simple random sampling. • Ranked set sampling. • Examples. • Partial ranked set sampling. • Simulation and case study. • Main findings.

3. Estimate the mean height of Arabidopsis Thaliana (AT) plants

4. AT Population

5. Simple Random Sampling (SRS) 1. Select randomly units from population. 2.Get careful measurements of selected plants. 3. Estimate population mean and variance based on this sample. A simple random sample of size

6. Simple random sampling(Estimation of population mean) A simple random sample of size is drawn with replacement from the population having mean and variance say , then the sample mean is 1. is an unbiased estimator of i.e. . 2. .

7. Ranked set sampling(Estimation of population mean) • Actual measurements are expensive. • Ranking of sampling units can be done visually and cheaper. • It provides more representative sample. • Examples: • Estimating average height of students in NZ university. • Estimating average weight of students in NZ university. • Estimating average milk yield from cows in a farm. • Bilirubin level in jaundiced neonatal babies.

8. Ranked set sampling procedure • Identify units, randomly. • Randomly divide these units into sets, each of size . • Rank units within each set. • Select smallest ranked unit from first set of units, second smallest ranked unit from second set, and so on, select largest ranked unit from last set. This gives a ranked set sample of size . • The above steps can be repeated for larger samples.

9. First set of units After ranking Second set of units After ranking Third set of units After ranking

10. Diagram Now apply the RSS procedure to these 3 sets of 2 cycles.

11. Diagram Here is a ranked set sample of size Notes: For each measured unit, we need units. All measured units are independent. If ranking procedure is uniform for all cycles, then measurements from the same judgment class are i.i.d. but the selected units within each cycle are independent but NOT identically distributed.

12. Some Elementary Results • The population mean can be written as • . • The RSS mean estimator is • . • is an unbiased estimator of and more efficient than i.e. • .

13. Partial Ranked Set Sampling (PRSS) Design • PRSS scheme is a mixture of both SRS and RSS designs. • It involves less number of units compared with RSS. • RSS design becomes a special case of PRSS design. PRSS Procedure Step 1: Define a coefficient such that , where 0 . Step 2: firstly select simple random samples each of size one. Step 3: For remaining units, identify sets each of size . Apply RSS on these sets. Step 4: Above steps can be repeated times for large samples. PRSS represents PRSS design.

14. Diagram: Partial ranked set sample with 36 units

15. Diagram: Partial ranked set sample with 26 units

16. Diagram: Partial ranked set sample with 16 units

17. Estimation of population mean • The PRSS mean estimator is • . • Its variance is • . • For symmetric populations • is an unbiased estimator of . • . • i.e. • .

18. Simulation study: Symmetric populations (perfect ranking)

19. Simulation study: Asymmetric populations (perfect ranking)

20. Simulation study: Bivariate Normal Distribution (imperfect ranking)

21. An application to Conifer trees data Study variable : Height of trees (ft). Auxiliary variable : Diameter of trees at chest level (cm). Correlation coefficient 0.908 See Platt et al. (1988).

22. Main Findings • PRSS requires less number of units, which helps in saving time and cost. • RSS is special case of PRSS design. • Mean estimators under PRSS are better than SRS for perfect and imperfect rankings. • PRSS can be used as an efficient alternative to SRS design.