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Partial Ranked SetSampling Design
By
Abdul Haq
Ph.D. Student,
Department of Mathematics and Statistics,
University of Canterbury, Christchurch, NZ.
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Outline
• Simple random sampling.• Ranked set sampling.• Examples.• Partial ranked set sampling.• Simulation and case study.• Main findings.
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Estimate the mean height of Arabidopsis Thaliana (AT) plants
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AT Population
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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
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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. .
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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.
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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.
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First set of units
After ranking
Second set of units
After ranking
Third set of units
After ranking
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DiagramSample values
Cycle 1 2 3
1
2
Now apply the RSS procedure to these 3 sets of 2 cycles.
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DiagramJudgment ranks
Cycle 1 2 3
1
2
Notes:1. For each measured unit, we need units.2. All measured units are independent.3. 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.
Here is a ranked set sample of size
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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.
.
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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.
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Diagram: Partial ranked set sample with 36 units
PRSS Judgment ranks
Cycle 1 2 3 4 5 6
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PRSS Judgment ranks
Cycle 1 2 3 4 5 6
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Diagram: Partial ranked set sample with 26 units
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PRSS Judgment ranks
Cycle 1 2 3 4 5 6
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Diagram: Partial ranked set sample with 16 units
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Estimation of population mean
The PRSS mean estimator is.
Its variance is.
For symmetric populations
• is an unbiased estimator of .
• .i.e.
.
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Simulation study: Symmetric populations (perfect ranking)
10 20 30 40 50
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34
56
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Normal(0,1)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
12
34
56
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Uniform(0,1)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
12
34
56
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Logistic(0,1)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
12
34
56
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Beta(6,6)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
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10 20 30 40 50
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Exponential(1)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Weibull(0.5,1)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Lognormal(0,1)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Gamma(0.5,2)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
Simulation study: Asymmetric populations (perfect ranking)
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Simulation study: Bivariate Normal Distribution (imperfect ranking)
10 20 30 40 50
12
34
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Bivariate Normal(0,0,1,1,0.99)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
1.0
1.5
2.0
2.5
3.0
Bivariate Normal(0,0,1,1,0.80)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
1.0
1.1
1.2
1.3
1.4
1.5
Bivariate Normal(0,0,1,1,0.50)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
10 20 30 40 50
1.00
1.02
1.04
1.06
Bivariate Normal(0,0,1,1,0.20)
Number of units
Rel
ativ
e E
ffic
ienc
y
RSS k=0PRSS k=1PRSS k=2PRSS k=3
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An application to Conifer trees dataStudy variable : Height of trees (ft). Auxiliary variable : Diameter of trees at chest level (cm).Correlation coefficient 0.908
Relative efficiencies of the estimators of population mean
RSS PRSS PRSS PRSS
1 2 3
4 (ranking on ) 1.92037 1.38698 _______ _______
4 (ranking on ) 1.91247 1.38545 _______ _______
5 (ranking on ) 2.21737 1.51641 1.15711 _______
5 (ranking on ) 2.20384 1.51553 1.15685 _______
6 (ranking on ) 2.52342 1.61253 1.26187 _______
6 (ranking on ) 2.49267 1.60968 1.26067 _______
7 (ranking on ) 2.80967 1.68997 1.32322 1.106897 (ranking on ) 2.77011 1.68898 1.32021 1.10399
See Platt et al. (1988).
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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.
