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Ranked Set Sampling: Improving Estimates from a Stratified Simple Random SamplePowerPoint Presentation

Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample

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Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample

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Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample

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Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample

Christopher Sroka, Elizabeth Stasny, and Douglas Wolfe

Department of Statistics

The Ohio State University

- Purpose: Show that RSS can be incorporated into traditional sampling designs
- Compare RSS to traditional sampling designs
- Develop stratified ranked set sampling (SRSS)
- Computer simulation to evaluate relative standard error

- Select m random samples of size m with replacement from the population
- Order the m items within each set using auxiliary variable or visual judgment
- We do this before measuring our variable of interest

- Select one ranked unit from each set and quantify with respect to variable of interest

X[1]1

X[1]2

X[1]3

. . .

X[1]m

X[2]1

X[2]3

X[2]2

X[2]m

X[3]m

X[3]1

X[3]3

X[3]2

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X[m]m

X[m]1

X[m]3

X[m]2

Set m

Set 1

Set 3

Set 2

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X[1]k X[2]k X[3]k . . . X[m]k

- Repeat k times to get a total of mk measurements on our variable of interest

X[1]1 X[2]1 X[3]1 . . . X[m]1

X[1]2 X[2]2 X[3]2 . . . X[m]2

- Our estimator of the population mean for the variable of interest is the average of our mk quantified observations:

- For fixed sample size n = mk,

RSS estimator from before

Stratum weights

- Expect SSRS to be better than RSS, since uses more population info
- Can we improve on SSRS using RSS?
- Stratified ranked set sampling (SRSS):
Use RSS to select units from each stratum

- We estimate the population mean by

- USDA data on corn production in Ohio
- Treat the data set as a population
- Use computer simulation to estimate the precision of each technique
- Sample from data using each method
- Estimate mean accordingly
- Repeat 50,000 times

- Use the variance of the 50,000 mean estimates to approximate the standard error of the estimator

- Performed simulation multiple times, varying
- Sample size
- Number of strata
- Number of sets
- Combination of ranking variable and variable of interest (correlations vary)

- Reported standard error as percent of standard error under simple random sampling

- Number of sets in RSS equals number of strata in SSRS and SRSS
- Only one cycle within strata for SRSS
- For example, for 3 strata and sample size of 30
RSS: 3 sets of 3, repeat for 10 cycles

SSRS: 3 strata, 10 observations per stratum

SRSS: 3 strata, 10 sets of 10, 1 obs. per set

- SRSS is more precise than SSRS for almost all combinations of variables, set sizes, and sample sizes
- Increased precision of SRSS the highest when
- Strong correlation between ranking variable and variable of interest (i.e., accurate rankings)
- Large sample size

- SRSS less precise or not much more precise than SSRS when
- Low correlation
- Large number of strata combined with low sample size

WHITE = SSRS

RED = SRSS

WHITE = SSRS

RED = SRSS

- Can improve precision of survey estimation by using RSS in place of SRS
- SRSS will improve estimation for all variables in a survey, no matter how low the correlation
- SRSS may not require collecting additional information

- Use different variables for stratification and ranking
- Performance under optimal strata allocation
- Do results hold for any sampling design that uses SRS in its final stage?
- Cost considerations