Ranked Set Sampling

Ranked Set Sampling
A SAS ® Macro by Paul Johnson (JohnsonP12@ )
A comparison of values were obtained from the Ranked Set Sampling (RSS) (see McIntyre, 1952), Simple Random Sampling (SRS) methods and the population value for the mean of a lognormal variate.
PROCEDURE
Ranked Set Sampling (see Kim and Arnold, 1999) is a useful technique and method when items can be easily ranked and for which exact measurement is costly. n (in this case = 10)random samples of size n are drawn from a lognormal distribution. The lognormal variate was generated by using the RANNOR function of SAS (see SAS/BASE, 1985). This function generates an observation of a normal random variable with mean zero and variance one for any numeric seed value. The Box-Muller transformation of RANUNI uniform deviates is used for generation. If ∗= +exp X (σµ RANNOR(seed))then X is a lognormal variate with mean ()2/exp 2σµ+ and variance ()()1exp 2exp 22−∗+σσµ.For this particular lognormal variate from which a random sample of size 10 was selected,10 times µ was set equal to -1.5 and σ2 was set equal to 1.25. Thus X is a lognormal variate with mean 0.416862 and variance 0.432757.
For a (n *n =10*10) set of data points the RSS consists of sampling the 1st order statistic from the first sample, the second order statistic from the 2nd sample, ....., and the 10th order statistic from the 10th sample. The procedure is repeated by performing a Monte-Carlo (r =500) times to yield a total of 5,000 measured units from the 50,000 which are ranked (see Manly, 1997).
A simple random sample is also obtained to measure the accuracy of the RSS compared to the SRS.
The RSS estimate for the mean and variance of the lognormal variate was calculated using two methods:
a) mean(1) = exp(meany+vary/2) where y = log(X) and
var(1) = exp(2*meany+vary)*(exp(vary)-1).
b) mean(2) = exp(mediany+vary/2) where y = log(X) and
var(2) = exp(2*mediany+vary)*(exp(vary)-1).
RESULTS
Mean
Population
Variance
Population
REFERENCES
Kim, YH., and Arnold, B. C. (1999). "Estimation of a Distribution Function Under
Generalized Ranked Set Sampling, " Communications in Statistics: Simulation and
Computation, 28(3): 657-666.
Manly B.F. (1997). Randomization, Bootstrap and Monte Carlo Methods in Biology.
Chapman and Hall, London.
McIntyre, G.A. (1952). "A Method of Unbiased Selective Sampling, using Ranked Sets," Australian Journal of Agricultural Research 3, 385-390.
SAS Institute Inc. (1985), SAS User’s Guide: Basics, Version 5, Cary, NC:
SAS Institute, Inc.
ACKNOWLEDGMENTS
SAS is a registered trademark of SAS Institute Inc. in the USA and other countries.
® indicates USA registration.。