Dependence Between Order Statistics in Samples from Finite Population and its Application to Ranked Set Sampling
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Let X1, X2,..., Xm, Y1, Y2,..., Yn be a simple random sample without replacement from a finite population and let X(1) ≤ X(2) ≤...≤ X(m) and Y(1) ≤ Y(2) ≤...≤ Y(n) be the order statistics of X1, X2,..., Xm and Y1, Y2,..., Yn, respectively. It is shown that the joint distribution of X(i) and X(j) is positively likelihood ratio dependent and Y(j) is negatively regression dependent on X(i). Using these results, it is shown that when samples are drawn without replacement from a finite population, the relative precision of the ranked set sampling estimator of the population mean, relative to the simple random sample estimator with the same number of units quantified, is bounded below by 1.
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