DEPENDENCE BETWEEN ORDER STATISTICS IN SAMPLES FROM FINITE POPULATION AND ITS APPLICATION

TO RANKED SET SAMPLING

KOITI TAKAHASI^{1} AND MASAO FUTATSUYA^{2}

1Department of Basic Science, School of Science and Engineering, Ishinomaki Senshu University, Ishinomaki 986-80, Japan

2Department of Mathematical System Science, Faculty of Science and Technology, Hirosaki University, Hirosaki 036, Japan

(Received June 4, 1996; revised February 10, 1997)

Abstract. Let X_{1}, X_{2}, …, X_{m}, Y_{1}, Y_{2}, …, Y_{n} 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 X_{1}, X_{2}, …, X_{m} and Y_{1}, Y_{2}, …, Y_{n}, 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.

Key words and phrases: Ranked set sampling, finite population, order statistics, dependence.