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DEPENDENCE BETWEEN ORDER STATISTICS IN SAMPLES

FROM FINITE POPULATION AND ITS APPLICATION

TO RANKED SET SAMPLING

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

^{1} *Department of Basic Science, School of Science and Engineering,*

Ishinomaki Senshu University, Ishinomaki 986-80, Japan

^{2} *Department 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 *X*_{(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.

**Source**
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