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FINITE POPULATION CORRECTIONS

FOR RANKED SET SAMPLING

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G. P. PATIL, A. K. SINHA AND C. TAILLIE

*Center for Statistical Ecology and Environmental Statistics,
Department of Statistics,*

Pennsylvania State University, University Park, PA 16802-2112, U.S.A.
(Received April 7, 1993; revised March 22, 1995)

**Abstract.**
Ranked set sampling (RSS) for estimating a
population mean *mu* is studied when sampling is without replacement
from a completely general finite population *x*=(*x*_{1}, *x*_{2},
···, *x*_{N})*'*. Explicit expressions are obtained for the variance of
the RSS estimator ^{^}*mu*RSS and for its precision relative to that of
simple random sampling without replacement. The critical term in
these expressions involves a quantity *gamma* =
(*x* - *mu*)*'***Gamma**(*x* - *mu*) where **Gamma** is
an *N* × *N* matrix whose entries are functions of the population
size *N* and the set-size *m*, but where **Gamma** does not depend on
the population values **x**. A computer program is given to
calculate **Gamma** for arbitrary *N* and *m*. When the population
follows a linear (resp., quadratic) trend, then *gamma* is a
polynomial in *N* of degree 2*m* + 2 (resp., 2*m* + 4). The coefficients
of these polynomials are evaluated to yield explicit expressions for
the variance and the relative precision of ^{^}*mu*RSS for these
populations. Unlike the case of sampling from an infinite population,
here the relative precision depends upon the number of replications
of the set size *m*.

*Key words and phrases*:
Linear range, observational
economy, order statistics from finite populations, quadratic range,
relative savings, sampling efficiency, sampling from finite
populations, sampling without replacement.

**Source**
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