AISM 53, 631-646

© 2001 ISM

## On random sampling without replacement from a finite population

### Subhash C. Kochar^{1} and Ramesh Korwar^{2}

^{1}Indian Statistical Institute, 7 SJS Sansanwal Marg,
New Delhi-110016, India

^{2}Department of Mathematics and Statistics, Lederle
Graduate Research Center, University of Massachusetts, Amherst, MA 01003, U.S.A.

(Received June 28, 1999; revised April 4, 2000)

Abstract.
We consider the three progressively more general
sampling schemes without replacement from a finite population:
simple random sampling without replacement, Midzuno
sampling and successive sampling. We (i) obtain a lower
bound on the expected sample coverage of a successive
sample, (ii) show that the vector of first order inclusion
probabilities divided by the sample size is majorized by
the vector of selection probabilities of a successive
sample, and (iii) partially order the vectors of first order
inclusion probabilities for the three sampling schemes by
majorization. We also show that the probability of an
ordered successive sample enjoys the arrangement increasing
property and for sample size two the expected sample
coverage of a successive sample is Schur convex in its
selection probabilities. We also study the spacings of a
simple random sample from a linearly ordered finite
population and characterize in several ways a simple random
sample.

Key words and phrases:
Simple random sampling without replacement, Midzuno sampling, sample coverage, inclusion
probabilities, arrangement increasing functions, exchangeable random variables, majorization, successive sampling, Schur convexity.

**Source**
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