ON RANDOM SAMPLING WITHOUT REPLACEMENT FROM A FINITE POPULATION

SUBHASH C. KOCHAR^{1} AND RAMESH KORWAR^{2}

1Indian Statistical Institute, 7 SJS Sansanwal Marg, New Delhi-110016, India 2Department 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.