FINITE POPULATION CORRECTIONS FOR RANKED SET SAMPLING

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 μ 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 μ̂_{RSS} and for its precision relative to that of simple random sampling without replacement. The critical term in these expressions involves a quantity γ = (x − μ)′Γ(x − μ) where Γ is an N × N matrix whose entries are functions of the population size N and the set-size m, but where Γ does not depend on the population values x. A computer program is given to calculate Γ for arbitrary N and m. When the population follows a linear (resp., quadratic) trend, then γ is a polynomial in N of degree 2m + 2 (resp., 2m + 4). The coefficients of these polynomials are evaluated to yield explicit expressions for the variance and the relative precision of μ̂_{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.