Finite population corrections for ranked set sampling

Abstract

Ranked set sampling (RSS) for estimating a population mean μ is studied when sampling is without replacement from a completely general finite populationx=(x ₁,x ₂,...,x _N )′. Explicit expressions are obtained for the variance of the RSS estimator$\hat \mu _{RSS}$ and for its precision relative to that of simple random sampling without replacement. The critical term in these expressions involves a quantity γ=(x−γ)′Γ(x−μ) where Γ is anN × N matrix whose entries are functions of the population sizeN and the set-sizem, but where Γ does not depend on the population valuesx. A computer program is given to calculate Γ for arbitraryN andm. When the population follows a linear (resp., quadratic) trend, then γ is a polynomial inN 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$\hat \mu _{RSS}$ 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 sizem.