Ranked set sampling (RSS) is a cost-efficient alternative to simple random sampling (SRS) when units can be ordered cheaply with respect to an easily measured auxiliary variable. Existing work on the Morgenstern-type bivariate generalized uniform (MTBGU) distribution has derived unbiased and BLUE estimators for the scale parameter of the study variable under several RSS schemes, but always under the assumption of perfect ranking. In practice, rankings are almost never error-free: visual judgment, subjective scoring, and noisy proxy measurements all introduce misclassification of ranks. In this paper we investigate the impact of imperfect ranking on the estimation of the scale parameter θ2 of the MTBGU distribution, when the study variable is the concomitant of an ordered auxiliary variable. We introduce a general misranking model via a misclassification matrix for the judged ranks and obtain the distribution, mean, and variance of the concomitants selected under imperfect RSS. Using these results we (i) quantify the bias and mean squared error (MSE) of the standard unbiased RSS estimator derived under perfect ranking and (ii) construct a new minimum-variance unbiased linear estimator of θ2 when the misranking probabilities are known. A special case of a symmetric neighbour misplacement model is discussed in detail. Monte Carlo simulations compare relative efficiencies of SRS, perfect RSS, naive imperfect RSS, and the proposed corrected estimator for a range of parameter values, association levels, and misranking intensities.
Ranked set sampling (RSS) is a cost-efficient alternative to simple random sampling (SRS) when units can be ordered cheaply with respect to an easily measured auxiliary variable. Existing work on the Morgenstern-type bivariate generalized uniform (MTBGU) distribution has derived unbiased and BLUE estimators for the scale parameter of the study variable under several RSS schemes, but always under the assumption of perfect ranking. In practice, rankings are almost never error-free: visual judgment, subjective scoring, and noisy proxy measurements all introduce misclassification of ranks. In this paper we investigate the impact of imperfect ranking on the estimation of the scale parameter θ2 of the MTBGU distribution, when the study variable is the concomitant of an ordered auxiliary variable. We introduce a general misranking model via a misclassification matrix for the judged ranks and obtain the distribution, mean, and variance of the concomitants selected under imperfect RSS. Using these results we (i) quantify the bias and mean squared error (MSE) of the standard unbiased RSS estimator derived under perfect ranking and (ii) construct a new minimum-variance unbiased linear estimator of θ2 when the misranking probabilities are known. A special case of a symmetric neighbour misplacement model is discussed in detail. Monte Carlo simulations compare relative efficiencies of SRS, perfect RSS, naive imperfect RSS, and the proposed corrected estimator for a range of parameter values, association levels, and misranking intensities.
