On the efficiency of ranked set sampling relative to simple random sampling for estimating the ordinary least squares parameters of the simple linear regression model

When the actual measurement cost of a sampling unit is expensive compared with the acquisition cost of the unit, considerable savings may be possible if only a few quantifications are made, but all acquired units contribute information. The ranked set sampling technique suggested by McIntyre (1952) for circumstances such as this takes advantage of ranking information on all acquired units to provide a smaller stratified sample of independent order statistics. While it has long been known that ranked set sampling is an efficient method for estimating the population mean, the use of ranked set sampling for estimating the parameters of an ordinary least squares regression has only recently been examined in the literature. In this dissertation, it is shown that the dependence of the relative efficiencies of the slope and intercept estimators on the underlying population correlation is affected by whether ranking is performed on the independent (RSSX) or dependent (RSSY) variable. It is also demonstrated that for population correlations close to one in magnitude, the relative efficiencies of the estimated regression parameters for RSSY behave similarly to those for RSSX. Additionally, it is established that estimating the intercept using RSSY is equivalent to estimating the population mean of the dependent variable. Furthermore, it is shown that as the number of cycles or the number of groups in a cycle in the ranked set sample increases, the relative efficiencies of both the slope and the intercept estimators asymptotically approach one for the RSSX slope, RSSX intercept and RSSY slope estimators. Boundaries are obtained for the relative efficiencies of the slope and intercept estimators under RSSX. Finally, the improvement in the relative efficiency of the RSSX and RSSY slope estimators are recast in terms of an equivalent simple random sample size, where it is seen that the gains are necessarily small.