A Probability Sampling Approach for Variance Minimization

A number of techniques for probability sampling without replacement (SWOR) have been suggested, although it is not clear which method is consistently superior in terms of statistical efficiency. Rao and Bayless (1969) empirically studied the stability of estimators of the population total for a variety of methods of unequal probability SWOR when selecting two units per stratum. One of their major conclusions is that when a stable estimator is required, Murthy’s (1957) method is preferred over the methods of Lahiri (1951), Raj (1956), Rao, Hartley and Cochran (1962), Brewer (1963), Fellegi (1963), Hanurav (1967), and probability proportional to size (PPS) sampling with replacement. One the other hand, Jessen (1969) proposed four interesting sampling schemes. One of them, labeled method 4, shows high efficiency in comparisons of variances of estimators relative to those of alternative SWOR selection schemes, including some of the above-mentioned methods. However, Jessen’s method may be difficult to employ in practical problems due to the arbitrariness and complexities of trials to determine the inclusion probabilities that are required for the variance formula for the estimator of the total population. In this paper, we first review Jessen’s method. Second, we suggest two probability sampling schemes using non-linear programming approaches to overcome certain disadvantages in carrying out Jessen’s method. Finally, we illustrate the practicality and statistical efficiency of our methods through application to several examples from the literature.