A comparison of two alternative procedures for the classical scenario of probability proportional to size

There are many alternative estimation procedures for the classical scenario of probability proportional to size given by Horvitz-Thompson. In this paper, two well-known alternative procedures are considered namely the ratio and linear regression estimators for estimating the population total of the variable of interest in the unequal probability sampling designs. Both procedures used auxiliary information from a suitable variable that is known for all units in the population. This study establishes the primary differences between the two alternative methods for estimating the population total and how the data of auxiliary variable is used in the estimation stage. The two estimators are compared theoretically and empirically by calculating the population total estimate, variances and relative efficiency between the estimators. The results show that under simple random sampling design with moderate positive correlation, the small and medium sample sizes lead to linear regression estimator which is more efficient to estimate the population total than the ratio estimator, but for large sample size, the two estimators have no significant difference in the variance estimates. For moderate negative correlation, the linear regression is more efficient than ratio estimator for all sample sizes