Estimation of Finite Population Ratio When Other Auxiliary Variables are Available in the Study

The estimation of the population total ty, by using one or more auxiliary variables, and the population ratio θxy = ty/tx, tx is the population total for the variable X, for a finite population are heavily discussed in the literature. In this paper, the idea of estimating the finite population ratio θxy is extended to use the availability of auxiliary variable Z in the study. The availability of such variable can be used to increase the precision of estimating the population ratio θxy. Our idea is supported by the fact that the variable Z may be more correlated with the variable Y than the correlation between the variables X and Y. To our knowledge, this idea is not discussed in the literature, and may be extended to use the availability of p auxiliary variables. The bias, variance and the mean squares error are given for our approach. Simulation from real data set, the empirical relative bias and the empirical relative mean squares error are computed for our approach and for different estimators proposed in the literature for estimating the population ratio θxy. Analytically and the simulation results show that, by suitable choices, our approach gives negligible bias and has less mean squares error. Further, under simple random sampling without replacement, the population variances of the estimators that are used in this paper are computed. Based on the random samples, that are used for estimating the population ratio θxy, the sample variances for the different estimators that are used in our approach are compared with the population variances for each estimators i.e. the empirical mean, the empirical relative bias, and the empirical relative mean squares error for the sample variances are reported. As a result of this simulation study, our approach is more efficient than other estimators proposed in the literature.