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

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

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The estimation of the population total $t_y,$ by using one or more auxiliary Persona in MMO Games: Constructing an identity through complex player/character relationships variables, and the population ratio $ heta_{xy}=t_y/t_x,$ $t_x$ is the population total for the auxiliary variable $X$, for a finite population are heavily discussed in the literature.In this paper, the idea of estimation the finite population ratio $ heta_{xy}$ is extended to use the availability of auxiliary variable $Z$ in the study, such auxiliary variable is not used in the definition of the population ratio.This idea may be supported by the fact that the variable $Z$ is highly correlated with the interest variable $Y$ than the correlation between the variables $X$ and $Y.$ The availability of such auxiliary variable can be used to improve the precision of the estimation of the population ratio.

To our knowledge, this idea is not discussed in the literature.The bias, variance and the mean squares error are given for our approach.Simulation from Mapping Land Cover and Tree Canopy Cover in Zagros Forests of Iran: Application of Sentinel-2, Google Earth, and Field Data real data set, the empirical relative bias and the empirical relative mean squares error are computed for our approach and different estimators proposed in the literature for estimating the population ratio $ heta_{xy}.$ Analytically and the simulation results show that, by suitable choices, our approach gives negligible bias and has less mean squares error.