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

Abstract

The estimation of the population total $t_y,$ by using one or more
auxiliary variables, and the population ratio $\theta_{xy}=t_y/t_x,$
$t_x$ 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
$\theta_{xy}$ 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 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 $\theta_{xy}.$ Analytically and the simulation results show that, by
 suitable choices, our approach gives negligible bias and has less
 mean squares error.