In Survey Sampling, we often use information on some auxiliary variable, to improve our estimator of the finite population parameter, by giving our estimator protection against selection of bad sample. One such estimator is the ratio estimator introduced as follows:

In practice, we are often interested to estimate the ratio of the type:

For example, in different socio-economic survey, we may be interested in per-capita expenditure on food-items, infant mortality rate, literacy rate, etc. So, the estimation of R itself is of interest to us and beside that, we can get an improved estimate of Y ̄, as follows:

NOTATIONS:

Population Size

N

Population

U=(U_1,U_2,\dots,U_N)

Study variable

Y=(Y_1,Y_2,\dots,Y_N)

Auxilliary Variable

X=(X_1,X_2,\dots,X_N)

Mean of Y(study variable)

\bar{Y} = \frac{1}{N} \sum Y_\alpha

Mean of X( auxilliary variable)

\bar{X} = \frac{1}{N} \sum X_\alpha

Sample size

n

Sample (drawn by SRSWOR from U)

s=(i_1,i_2,\dots,i_n)

Sample Mean of Y(study variable)

\bar{y} = \frac{1}{n} \sum y_i

Sample Mean of X( auxilliary variable)

\bar{x} = \frac{1}{n} \sum x_i

Since, \bar{Y} = R \bar{X} , where R is unknown but \bar{X} is known, we can take