Large scale surveys very often involve multi-stage sampling design,
where the first-stage units are selected with varying probability sampling
without replacement method and the second and subsequent
stages units are selected with varying or equal probability sampling
schemes. It is well known (vide Chaudhuri and Arnab (1982)) that
for such sampling designs it impossible to find an unbiased estimator
of the variance of the estimator of the population total (or mean) as a
homogeneous quadratic function of the estimators of the totals (means)
of second-stage units without estimating variances of the estimators of
the totals (means) of the second and sub-sequent stages of sampling.
Wolter (1985) has shown that the Jackknife estimators of the population
total based on unequal probability sampling overestimates the
variance. In this paper we have proposed an alternative Jackknife estimator
after reduction of bias from the original Jackknife estimator.
The performances of the proposed Jackknife estimator and the original
estimator are compared through simulation studies using Household Income
and Expenditure Survey (HIES) 2002/03 data collected by CSO,
Botswana. The simulation studies reveal that the proposed estimator
fares better than the original Jackknife estimator in terms of relative
bias and mean-square error.