Coverage of the Horvitz-Thompson Estimator

Sen(1953) and Yates and Grundy(1953) suggested independently the variances of Horvitz-Thompson (HT) estimator. But it have difficulties to compute. Donadio have shown that the variance of HT estimator depends almost entirely on first order inclusion probabilities. For these approximation, Brewer proposed a simple approximation to second order inclusion probability. Tille and Brewer and Donadio used a standard Monte Carlo simulation approach to examine the performance of variance estimators. They used relative biases and CV of variance estimators and concludes that under conditions of high entropy, the calculation of second order inclusion probability is not essential for obtaining nearly unbiased variance estimators . But for small size of samples I found 95% confidence interval failed to cover population total. And showed adjusted confidence interval and simulation results. It was works well for real data.