Improved Horvitz-Thompson estimator in survey sampling

The Horvitz-Thompson (HT) estimator is widely used in survey sampling. However, the variance of the HT estimator becomes large when the inclusion probabilities are highly heterogeneous. To overcome this shortcoming, in this paper we propose a hard-threshold method for the first-order inclusion probabilities. Specifically, we carefully choose a threshold value, then replace the inclusion probabilities smaller than the threshold by the threshold. Through this shrinkage strategy, we construct a new estimator called the improved Horvitz-Thompson (IHT) estimator to estimate the population total. The IHT estimator increases the estimation accuracy much although it brings a bias which is relatively small. We derive the IHT estimator's mean squared error and its unbiased estimator, and theoretically compare the IHT estimator with the HT estimator. We also apply our idea to construct an improved ratio estimator. We numerically analyze simulated and real data sets to illustrate that the proposed estimators are more efficient and robust than the classical estimators.