Effective sample size aggregation

In settings with independent observations, sample size is one way to quickly characterize the precision of an estimate. But what if your estimate is based on weighted data, where each observation doesn’t necessarily contribute to equally to the estimate? Here, one useful way to gauge the precision of an estimate is the effective sample size or ESS. Suppose that we have (N) independent observations (Y_1,…,Y_N) drawn from a population with standard deviation (\sigma), and that observation (i) receives weight (w_i).