Question

Suppose you have N items with values x_i from which we want to sample n items with replacement. Each item has its own probability p_i of being selected across all times we pick items.

What's the estimated variance of the Horvitz-Thompson estimator of T?

Let T = > 1, 1=1

Answer 1

In this sample, we are picking the n items with replacement.

Let be the probability that the i^th item is selected atleast once. So, the probability that it is selected in terms of p_i is:

Let be the probability that both i^th and j^th item are included. Since:

We have:

P(i^th and j^th item included) = P(i^th included) + P(j^th item included) - (1 - P(i^th and j^th item excluded))

Substituting the formulae:

The estimated variance of the Horvitz-Thompson estimator of T is given by: