Question

I need a little more help with this question: Give an unbiased estimate of the variance of the estimator using the Horvitz-Thompson estimator. (This may be compared to the value of the simpler alternative estimator.)

The answer to this question is 62.46, but I need to see a proof of the variance of the estimator using the Horvitz-Thompson variance estimator to understand how to calculate it myself. I have included an image of the equation below:

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Note - Below is the previous information(i.e answers) to help with the problem. Please do not provide me with answers or proofs to the questions below. I already have calculated those. I am only interested in seeing a proof of the answer above. I have no idea what a proof of this kind of question looks like and the textbook is unhelpful.

An unequal probability sample of size 3 is selected from a population of size 10 with replacement. The y-values of the selected units are listed along with their draw-by-draw selection probabilities: y1 = 3, p1 = 0.06; y2 = 10, p2 = 0.20; y3 =7, p3 =0.10.

The population total using the Hansen–Hurwitz estimator is 56.667

The variance of the estimator is 44.444

The population total using the Horvitz–Thompson estimator is: Pi 1 = .1694, Pi 2 = .488, Pi 3 = .271, Population = 64.03

Answer 1