Bias Using Order Statistics | MAS 1 Fall 2018 Q25

Does using an order statistic from a uniform distribution to estimate the mean create a biased estimator? Depends on the sample size and order statistic!

For additional context/derivation of the provided order statistic equation within the problem itself, I found the second half of this Harvard lecture really useful:

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▬▬▬▬▬▬▬▬▬▬ Music ▬▬▬▬▬▬▬▬▬▬
- Artist Attribution
Music By: "KaizanBlu"
Track Name: "Remember"

▬▬▬▬▬▬▬▬▬▬ The Question ▬▬▬▬▬▬▬▬▬▬
You draw a large number of independent samples, each of size n=4, from a uniform distribution on (0,θ). You want to use the second smallest value in each sample as an estimate for the mean.

The density for the kth order statistic of a sample is given as:

gk(yk)=n!(k−1)!(n−k)![F(yk)]k−1[1−F(yk)]n−kf(yk)

Calculate the expected bias of this estimate.

A. −4θ5

B. −4θ7

C. −θ10

D. −θ30

E. The answer is not given by (A), (B), (C), or (D)