Let ?1,?2,…,??be independent, identically distributed random variables with p.d.f. ?(?) = ??^?−1,0 ≤ ? ≤ 1...

Let ?1,?2,…,??be independent, identically distributed random variables with p.d.f. ?(?) = ??^?−1,0 ≤ ? ≤ 1 .

c. Show that the maximum likelihood estimator for ? is biased, and find a function of the mle that is unbiased. (Hint: Show that the random variable −ln (??) is exponential, the sum of exponentials is Gamma, and the mean of 1/X for a gamma with parameters ? and ? is 1 / (?(? − 1)).

d. Is the estimator you found in part c. a minimum variance unbiased estimator?

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orchestra answered 1 year ago

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