Question

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?

Answer 1

Answer:-

Given That:-

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)).)

Likelihood function

  

  

Now,

  

  

  

  

  

Hence 1/z is u.e

Hence (1) is biased while (2) is unbiased.

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

similarly,

Minimum sufficient statistics

= 0

E(T) = 0

P(T = 0) = 1

E(T) = -(E(z))

UMVUE of unbiased estimator of

Thank you for your supporting.Please upvote my answer...