Question

Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood estimator of 8 is ΣΤ. (c) Compute the Cramér-Rao lower bound for the variance of unbiased estimators of 0 (d) What functions of have unbiased estimators that achieve the relevant Cramér-Rao lower bounds? (e) Show that is a biased estimator of but that its bias decreases to 0 as nc. Hint: What is the distribution of - log X;"]

Answer 1

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