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2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) = ox-(0-1) where : > 1
2.1 Show that the maximum likelihood estimator of a is ômle = Ei log Xi
2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency?
2.4 Show that the fisher information in the whole sample is: 1(a)=
2.5 What Cramer Rao lower bound for unbiased estimators of a?
2.6 Suppose n is large. We know (by properties of maximum likelihood estimators) that @MLE is approximately normally distri
Q2.7 1 Point 2.7 Consider estimating the unknown quantity: 9(a) = -1,1+. Determine the MLE of gla). What property are you usi
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26. Assume nis large, then by assymptotic property of MLE, LMLE ~ N(X, 1 2 e In(x), Mean a x Valianie ! Inca) řkl lit he 1/2=) els کے log X, Solution We are given XX2, X2)X2,.-. In que independent random variables from the distribur following tron.2 n - 2 a 2 Since h, depends á only through Now a?log L dx² Tex) = xi , then by Neymann. Factorisation theorem, So In (a)= 16

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