Problem 2 This Problem Is Taken From Ex 6 2 2 In The Textbook Consider F X 0 1 0 0 X 0 Zero Elsewhere With 1 (127.61 KiB) Viewed 19 times
Problem 2. This problem is taken from Ex 6.2.2 in the textbook. Consider f(x; 0) = 1/0, 0 < x < 0, zero elsewhere, with 0 > 0. 2 j. Hint: the expectation of a function is a Lebesgue integral of ae • Calculate E a ln f(X;0) the function. Therefore changing the value of the function at one point of 2 (which has zero measure) does not affect the value of the expectation, even if the function is o at that x. • Suppose that X1, ..., Xn are i.i.d. following f(x; 0). Let Yn be the largest value of these observations. Is n+1 Yn an unbiased estimator of ? Hint: use the cdf method for transformation of r.v. n • What is the maximum likelihood estimator of ? • Compare var (mutY.) and the reciprocal of El( 21 (839) *; which is larger? Does this n+1 n ae a ln fX;0 (YnnE[ 1 violate the Rao-Cramer bound? Hint: recall the a few regularity conditions required for the Rao-Cramer bound; are they all satisfied?
• (3 extra points) Consider the alternative expression of the Fisher information we provided in the class based on the KL divergence lims40 28-2D(F(;0 + 5) || F(;0)); what is the value of this limit? Is there any violation of RCB this time concerning the estimator n+lYn? n
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