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2. Proof of the Cramér-Rao Minimum Variance Bound 30 points Consider a likelihood L = where 0 is an unknown parameter and x = {x1, xn} is a set of data that we use to construct a maximum likelihood estimator 0. The Cramér-Rao inequality gives us a lower bound on the reliability of the estimator: -1 var () C , where b = (0) - 0 is the bias of the estimator. (a) (5 points) Prove the Cauchy-Schwarz inequality, which states that for any two random variables u and v, var (u) var (v) > (cov (11,0))? Use the fact that var (au + v) > 0 for any value of a, and then consider the special case a = var (v) / var (u). (b) (5 points) Use the Cauchy-Schwarz inequality with u = 0, U = In ae L Write the inequality so as to express a lower bound on var (0). (c) (10 points) Show that aln L ao ) = 0. Note that we are not evaluating this at 0. (Hint: take advantage of your ability to inter- change the order of differentiation and integration.) Combine this with the results from part (a) and (b) to show that 2 1 var () 2 a In ao L

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