Exercise 4. Let T₁ and T₂ be uncorrelated unbiased estimators of an unknown parameter 0, with known variances Var[T₁] = 0} and Var[T₂] = σ². aT₁bT2, a, b = R}, that is the class of estimators of given by a linear Consider T = {Ta,b: Ta,b combination of T₁ and T₂. = (a) Find, if any, all the unbiased estimators for in T. (b) Compute E Ta,b] and Var[Ta,b], for a generic estimator Tɑ‚ € T. (c) Find, if it exists, the estimator T* ET such that - - E[T*] = 0, i.e. T* is unbiased for - T* has the smallest variance in the subset of T composed by unbiased estimators. (d) Generalize the results in (c) to the case of k pairwise uncorrelated unbiased estimators of 0, Tj, with Var[T]=o, for j = 1,..., k. (Hint: guess the optimal weights, according to the expression found in c). It might be useful to express the optimal weights in terms of the precisions of the estimators. If Tj has variance σ, then 7, its precision, is given by the reciprocal of its variance: 7 = 1/0. For a formal check, use induction with starting point the result in c), that is, if T* is the optimal combination of T1 and T2, try to find the optimal combination of T* and a third estimator T3. Alternatively solve the minimization problem for the variance of Σ; a;T;, with respect to (a1, ..., ak) under the unbiasedness constraint.)