Efficiency Comparison of Unbiased Estimators
Suppose you have two independent unbiased estimators \(T_1\) and \(T_2\) for a parameter \(\theta\) with variances \(\operatorname{Var}(T_1)=\sigma_1^2\) and \(\operatorname{Var}(T_2)=\sigma_2^2\) respectively. Consider a combined estimator defined as \(T = aT_1 + (1-a)T_2\). Since \(T_1\) and \(T_2\) are unbiased, any linear combination with weights summing to 1 is also unbiased. (a) Write an expression for the variance […]