Suppose that E(θˆ1) = E(θˆ2) = θ, V(θˆ 1) = σ2 1 , and V(θˆ2) = σ2 2 . Consider the estimator θˆ 3 = aθˆ 1 + (1 − a)θˆ 2. a Show that θˆ 3 is an unbiased estimator for θ. b If θˆ1 and θˆ2 are independent, how should the constant a be chosen in order to minimize the variance of θˆ3?

Step-by-step explanation:

Given that:

If we are to consider the estimator

a. Then, for   to be an unbiased estimator ; Then:

b) If are independent

Thus; in order to minimize the variance of ; then constant a can be determined as :

Using differentiation:

⇒

This implies that

So, is minimum when

As such;       if  

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step-by-step explanation: