Question

3. You have two independent random samples: XiXX from a population with mean In and variance σ2 and Y, Y2, , , , , Y,n from a population with mean μ2 and variance σ2. Note that the two populations share a common variance. The two sample variances are Si for the first sample and Si for the second. We know that each of these is an unbiased estimator of the common population variance σ2, we also know that both of these are consistent estimators of σ2 (a) Show that any linear combination aSi (1-a)S? for a number 0 a is an unbiased and consistent estimator of σ2 (b) If both populations are normally distributed, you can find the variances of S and S from the fundamental sampling theorem Do this. Then find the value of a that produces the smallest variance for the linear combination aSi (1-a)Si. This leads to the pooled estimator of the common variance you have seen in class. From part (a), we know that this estimator is unbiased and consistent.

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