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Suppose X1 is from a population with mean µ and variance 1 and X2 is from...

Suppose X1 is from a population with mean µ and variance 1 and X2 is from a population with mean µ and variance 4 (X1, X2 are independent). Construct an estimator of ? as ?̂=??1+(1−?)?2. Show that ?̂ is unbiased for ?. Find the most efficient estimator in this class, that is, find the value of a such that the estimator has the smallest variance.

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Answer #1

Here

Now,

Hence, it is unbiased.

Now, the most efficient estimator would have the smallest variance.

Now,

So,

Equating this to 0, we get,

Also, which is greater than 0. So, is minimum at a = 0.8

Required estimator =

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