Question

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.

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 =

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.

Let X1, X2 be two normal random variables each with population
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Consider the following
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= (5/4)X - (1/4)Y
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(a) [6] Find the pdf of Yn, the nth order statistic of the
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(b) [4] Find E[Yn].
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(2) Determine whether the maximum likelihood estimator is
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(3) Find the mean squared error of the maximum likelihood
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(4) Find the Cramer-Rao lower bound for the variances of
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a Show that each is unbiased, and that one is more efficient
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