Let X1,X2X7 denote a random sample from a population having mean μ and variance σ. Consider the following estimators of μ:

θ1=X1+X2++X7 / 7
θ2= (2X1-X3+X5) / 2

a. Is either estimator unbiased?
b. Which estimator is best? In what sense is it best? Calculate the relative efficiency of the 2 estimtors.

a) In order to check if an estimator is unbiased we need to check this condition:

And we can find the expected value of each estimator like this:

So then we conclude that is unbiased.

For the second estimator we have this:

And then we conclude that is unbiaed too.

b) For this case first we need to find the variance of each estimator:

And for the second estimator we have this:

And the relative efficiency is given by:

Step-by-step explanation:

For this case we assume that we have a random sample given by: and each

Part a

In order to check if an estimator is unbiased we need to check this condition:

And we can find the expected value of each estimator like this:

So then we conclude that is unbiased.

For the second estimator we have this:

And then we conclude that is unbiaed too.

Part b

For this case first we need to find the variance of each estimator:

And for the second estimator we have this:

And the relative efficiency is given by:

answer: the common difference is -0.10