Question

Suppose that E[X]= E[Y] = mu, where mu is a fixed unknown number. We have independent simple random samples of size n each from the distribution of X and Y, respectively. Suppose that Var[X] = 2*Var[Y]. Consider the following estimators of mu:

m1 = bar{X}

m2 = bar{Y}/2

m3 = 3*bar{X}/4 + 2*bar{Y}/8

where bar{X} and bar{Y} are the sample mean of X and Y values, respectively. Which of the estimators are unbiased?

Answer 1

Given that

1)

So,

So, the first estimator is unbiased.

2)

So,

So, the second estimator is not unbiased.

3)

So,

So, the third estimator is unbiased.

So, the answer is m1 and m3 are unbiased estimators.