Question

Let Y, Y2, Yz and Y4 be independent, identically distributed random variables from a population with mean u and variance o. Let Y = -(Y, + Y2 + Y3 +Y4) denote the average of these four random variables. i. What are the expected value and variance of 7 in terms of u and o? ii. Now consider a different estimator of u: W = y + y + y +Y4 This an example of weighted average of the Y. Show that W is also an unbiased estimator of u. Find the variance of W. iii. Based on your answers to parts i. and ii., which estimator of u do you prefer, or W?

Answer 1

Here $Y_i$ has mean $\mu$ and variance $\sigma ^2$ .

Now the sample mean is

(Y1 + Y2 + Y3 +Y4)

i) The expected value of $\overline{Y}$ is

(Y1) + E (Y2) + E(Y3) + E(Y4)) E (T) = E () = 11

The variance of $\overline{Y}$ is

\

(Var (Y1) + Var (Y2) + Var (Y3) + Var (Y4)) var (7) - 16 var (7) =ๆ

ii) Consider the estimator is

00- 以 + + 111 + ト

The expected value of is

E(W) = E(VL) + F(X2) + E() + (Y) E (W) = x + + E(W) =

Hence
00- 以 + + 111 + ト
is an unbiased estimator of $\mu$.

The variance is

Var (W) = az Var (Y4) + ez Var (Y2) + dvar (Y3) + zz Var (Y4) Var (W) = 110?

Since the variance of
(Y1 + Y2 + Y3 +Y4)
is lesser $\left (\frac{\sigma ^2}{4} <\frac{11\sigma ^2}{32} \right )$ than that of
00- 以 + + 111 + ト

we prefer the estimator $\overline{Y}$ .