Relative efficiency

**lllll** · December 6th 2008, 10:34 PM

Suppose that $Y_1, \ Y_2, \ \dotso,\ Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$ . Given two unbiased estimators, find their relative efficiency.

$\sigma^2_1 = S^2 = \frac{1}{n-1} \sum^n_{i=1} (Y_i-\overline{Y})^2$ and $\sigma^2_2 =\frac{1}{2}(Y_1-Y_2)^2$

by definition the relative efficiency is $\frac{V[Y_2]}{V[Y_1]}$ so in this case I should have:

$\frac{\sigma^2_2}{\sigma^2_1} = \frac{\frac{1}{2}(Y_1-Y_2)^2}{\frac{1}{n-1} \sum^n_{i=1} (Y_i-\overline{Y})^2}$

I was thinking of expanding it out but all I got was $\frac{\sigma^2_2}{\sigma^2_1}$ .

**mr fantastic** · December 7th 2008, 12:10 AM

Originally Posted by lllll

Suppose that $Y_1, \ Y_2, \ \dotso,\ Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$ . Given two unbiased estimators, find their relative efficiency.

$\sigma^2_1 = S^2 = \frac{1}{n-1} \sum^n_{i=1} (Y_i-\overline{Y})^2$ and $\sigma^2_2 =\frac{1}{2}(Y_1-Y_2)^2$

by definition the relative efficiency is $\frac{V[Y_2]}{V[Y_1]}$ so in this case I should have:

$\frac{\sigma^2_2}{\sigma^2_1} = \frac{\frac{1}{2}(Y_1-Y_2)^2}{\frac{1}{n-1} \sum^n_{i=1} (Y_i-\overline{Y})^2}$

I was thinking of expanding it out but all I got was $\frac{\sigma^2_2}{\sigma^2_1}$ .

$Var(\hat{\sigma_1}^2)$ :

It's well known that $\frac{(n-1)S^2}{\sigma^2}$ has a $\chi^2$ distribution with $n - 1$ degrees of freedom. It's also well known that the variance of a $\chi^2$ distribution with $\nu$ degrees of freedom is $2\nu$ . Therefore:

$Var\left(\frac{(n-1)S^2}{\sigma^2}\right) = 2(n-1)$

$\Rightarrow \frac{(n-1)^2}{\sigma^4} Var(S^2) = 2(n-1)$

$\Rightarrow Var(S^2) = \frac{2 \sigma^4}{n-1}$ .

An approach similar to the one used below can also be used but it's more tedious.

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$Var(\hat{\sigma_2}^2)$ :

$Var\left( \frac{1}{2} (Y_1 - Y_2)^2 \right) = \frac{1}{4} Var((Y_1 - Y_2)^2) = \frac{1}{4} \left( E((Y_1 - Y_2)^4) - [E((Y_1 - Y_2)^2)]^2\right)$ .

$E((Y_1 - Y_2)^2) = E(Y_1^2 - 2Y_1 Y_2 + Y_2^2) = E(Y_1^2) - 2 E(Y_1) \cdot E(Y_2) + E(Y_2^2)$ $= (\sigma^2 + \mu^2) - 2 \mu^2 + (\sigma^2 + \mu^2) = 2 \sigma^2$ .

$E((Y_1 - Y_2)^4) = E(Y_1^4 - 4Y_1^3 Y_2 + 6 Y_1^2 Y_2^2 - 4 Y_1 Y_2^3 + Y_2^4)$ $= E(Y_1^4) - 4E(Y_1^3) E(Y_2) + 6 E(Y_1^2) E(Y_2^2) - 4 E(Y_1) E(Y_2^3) + E(Y_2^4)$

I'll cheat and get the moments $E(Y^4)$ and $E(Y^3)$ from here: http://en.wikipedia.org/wiki/Normal_distribution (the other moments were shown in my reply to your previous unbiased estimator question)

$= (\mu^4 + 6\mu^2 \sigma^2 + 3\sigma^4) - 4(\mu^3 + 3\mu \sigma^2) \mu + 6 (\mu^2 + \sigma^2)^2$ $- 4 \mu (\mu^3 + 3\mu \sigma^2) + (\mu^4 + 6\mu^2 \sigma^2 + 3\sigma^4) = 12 \sigma^4$ .

Therefore $Var\left( \frac{1}{2} (Y_1 - Y_2)^2 \right) = \frac{1}{4} \left(12 \sigma^4 - 4 \sigma^4 \right) = 2 \sigma^4$ .

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Therefore $\frac{Var(\hat{\sigma_2}^2)}{Var(\hat{\sigma_1}^2) } = n - 1$ .

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