Need help understanding consistency

**jass10816** · February 21st 2010, 12:37 PM

I am having a hard time understanding the way my textbook explains how to check consistency. I understand the concept of how it refers to the shape of the pdf of an estimator as a function of $n$ and the two examples are clear, but as soon as I try an exercise I am lost.

The first one I am looking at is
Let $Y_1, Y_2,...,Y_n$ by a random sample of size $n$ from a normal pdf having $\mu=0$ . Show that $S_n^2=\frac{1}{n}\sum_{i=1}^nY_i^2$ is a consistent estimator for $\sigma^2=Var(Y)$

I tried this:
$\lim_{n\to\infty}P(\frac{1}{n}\sum_{i=1}^nY_i^2-\sigma^2<\epsilon)=1$

I am not sure how to write this probability though. I tried to write it as an integral, but then I don't know how to solve the integral I got. I am really just guessing at this point, so if I could have any advice that would be great.

**matheagle** · February 21st 2010, 02:06 PM

This is just the Weak Law of Large Numbers

${\sum_{i=1}^nW_i\over n}\buildrel P\over\to E(W)$

So ${\sum_{i=1}^nY^2_i\over n}\buildrel P\over\to E(Y^2)=\sigma^2$ since the MOO is zero

**jass10816** · February 21st 2010, 02:19 PM

sorry, what is MOO and what is $\buildrel P\over\to$ ? Is it a function P?

**matheagle** · February 21st 2010, 02:23 PM

Originally Posted by jass10816

sorry, what is MOO and what is $\buildrel P\over\to$ ? Is it a function P?

MOO is the mean(ie) here
and that symbol means convergence in probability.

**jass10816** · February 22nd 2010, 03:50 PM

So I don't see how your first line implies the second line of your solution. Here's what I did if you don't mind checking my work:

I tried to show that $\frac{Var(S_n^2)}{\epsilon^2}<\delta$ for arbitrary $\epsilon$ and $\delta$ .

Then, $Var(S_n^2)=\frac{1}{n}Var(Y^2)$ and $Var(Y^2)=E(Y^4)-E(Y^2)^2$

Since $\mu=0$ , $E(Y^4)=M_y^{(4)}(0)$

I found the fourth moment about the origin to be $3\sigma^4$

Then $Var(Y^2)=3\sigma^4-\sigma^4=2\sigma^4$

So $Var(S_n^2)=\frac{2\sigma^4}{n}$

So for large enough n, $\frac{2\sigma^4}{n\epsilon^2}<\delta$

which shows that $S_n^2$ is consistent for $\sigma^2$

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