I am trying to prove that $s^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2$ is a consistent estimator of $\sigma^2$ (variance), meaning that as the sample size $n$ approaches $\infty$ , $\text{var}(s^2)$ approaches 0 and it is unbiased.

I understand how to prove that it is unbiased, but I cannot think of a way to prove that $\text{var}(s^2)$ has a denominator of n. Does anyone have any ways to prove this?

The question asks:

A random sample of size n is taken from a normal population with variance $\sigma^2$. Show that the statistic $s^2$ is a consistent estimator of $\sigma^2$

So far I have gotten:
$\text{var}(s^2) = \text{var}(\frac{1}{n-1}\Sigma X^2-n\bar X^2)$
$= \frac{1}{(n-1)^2}(\text{var}(\Sigma X^2) + \text{var}(n\bar X^2))$
$= \frac{n^2}{(n-1)^2}(\text{var}(X^2) + \text{var}(\bar X^2))$

But as I do not know how to find $Var(X^2) $and$ Var(\bar X^2)$, I am stuck here (I have already proved that $S^2$ is an unbiased estimator of $Var(\sigma^2)$)

Source : Edexcel AS and A Level Modular Mathematics S4 (from 2008 syllabus) Examination Style Paper Question 1. This is for my own studies and not school work.

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    $\begingroup$ Please add the [self-study] tag & read its wiki. Also, you can use $\LaTeX$ markup on this site, see math.meta.stackexchange.com/questions/5020/…. $\endgroup$ Nov 14 '20 at 9:51
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    $\begingroup$ The decomposition of the variance is incorrect in several aspects. $\endgroup$
    – Xi'an
    Nov 14 '20 at 10:55
  • $\begingroup$ @Xi'an On the third line of working, I realised I did not put a ^2 on the n on the numerator of the fraction $\endgroup$ Nov 14 '20 at 11:02
  • $\begingroup$ @MrDerpinati, please have a look at my answer, and let me know if it's understandable to you or not. $\endgroup$
    – Kolmogorov
    Nov 14 '20 at 11:47
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    $\begingroup$ @Xi'an My textbook did not cover the variation of random variables that are not independent, so I am guessing that if $X_i$ and $\bar X_n$ are dependent, $Var(X_i +\bar X_n) = Var(X_i) + Var(\bar X_n)$ ? $\endgroup$ Nov 14 '20 at 14:18

It's a very well known result that :

If $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$ , then $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$ Thus, $ \mathbb{E}(Z_n) = n-1 $ and $ \text{var}(Z_n) = 2(n-1)$ .

If you wish to see a proof of the above result, please refer to this link.

Now, since you already know that $s^2$ is an unbiased estimator of $\sigma^2$ , so for any $\varepsilon>0$ , we have :

\begin{align*} &\mathbb{P}(\mid s^2 - \sigma^2 \mid > \varepsilon )\\ &= \mathbb{P}(\mid s^2 - \mathbb{E}(s^2) \mid > \varepsilon )\\ &\leqslant \dfrac{\text{var}(s^2)}{\varepsilon^2}\\ &=\dfrac{1}{(n-1)^2}\cdot \text{var}\left[\sum (X_i - \overline{X})^2)\right]\\ &=\dfrac{\sigma^4}{(n-1)^2}\cdot \text{var}\left[\frac{\sum (X_i - \overline{X})^2}{\sigma^2}\right]\\ &=\dfrac{\sigma^4}{(n-1)^2}\cdot\text{var}(Z_n)\\ &=\dfrac{\sigma^4}{(n-1)^2}\cdot 2(n-1) = \dfrac{2\sigma^4}{n-1} \stackrel{n\to\infty}{\longrightarrow} 0 \end{align*}

Thus, $ \displaystyle\lim_{n\to\infty} \mathbb{P}(\mid s^2 - \sigma^2 \mid > \varepsilon ) = 0$ , i.e. $ s^2 \stackrel{\mathbb{P}}{\longrightarrow} \sigma^2 $ as $n\to\infty$ , which tells us that $s^2$ is a consistent estimator of $\sigma^2$ .

Note : I have used Chebyshev's inequality in the first inequality step used above. Hope my answer serves your purpose. Thank you.

  • $\begingroup$ Since the OP is unable to compute the variance of $Z_n$, it is neither well-know nor straightforward for them. I thus suggest you also provide the derivation of this variance. $\endgroup$
    – Xi'an
    Nov 14 '20 at 12:15
  • $\begingroup$ I feel like I have seen a similar answer somewhere before in my textbook (I couldn't find where!) but the method is very different. As I am doing 11th/12th grade (A Level in the UK) maths, to me, this seems like a university level answer, and thus I do not really understand this. Thank you for your input, but I am sorry to say I do not understand. $\endgroup$ Nov 14 '20 at 14:07
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    $\begingroup$ I guess there isn't any easier explanation to your query other than what I wrote. Also, what @Xi'an is talking about surely needs a proof which isn't very elementary (I've mentioned a link). In fact, the definition of Consistent estimators is based on Convergence in Probability. Do you know what that means ? $\endgroup$
    – Kolmogorov
    Nov 14 '20 at 19:59

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