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I've got this question that I'm really struggling with.
we have $X \sim N(\mu_X,\sigma^2)$, $Y \sim N(\mu_Y,\sigma^2)$
$S_X^2=$ $\sum_{i=1}^n (X_i-\overline X) \over (n-1) $, and $S_Y^2=$ $\sum_{i=1}^m (Y_i-\overline Y) \over (m-1) $

The pooled variance is $S_p^2=$ $(n-1)S_X^2+(m-1)S_Y^2\over n+m-2$
we also know that $S_X^2, S_Y^2, S_p^2$ are all unbiased estimators of $\sigma^2$.

We want to show that the pooled standard deviation $S_p = \sqrt {S_p^2}$ is a biased estimator of $\sigma$.

We're asked to start with the distribution of $S_p^2$, then show the bias of $S_p$

I started with $S_p^2=$ $(n-1)S_X^2+(m-1)S_Y^2 \over n+m-2$
$S_p=$ $\sqrt{(n-1)S_X^2+(m-1)S_Y^2 \over n+m-1}$
$\mathrm{E}(S_p)=$ $1 \over \sqrt{n+m-2}$ $\mathrm{E}(\sqrt{(n-1)S_X^2+(m-1)S_Y^2})$

I tried introducing $\sigma^2 \over \sigma^2$ into the equation, in order to get to $\chi^2$ and get the expected values from there, but I'm not able to get rid of the square root to get the expected values as $\chi^2$ values.

what's my mistake?
really appreciate your help.

user158565
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suspect zero
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  • Given $T$ is unbiased estimate of $\theta$, $g(T)$ is biased estimate of $g(\theta)$, given that $g$ is not linear function. – user158565 Jun 11 '17 at 15:50
  • @a_statistician that correct and fair, but how do we prove it here? – suspect zero Jun 11 '17 at 16:02
  • You are asked to start from distribution of $S_p^2$. But you come to $S_p$ before you get the distribution of $S_p^2$. I guess the teacher want you to show that $S_p^2$ is unbiased estimate of variance based on its distribution. Then $S_p$ is biased estimate of $\sigma$ because sqrt is not linear function. – user158565 Jun 11 '17 at 16:05
  • Going further to what @a_statistician stated, you can apply Jensen's to $g(\cdot)$ since $\sqrt{\cdot}$ is strictly concave. – Josh Jun 11 '17 at 16:08
  • @a_statistician, proving that $S_p^2$ is unbiased to estimate $\sigma^2$ is pretty easy, and we have done it in many other exercises. but tbh, I am not sure what its distribution is... in the process of proving $E(S_p^2)=\sigma^2$, we ended up using $\chi^2$ distribution. – suspect zero Jun 11 '17 at 16:14
  • @Josh, we haven't covered Jensen's (inequality?) yet... I'm not sure how to apply it here – suspect zero Jun 11 '17 at 16:15
  • Okay (yeah, I meant Jensen's inequality). Look at $S_p^2$. If you take out the denominator, and divide both terms by $\sigma^2$ what is the distribution (Hint: It is $\chi^2$, but you need the df as well)? Can you use find the distribution of the sqrt of a $\chi^2$ (use the Jacobian). Can you find the expected value of that distribution? – Josh Jun 11 '17 at 16:20
  • @Josh that will be $\chi^2_{(n-1)}$ + $\chi^2_{(m-1)}$, which I'm assuming that is a $\chi^2_{(n+m-2)}$ distribution? and no, we haven't covered the jacobian either; hence, being stuck here for hours. – suspect zero Jun 11 '17 at 17:51
  • That is right. I find it weird that they did not at least discuss the mean of a $\sqrt{\chi^2}$ random variable. Are you sure there is no notes on that? Did you prof ever show that $s$ was biased for just the old fashioned standard deviation? I am not certain what methods you would be expected to use here. – Josh Jun 11 '17 at 19:36
  • @suspectzero Are you allowed to use calculus? – Josh Jun 11 '17 at 23:13
  • Duplicate : https://stats.stackexchange.com/questions/284208/what-is-the-expectation-of-pooled-standard-deviation ... you guys should figure out *one* person per class to post when you post your homework - multiple posts of the exact same question in a short period of time is a dead giveaway. Even though the other is posted first I may close the other since this one shows at least *some* beginning of an attempt. (Edit: I have closed the other one as a dupe of this) – Glen_b Jun 12 '17 at 02:43
  • @Josh, thanks for your help, yes I am allowed to use calculus and found a similar question on https://stats.stackexchange.com/questions/11707/why-is-sample-standard-deviation-a-biased-estimator-of-sigma which made sense, as we've covered all the material in solving that... from there on, it worked perfectly in my question... thanks again for your suggestions. – suspect zero Jun 12 '17 at 06:13
  • @Glen_b, teachers across the world might use similar questions, and students being hit with the same wall ':) $~~~~$ thanks anyways – suspect zero Jun 12 '17 at 06:18

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