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I have to compute the absolute value of an estimator defined as $T_5=\frac{1}{2}E[|X_1-X_2|]$ in order to state if it is unbiased for $\sigma$, where $X$ is distributed as a $N(0,\sigma^2)$.

I am stuck in computing the expected value of the absolute value of the difference of two normal random variables. Any hint?

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  • $\begingroup$ $X_1 -X_2$ has a normal distribution with mean $0$ and variance $2\sigma^2$. You might consider the mean of a half-normal distribution (which involves $\sqrt{\pi}$ so the answer to your question is likely to be "yes") $\endgroup$ – Henry Nov 10 '15 at 8:46
  • $\begingroup$ You're right! So maybe I can proceed in order to obtain a half-normal distribution with mean $0$ and variance $\sigma^2$ in order to use the known values of the distribution. $\endgroup$ – PhDing Nov 10 '15 at 8:51
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If $X_1$ and $X_2$ are iid random variables such that $X_1\sim\mathcal N(0,\sigma^2)$ and $X_2\sim\mathcal N(0,\sigma^2)$, then $$ X_1-X_2\sim\mathcal N(0,2\sigma^2). $$ If $X\sim\mathcal N(0,\sigma^2)$, then $Y=|X|$ has the half-normal distribution and $$ \operatorname EY=\frac{\sqrt2\sigma}{\sqrt\pi}. $$ Hence, we have that $$ \frac12\operatorname E|X_1-X_2|=\frac12\frac{\sqrt2\sqrt2\sigma}{\sqrt\pi}=\frac{\sigma}{\sqrt\pi}. $$

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