# Fisher Information for $\bar{X}^2 - \frac{\sigma^2}{n}$ with $X_1, \dots, X_n$ normally distributed

I need to find the Fisher Information for $$T = \bar{X}^2 - \frac{\sigma^2}{n}$$ with $$X_1, \dots, X_n$$ normally distributed sample with unknow mean $$\mu$$ and know variance $$\sigma^2$$. For this I'm trying to find the pdf of $$T$$ but I cant make it. I'm losted since the normal distribution is not standard and not even have $$\mu = 0$$.

Tried with the CDF technique from this using the fact that the sample sum will be also normally distributed but can't get work.

Any help is appreciated.

• Do you mean Fisher information for $\mu$ based on $T$? Commented Jul 8, 2023 at 18:56
• @StubbornAtom yes. Commented Jul 8, 2023 at 21:54
• Since $T$ is an unbiased estimator of $\mu^2$, it probably makes more sense to talk about Fisher information for $\mu^2$ instead. Commented Jul 9, 2023 at 6:04
• This is an old question but FWIW, I wish to point out that the following can help. stats.stackexchange.com/q/149468/183497 Commented Feb 4 at 18:34

The CDF of $$T \mathrel{:=} \bar X^2 - \frac{\sigma^2}{n}$$ is, by definition, given by $$\mathop{F_T}\left(t\right) =\mathop{\mathbb P}\left(\bar X^2 - \frac{\sigma^2}{n} \leq t\right)$$ for all $$t \in \mathbb R$$.
For $$t > -\sigma^2/n$$ we have \begin{align} \mathop{F_T}\left(t\right) &= \mathop{\mathbb P}\left(|\bar X| \leq \sqrt{t + \frac{\sigma^2}{n}}\right) \\ &= \mathop{\mathbb P}\left(-\sqrt{t + \frac{\sigma^2}{n}} \leq \bar X \leq \sqrt{t + \frac{\sigma^2}{n}}\right) \\ &= \mathop{F_{\bar X}}\left(\sqrt{t + \frac{\sigma^2}{n}}\right) - \mathop{F_{\bar X}}\left(-\sqrt{t + \frac{\sigma^2}{n}}\right); \end{align} and $$\mathop{F_T}\left(t\right) = 0$$ for $$t \leq -\sigma^2/n$$.
Differentiating w.r.t $$t$$ yields the PDF of $$T$$ $$\mathop{f_T}\left(t\right) = \frac{1}{2\sqrt{t + \frac{\sigma^2}{n}}} \left[\mathop{f_{\bar X}}\left(\sqrt{t + \frac{\sigma^2}{n}}\right) + \mathop{f_{\bar X}}\left(-\sqrt{t + \frac{\sigma^2}{n}}\right)\right]$$ for $$t > -\sigma^2/n$$ (and $$\mathop{f_T}\left(t\right) = 0$$ otherwise), where $$\mathop{f_{\bar X}}$$ is the PDF of $$\bar X$$.
Note that $$\bar X \sim \mathop{\mathcal N}\left(\mu, \frac{\sigma^2}{n}\right)$$ under the assumption of i.i.d. normal random variables.
• First of all thank you! And tried to finish but get stuck trying to calculate the fisher information doing $-E (\frac{\partial^2}{\partial\mu^2} ln \circ f_T)$ since have the logaritm of a sum. There is a more convenient way? Or just have to mess with that? Commented Jul 3, 2023 at 19:20