Finding Fisher information

Atstovas

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Feb 27, 2019
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Let $X$ distribution belongs for the family\(\displaystyle \mathcal{P}\{P_{\theta}, \theta \in \Theta \}\). We need to find Fisher information\(\displaystyle I(\theta)\) according\(\displaystyle n\) simple sample, when \(\displaystyle P_{\theta}\) is\(\displaystyle N(\mu,\sigma^2)\) distribution, \(\displaystyle \theta=(\mu,\sigma^2)^T.\)


I know that \(\displaystyle I(\theta)= -\mathbb{E} l''(\theta). \)
Here\(\displaystyle l(\theta)=log L(\theta).\)

Also I know \(\displaystyle f(x; \mu, \sigma^2)= \frac{1}{\sqrt{2\pi \sigma^2}} \exp \Big(
- \frac{(x-\mu)^2}{2\sigma^2}\Big)\)

and the likelihood function is

\(\displaystyle L(\mu,\sigma^2|x_i)= \prod_{i=1}^n f(x_i;\mu,\sigma^2).\)

So I can find

\(\displaystyle l(\theta)=log L(\theta)=-\frac{n}{2}log(2\pi \sigma^2)-\frac{\sum_{i=1}^n(x_i-\mu)^2}{2\sigma^2}.\)

Is it right? But what is next? I need to find \(\displaystyle l''(\theta)\) but how to do this?
 
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