Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\mathcal{N}(\mu,\sigma^{2})$, where both parameters are unknown.

(a) Prove the normal probability density function satisfies the Cramer-Rao theorem hypothesis.

(b) Prove that $T(\textbf{X}) = \hat{\sigma}^{2}$ reaches the Cramer-Rao bound.

The exercise propose using the following result


Suppose that $X_{1},X_{2},\ldots,X_{n}$ are IID whose joint probability density function is given by $f(\textbf{x}|\theta)$. If $T(\textbf{X})$ represents any unbiased estimator for $\psi(\theta) = \textbf{E}_{\theta}(T(\textbf{X}))$, then $T(\textbf{X})$ reach the Cramer-Rao bound if and only if \begin{align*} \frac{\partial\ln f(\textbf{x}|\theta)}{\partial\theta} = h(\theta)[T(\textbf{X}) - \psi(\theta)] \end{align*}

for some function $h(\theta)$.

I am really having trouble in proceeding with the exercise. Any help is appreciated. Thanks in advance!


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