(Solved): Let X1,,XnidN(,) where the parameters and are both unknown. a) Show that ...

Let $X_1,\dots ,X_n\stackrel{\text{ id }}{\sim }\mathrm{N}(\mu ,\varphi )$ where the parameters $\mu$ and $\varphi$ are both unknown. a) Show that $\left(\sum X_i,\sum X_i^2\right)$ is the sufficient statistic for $(\mu ,\varphi )$. b) Show that the observed information matrix is $I(\theta )=\left[\begin{array}{cc}\frac{a}{6}& \frac{1}{{\rho }^2}\left({\sum }_{i=1}^n{X}_i-n\mu \right)\\ \frac{1}{{\rho }^2}\left({\sum }_{i=1}^n{X}_i-n\mu \right)& \frac{1}{{\varphi }^2}{\sum }_{i=1}^n{\left(X_i-\mu \right)}^2-\frac{n}{2{\rho }^2}\end{array}\right].$ c) Derive the expected information matrix, $\mathcal{I}(\theta )$. Hint: $\frac{{\sum }_{i=1}^n{\left(X_i-\mu \right)}^2}{\varphi }\sim {\chi }_n^2$. d) Consider the usual sample mean and variance estimators: $\bar{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i,S^2=\frac{1}{n-1}{\sum }_{i=1}^n{\left(X_i-\bar{X}\right)}^2.$ Find the efficiency for both $\bar{X}$ and $S^2$. Hint: $\frac{{\sum }_{i=1}^n{\left(X_i-\bar{X}\right)}^2}{\varphi }\sim {\chi }_{n-1}^2$