(a) Formulas for the elements of the gradient and the Hessian of l(mu, sigma).

\[ dl(d\mathbf{\mu},\Sigma) = \sum_{i = 1}^{n} trace [\Sigma^{-1}(x_i-\mu)d\mu^T] = trace[\sum_{i = 1}^{n}\Sigma^{-1}(x_i-\mu)d\mu^T] = trace[Ad\mu^T]\] \[ A = \sum_{i = 1}^{n}\Sigma^{-1}(x_i-\mu) \] \[ \displaystyle\frac{\partial l}{\partial \mu_i}= A_i\]

\[ ddl(d\mu,d\Sigma) = trace [\sum_{i = 1}^{n}-\Sigma^{-1}d\Sigma\Sigma^{-1}(x_i-\mu)d\mu^T] = -trace [-\Sigma^{-1}d\Sigma\Sigma^{-1}\sum_{i = 1}^{n}(x_i-\mu)d\mu^T] = -trace [B d\Sigma A d\mu^T] \] \[ B=\Sigma^{-1}, \text{ }\text{ }\text{ } A=\Sigma^{-1}\sum_{i = 1}^{n}(x_i-\mu)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ii} \partial\mu_{i}} = - A_iB_{ii} \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j=k)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}} = - (A_iB_{kj}+A_jB_{ki})\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j\neq k) \] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}} = - A_iB_{ki}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j\neq k)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}} = - (A_iB_{ij}+A_jB_{ii})\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, i=k)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}} = - (A_iB_{jj}+A_jB_{ji})\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, j=k)\]

\[ dl(\mu,d\Sigma) = -\frac{1}{2}trace[n\Sigma^{-1}-\Sigma^{-1}C(\mu)\Sigma^{-1})d\Sigma = -\frac{1}{2}trace [C d\Sigma]\]
\[ C = n\Sigma^{-1}-\Sigma^{-1}C(\mu)\Sigma^{-1} \] \[ \displaystyle\frac{\partial l}{\partial \sigma_{ii}}= -\frac{1}{2}C_{ii}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j)\] \[ \displaystyle\frac{\partial l}{\partial \sigma_{ij}}= -\frac{1}{2}(C_{ij}+C_{ji})\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j)\]

\[ ddl(d\mu_i,d\mu_j) = trace[\sum_{i = 1}^{n}\Sigma^{-1}(-d\mu_j)d\mu_i^T]= trace[D(-d\mu_j)d\mu_i^T]\]
\[ D=\sum_{i = 1}^{n}\Sigma^{-1} \] \[ \frac{\partial^2 l}{\partial \mu_i \partial\mu_i} = - D_{ii}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j)\] \[ \frac{\partial^2 l}{\partial \mu_i \partial\mu_j} = - D_{ij}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j)\]

\[ ddl(d\Sigma,d\Sigma) = -\frac{1}{2}trace[-n\Sigma^{-1}+C(\mu)\Sigma^{-1}\Sigma^{-1}+\Sigma^{-1}C(\mu)\Sigma^{-1}]d\Sigma\Sigma^{-1}d\Sigma \]
\[= -\frac{1}{2}trace[-n\Sigma^{-1}+2C(\mu)\Sigma^{-1}\Sigma^{-1}]d\Sigma\Sigma^{-1}d\Sigma =-\frac{1}{2}traceGd\Sigma Bd\Sigma\]
\[ G = -n\Sigma^{-1}+2C(\mu)\Sigma^{-1}\Sigma^{-1},\text{ } B = \Sigma^{-1}\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}} = - \frac{1}{2}(G_{ik}B_{ki})\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j, k=l)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}} = - \frac{1}{2}(G_{il}B_{kj}+G_{jl}B_{ki}+G_{ik}B_{lj}+G_{jk}B_{li}) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, k\neq l)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}} = - \frac{1}{2}(G_{ik}B_{jk}+G_{kj}B_{ki}) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, k=l)\] \[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}} = - \frac{1}{2}(G_{il}B_{ki}+G_{ik}B_{li})\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j, k\neq l)\]

(b) Formulas for the elements of the information matrix.

\[ Note\text{ }that\text{ } \Sigma_{ij}^{-1} = (i,j)th\text{ } element\text{ }of\text{ }the\text{ } \Sigma^{-1} matrix\] \[ - E[\frac{\partial^2 l}{\partial \mu_i \partial\mu_j} ] = E[D_{ii}] = E[\sum_{i = 1}^{n}\Sigma_{ii}^{-1}] = n\Sigma_{ii}^{-1}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j)\] \[ - E[\frac{\partial^2 l}{\partial \mu_i \partial\mu_j} ] = E[D_{ij}] = E[\sum_{i = 1}^{n}\Sigma_{ij}^{-1}] = n\Sigma_{ij}^{-1} \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j)\]

\[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij}\mu_{k}}]=E[A_iB_{ii}]= 0 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j=k)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}}]=E[A_iB_{kj}+A_jB_{ki}]= 0 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j\neq k)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}}]=E[A_iB_{ki}]= 0 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j\neq k)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}}]=E[A_iB_{ij}+A_jB_{ii}]= 0 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, i=k)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\mu_{k}}]=E[A_iB_{jj}+A_jB_{ji}]= 0 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, j=k)\]

\[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}}]=-\frac{1}{2}E[G_{ik}B_{ki}]= \frac{n}{2}\Sigma_{ik}^{-1}\Sigma_{ki}^{-1}=\frac{n}{2}(\Sigma_{ik}^{-1})^2 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j, k=l)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}}]=-\frac{1}{2}E[G_{ik}B_{ki}+G_{jl}B_{ki}+G_{ik}B_{lj}+G_{jk}B_{li}]=\frac{n}{2}(\Sigma_{il}^{-1}\Sigma_{kj}^{-1}+\Sigma_{jl}^{-1}\Sigma_{ki}^{-1}+\Sigma_{ik}^{-1}\Sigma_{lj}^{-1}+\Sigma_{jk}^{-1}\Sigma_{li}^{-1})\] \[=n(\Sigma_{il}^{-1}\Sigma_{kj}^{-1}+\Sigma_{jl}^{-1}\Sigma_{ki}^{-1} ) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, k\neq l)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}}]=-\frac{1}{2}E[G_{ik}B_{kj}+G_{jk}B_{ki}] = \frac{n}{2}(\Sigma_{ik}^{-1}\Sigma_{kj}^{-1}+\Sigma_{jk}^{-1}\Sigma_{ki}^{-1})=n\Sigma_{ik}^{-1}\Sigma_{kj}^{-1} \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i\neq j, k=l)\] \[ - E[\displaystyle \frac{\partial^2 l}{\partial \sigma_{ij} \partial\sigma_{kl}}]=-\frac{1}{2}E[G_{il}B_{ki}+G_{ik}B_{li}] =\frac{n}{2}(\Sigma_{il}^{-1}\Sigma_{ki}^{-1}+\Sigma_{ik}^{-1}\Sigma_{li}^{-1}) =n\Sigma_{il}^{-1}\Sigma_{ki}^{-1} \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }(i=j, k\neq l)\]