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Let $X_{1},…,X_{n}∼idN(μ,ϕ)$ where the parameters $μ$ and $ϕ$ are both unknown. a) Show that $(∑X_{i},∑X_{i})$ is the sufficient statistic for $(μ,ϕ)$. b) Show that the observed information matrix is $I(θ)=[6a ρ_{2}1 (∑_{i=1}X_{i}−nμ) ρ_{2}1 (∑_{i=1}X_{i}−nμ)ϕ_{2}1 ∑_{i=1}(X_{i}−μ)_{2}−2ρ_{2}n ].$ c) Derive the expected information matrix, $I(θ)$. Hint: $ϕ∑_{i=1}(X_{i}−μ)_{2} ∼χ_{n}$. d) Consider the usual sample mean and variance estimators: $Xˉ=n1 ∑_{i=1}X_{i},S_{2}=n−11 ∑_{i=1}(X_{i}−Xˉ)_{2}.$ Find the efficiency for both $Xˉ$ and $S_{2}$. Hint: $ϕ∑_{i=1}(X_{i}−Xˉ)_{2} ∼χ_{n−1}$

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