# Question

Consider the linear model with n observations y ~ N (Xβ, σ2In), since we have many predictors, β is a vector.
(a) Write down the log-likelihood ℓ(β; y) of the model and the score U(β) = ∂ℓ/∂ β.
(b) Under certain regularity conditions it can be shown that the expected score is zero and thus that Fisher information I(β), the variance of the score, is


Show that I(β)–1 is the same as the variance of the MLE β of β, and thus that the information does not depend on β.
(c) The same regularity conditions imply that


That is, that the information is also the negative Hessian of the log-likelihood. Verify that this is true by computing the negative Hessain and showing it equals the inverse of the variance ofβ.

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