Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 18H / a

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- Solution a

Solution

The log likelihood, up to constants, is

- \frac{n}{2} lo g σ^{2} - \frac{1}{2 σ ^{2}} (Y - Xβ)^{T} Σ_{0}^{- 1} (Y - Xβ) .

Differentiation gives the generalized least squares estimator

β = (X^{T} Σ_{0}^{- 1} X)^{- 1} X^{T} Σ_{0}^{- 1} Y,

and maximizing over the scale gives

σ^{2} = \frac{1}{n} (Y - X β)^{T} Σ_{0}^{- 1} (Y - X β) .

n

is appropriate for maximum likelihood rather than unbiased estimation.

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