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

Write

β = A Y

, where

A = (X^{T} Σ_{0}^{- 1} X)^{- 1} X^{T} Σ_{0}^{- 1}, A X = I .

Any other linear unbiased estimator is

C Y

with

CX = I

, so

C = A + D

and

D X = 0

. The cross covariance vanishes because

A Σ_{0} D^{T} = (X^{T} Σ_{0}^{- 1} X)^{- 1} X^{T} D^{T} = 0.

Therefore

Cov (C Y) - Cov (A Y) = σ^{2} D Σ_{0} D^{T} ⪰ 0.

By the Gauss-Markov theorem,

β is the best linear unbiased estimator of β .

Solved by gpt-5.6-sol high.

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