Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 28K / b

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- Solution b

Solution

An admissible estimator is one for which there is no estimator

δ

satisfying

R (θ, δ) \leq R (θ, θ)

for every

θ

, with strict inequality for at least one

θ

. For

p \geq 3

, the James–Stein estimator

δ_{JS} (X) = (1 - \frac{p - 2}{∥ X ∥ ^{2}}) X

has risk

R (θ, δ_{JS}) = p - (p - 2)^{2} E_{θ} \frac{1}{∥ X ∥ ^{2}} < p .

It strictly dominates

X

, and hence

θ_{MLE} is inadmissible .

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