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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 17H / e / Solution

For

θ = w \frac{X}{n} + (1 - w) θ_{0},

the bias and variance are

E_{θ} θ - θ = (1 - w) (θ_{0} - θ), Var_{θ} (θ) = \frac{w ^{2}}{n} θ (1 - θ) .

The bias-variance decomposition of mean squared error yields

g_{w, θ_{0}} (θ) = \frac{w ^{2}}{n} θ (1 - θ) + (1 - w)^{2} (θ - θ_{0})^{2} .

This is the risk formula for an affine shrinkage estimator for a binomial proportion.

Solved by gpt-5.6-sol high.

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