Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 18H / e

Solution

The Rao-Blackwell theorem suggests conditioning

S

T

. Part (d) gives

S = E [X_{1}^{2} - 1 ∣ T] = \frac{1}{2} + \frac{T ^{2}}{4} - 1 = \frac{T ^{2}}{4} - \frac{1}{2} .

It remains unbiased. Since

T \sim N (2 μ, 2)

and a normal variable with mean

m

v

satisfies

var (T^{2}) = 2 v^{2} + 4 m^{2} v

MSE (S) = \frac{1}{16} var (T^{2}) = \frac{1}{16} (8 + 32 μ^{2}) = \frac{1}{2} + 2 μ^{2} .

This is strictly below

2 + 4 μ^{2}

for every

μ

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