Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 29L / b / i

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- Solution i

Solution

Write

v = σ^{2}

. For one observation, the score is

s_{μ} (x; μ, v) = \frac{x - μ}{v}, s_{v} (x; μ, v) = - \frac{1}{2 v} + \frac{( x - μ ) ^{2}}{2 v ^{2}},

and

I_{1} (μ, v) = (v^{- 1} 0 0 (2 v^{2})^{- 1}) .

Under the restriction

v = 1

, the restricted maximum-likelihood estimator is

μ = \overset{ˉ}{X}

. At

(\overset{ˉ}{X}, 1)

the first score component is zero, while the second is

\frac{1}{2} \sum_{i = 1}^{n} ((X_{i} - \overset{ˉ}{X})^{2} - 1) .

Since

I_{1} (\overset{ˉ}{X}, 1)^{- 1} = diag (1, 2)

, substitution into the score statistic gives

T_{n} = [\frac{1}{2 n} i = 1 \sum n ((X_{i} - \overset{ˉ}{X})^{2} - 1)]^{2} .

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