Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 17H / a

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- Solution a

Solution

The unrestricted maximum-likelihood estimates are

(\overset{ˉ}{X}, \overset{ˉ}{Y})

, while the null fixes both means at zero. Hence

- 2 lo g Λ = m \overset{ˉ}{X}^{2} + n \overset{ˉ}{Y}^{2} .

Under the null this is

χ_{2}^{2}

, so the size-

α

generalized likelihood-ratio test rejects exactly when

m \overset{ˉ}{X}^{2} + n \overset{ˉ}{Y}^{2} > F_{2}^{- 1} (1 - α) .

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