Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 18H / a

Solution

A possibly randomized test is a function

φ (X) \in [0, 1]

, interpreted as the conditional probability of rejecting

H_{0}

π_{φ} (θ) = E_{θ} [φ (X)] .

It has size at most

α

when

sup_{θ \in Θ_{0}} π_{φ} (θ) \leq α .

α

if, for every other test

ψ

of size at most

α

π_{φ} (θ) \geq π_{ψ} (θ) for every θ \in Θ_{1} .

Thus one test maximizes power simultaneously at every parameter value in the alternative.

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