Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 4 / c

Table of contents
- Solution c

Solution

If

B = \emptyset

, then for every

j

at least one of

H_{j}^{β}

and

H_{j}^{η}

is true. Since

{q_{j} \leq t} = {q_{j}^{β} \leq t} \cap {q_{j}^{η} \leq t},

validity of the p-value for whichever component null is true implies

P (q_{j} \leq t) \leq t

. Therefore the union bound gives

P (min_{1 \leq j \leq p} q_{j} \leq \frac{α}{p}) \leq \sum_{j = 1}^{p} P (q_{j} \leq \frac{α}{p}) \leq α .

This is the Bonferroni correction for the composite intersection alternatives.

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