Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 29L / b / Solution

With ${\ell }_n(\theta )={\sum }_i\log f(X_i,\theta )$, define the likelihood-ratio statisticWilks' theorem gives ${\Lambda }_n\Rightarrow {\chi }_p^2$ under $H_0$, so reject whenThis again has asymptotic type-I error $\alpha$.