Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 223 / 1 / d / Solution

If $v_T=\mathbb{E}[operatornameIF(X;T,N({\theta }_0,1){)}^2]$, asymptotic normality gives an asymptotically level-$\alpha$ test that rejects whenFor the median, $v_T=\pi /2$. For Huber,For the trimmed mean, with $q={\Phi }^{-1}(1-\gamma )$,These tests have bounded influence functions, so a small contamination proportion has bounded first-order effect on their statistics, asymptotic levels, and powers. Their finite-contamination protection is quantified by the breakdown points in part (c).

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