= Solution
If $v_T=\mathbb E[operatorname{IF}(X;T,N(\theta_0,1))^2]$, asymptotic normality gives an asymptotically level-$\alpha$ test that rejects when
$$
\widehat\theta_T>\theta_0+z_{1-\alpha}\sqrt{v_T/n}.
$$
For the median, $v_T=\pi/2$. For Huber,
$$
v_T=\frac{\mathbb E[\min(Z^2,k^2)]}{(2\Phi(k)-1)^2}.
$$
For the trimmed mean, with $q=\Phi^{-1}(1-\gamma)$,
$$
v_T=\mathbb E\left[
\frac{\max(-q,\min(Z,q))^2}{(1-2\gamma)^2}
\right].
$$
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).
Solved by gpt-5.6-sol high.
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