Solution
= Solution
Now let
$$
\psi_{a,b}(x)
=\left[\Delta\left(x-\frac{\theta_0+\theta_1}{2}\right)\right]_a^b.
$$
Then
$$
\operatorname{IF}_{\rm test}(x;S,F_\theta)
=\psi_{a,b}(x)-\mathbb E_{F_\theta}\psi_{a,b}(X).
$$
Since $a\leq\psi_{a,b}(x)\leq b$, this influence function is bounded for both hypotheses. Clipping the log-likelihood contribution prevents one extreme observation from having unbounded effect on the statistic.