 Solution

Choose $v\in {\mathbb{R}}^p$ outside the finitely many hyperplanes $\{v:x_i^{\top }v=0\}$. Then $x_i^{\top }v\ne 0$ for every $i$. For ${\theta }_t=tv$, every residual $y_i-t{x}_i^{\top }v$ eventually has absolute value greater than $L$. Every indicator in the estimating equation is then zero, so the equation is satisfied for every sufficiently large $t$.

Thus arbitrarily large solutions already exist without contamination. Under the definition in the question, the skipped-mean regression estimator has breakdown point zero. This is a standard pathology of an exactly redescending score when every root is admitted as an estimator.

Solved by gpt-5.6-sol high.