 Solution

Let replacement tolerance mean the greatest number of observations that can be replaced while the estimator remains bounded. For $n=2m$, a sample with $m$ zeros and $m$ copies of $a$ is within $m$ replacements of both the all-zero sample and its translate by $a$. If an equivariant estimator tolerated $m$ replacements, it would remain within bounded distance of both $T(0^n)$ and $T(0^n)+a$, which is impossible as $a\to \infty$. Thus at most $m-1$ replacements are tolerable.

For $n=2m+1$, a sample with $m$ zeros and $m+1$ copies of $a$ is obtained from the all-zero sample by $m+1$ replacements and from the all-$a$ sample by $m$ replacements. Translation equivariance again forces breakdown by $m+1$ replacements. In both cases the finite-sample replacement breakdown point is at most

Solved by gpt-5.6-sol high.