Replacement breakdown point (${\epsilon }^{\ast }$)

For a fixed sample $y$ of size $n$, let $b_n(m;y)$ be its finite-sample maximum bias after at most $m$ arbitrary replacements, with $m\in \{0,\dots ,n\}$. The largest-bounded convention used here isWhen breakdown is possible, the first-unbounded convention isThey differ by $1/n$. For five observations, the sample median remains bounded under any two replacements, while three replacements can make it unbounded. These conventions therefore give $2/5$ and $3/5$, respectively. Their common large-sample limit does not remove the need to specify the finite-sample convention. The finite-sample maximum bias article displays the first-unbounded form explicitly.