 Finite-sample maximum bias

Here $d_H$ counts the observations replaced. This curve measures the largest change after at most $m$ arbitrary replacements. Under the first-unbounded convention, the replacement breakdown point is $n^{-1}\min \{m:b_n(m;y)=\infty \}$. The largest fraction still guaranteed bounded is one grid step smaller; those conventions must not be confused. A sample mean breaks after one replacement, while a scalar sample median with a fixed ordinary tie convention needs about half the sample replaced.