Breakdown point

A breakdown point is the smallest contamination fraction capable of making an estimator arbitrarily unreliable.

Table of contents
- Finite-sample maximum bias Breakdown point
- Replacement breakdown point Breakdown point

Finite-sample maximum bias ( $b_{n} (m; y) = sup_{z : d_{H} (z, y) \leq m} ∣ T_{n} (z) - T_{n} (y) ∣$ )

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.

Replacement breakdown point ( $ε^{*}$ )

 0  0

The finite-sample replacement breakdown point is the largest replacement fraction under which an estimator remains bounded over arbitrary replacement values.