Breakdown point

Definition

Let $T$ be an estimator that maps each sample $x = (x_1, \dots, x_n)$ of points in $\R^p$ to a value $T(x) \in \R^k$ . For $0 \le m \le n$ , let $\mathcal{Y}_m(x)$ be the set of samples obtained from $x$ by replacing at most $m$ points by arbitrary points of $\R^p$ . The finite-sample breakdown point of $T$ at $x$ is $\varepsilon^*_n(T, x) = \min\Big\{ \frac{m}{n} \;:\; \sup_{y \in \mathcal{Y}_m(x)} \norm{T(y) - T(x)} = \infty \Big\}.$ The asymptotic breakdown point is the limit of $\varepsilon^*_n$ as $n \to \infty$ , when it exists.

Remarks

Example: the sample median of real data has $\varepsilon^*_n = \floor{(n+1)/2}/n$ . For $n = 2k + 1$ , changing $k$ points leaves an original point at the middle position, between original values; changing $k + 1$ points to a huge value $M$ makes the median $M$ . So the asymptotic breakdown point of the median is $1/2$ .

Non-example of robustness: the sample mean has $\varepsilon^*_n = 1/n$ , since replacing one point $x_1$ by $x_1 + nM$ moves the mean by $M$ . Its asymptotic breakdown point is $0$ .

The $k$ -trimmed mean, which drops the $k$ smallest and $k$ largest values, has $\varepsilon^*_n = (k+1)/n$ . For estimators of location that are translation equivariant, the asymptotic breakdown point is at most $1/2$ .

Reading path

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The rationals form an ordered field statement
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Expectation of a product of independent random variables statement
Statistic definition
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Estimator definition
Variance of a sum of independent random variables statement
Sample mean definition
Breakdown point definition

Tag	`DMFB5JQ2T7`
URI	`m2://DMFB5JQ2T7`
Kind	definition
Writing	W0 stub
MSC	62F35, 62G35
Defines	Breakdown point
Uses	Estimator, Sample mean, Median

Breakdown point

Definition

Remarks

Reading path

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