Robust statistics

Remarks

Qualitative robustness asks that the maps $G \mapsto \mathcal{L}_G(T_n)$ be continuous at $F$, uniformly in $n$. Closeness in $\pi$ allows both small rounding of all data and a small fraction of arbitrary outliers. The quantitative measures are the breakdown point and the influence function.

Example: if $F$ has a unique median $m$, the sample median is qualitatively robust at $F$. A small fraction of outliers can shift it only to nearby order statistics, so its law changes little.

Non-example: the sample mean is not qualitatively robust at any $F$. Let $0 < \eta < \delta$ and $G = (1 - \eta) F + \eta \, \delta_x$, with $\delta_x$ the point mass at $x$; then $\pi(F, G) \le \eta < \delta$. For large $n$, with high probability about $n\eta$ observations equal $x$, and the mean moves by about $\eta x$, which is arbitrarily large for large $x$.