Huber contamination model

Huber’s contamination model in robust statistics dates back to the work of Peter Huber in 1964 (see Robust Estimation of a Location Parameter). We assume we are trying to do inference on some distribution $P$, but we see a contaminated sample drawn from the mixture distribution

$$R=ϵQ+(1-ϵ)P,$$

for some $ϵ\ge 0$ (known to us), and some “contamination distribution” $Q$. This is a weaker contamination model than the adversarial contamination model.

We can also make Huber’s model slightly more general by assuming that the contaminated sample is drawn from $R$, where $R\in {\mathbb{B}}_{\text{TV}}(P,ϵ)=\{R:\text{TV}(R,P)\le ϵ\}$, where $\text{TV}$ is the total variation distance. This is a superset of the mixture distributions $ϵQ+(1-ϵ)P$.