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**The “automatic” robustness of minimum distance functionals.**
*(English)*
Zbl 0684.62030

If \(\{P_{\theta}\}\) is a model indexed by real parameter \(\theta\) and \(\mu\) is a metric between probabilities then the minimum distance (MD) functional \({\hat \theta}\)(P) is a solution of \(\mu (P,P_{{\hat \theta}})=\min_{\theta}\mu (P,P_{\theta}).\) The authors show that the functional is very stable with respect to the quantity being estimated in the following sense.

Denote by \(\delta\) a measure of distortion of an ideal distribution \(P_ 0\) and define the bias distortion curve for a functional T to be \[ b(\epsilon)=\sup \{| T(P)-T(P_ 0)|:\quad \delta (P,P_ 0)\leq \epsilon \}. \] The slope \(\gamma^*\) of b(\(\epsilon)\) at zero can be sometimes interpreted as the gross error sensitivity while \(\epsilon^*\) is equal to the smallest \(\epsilon\) such that the graph of b has no singularities in the ball of radius \(\leq \epsilon\) as the break down point of T.

The basic proposition of the paper claims that whenever the distortion measure \(\delta =\mu\) then for the MD functionals based on \(\mu\) we have \(b_ 0(\epsilon)\leq b(\epsilon)\leq b_ 0(2\epsilon),\) where \[ b_ 0(\epsilon)=\sup \{| \theta -\theta_ 0|:\quad \mu (P_{\theta},P_ 0)\}. \] This then implies \(\gamma^*(\theta)<2 \inf_{T} \gamma^*(T)\) and \(\epsilon^*(\theta)>\sup_{T} \epsilon^*(T)/2\), where the inf and sup are taken over Fisher consistent T. Thus the MD functional is shown to have within a factor of 2 the smallest sensitivity and the best breakdown point among Fisher consistent functionals.

These statements are then improved for Hilbertian distances and invariant cases. Generalizations are given for multidimensional parameter models, semiparametric models and minimum discrepancy functionals. Also consistency and finite sample robustness of MD estimators are studied.

Denote by \(\delta\) a measure of distortion of an ideal distribution \(P_ 0\) and define the bias distortion curve for a functional T to be \[ b(\epsilon)=\sup \{| T(P)-T(P_ 0)|:\quad \delta (P,P_ 0)\leq \epsilon \}. \] The slope \(\gamma^*\) of b(\(\epsilon)\) at zero can be sometimes interpreted as the gross error sensitivity while \(\epsilon^*\) is equal to the smallest \(\epsilon\) such that the graph of b has no singularities in the ball of radius \(\leq \epsilon\) as the break down point of T.

The basic proposition of the paper claims that whenever the distortion measure \(\delta =\mu\) then for the MD functionals based on \(\mu\) we have \(b_ 0(\epsilon)\leq b(\epsilon)\leq b_ 0(2\epsilon),\) where \[ b_ 0(\epsilon)=\sup \{| \theta -\theta_ 0|:\quad \mu (P_{\theta},P_ 0)\}. \] This then implies \(\gamma^*(\theta)<2 \inf_{T} \gamma^*(T)\) and \(\epsilon^*(\theta)>\sup_{T} \epsilon^*(T)/2\), where the inf and sup are taken over Fisher consistent T. Thus the MD functional is shown to have within a factor of 2 the smallest sensitivity and the best breakdown point among Fisher consistent functionals.

These statements are then improved for Hilbertian distances and invariant cases. Generalizations are given for multidimensional parameter models, semiparametric models and minimum discrepancy functionals. Also consistency and finite sample robustness of MD estimators are studied.

Reviewer: T.Bednarski

### MSC:

62F35 | Robustness and adaptive procedures (parametric inference) |

62F12 | Asymptotic properties of parametric estimators |