Robust Econometrics

F ? Quantitative robustness would concentrate on: will T (Fn) be bounded if
some observations x_i → ∞? In fact, the last question is easy to answer: if
x_i → ∞ for some i, T(Fn) = x → ∞ as well. So we can say here in a loose
sense that x is not quantitatively robust.

Formalities

In the following we present a mathematical setup that allows us to formalize
the robustness thoughts.

The notion of the sensitivity of an estimator T is put into theory by
considering a model characterized by a cdf F and its neighborhood F_ε,G :
distributions (1 - ε)F + εG, where ε ∈ (0, 1/2) and G is an arbitrary proba-
bility distribution, which represents data contamination. Hence, not all data
necessarily follow the pre-specified distribution, but the ε-part of data can
come from a different distribution G. If H ∈ F_ε,G , the estimation method
T is then judged by how sensitive or robust are the estimates T(H) to the
size of F_ε,G , or alternatively, to the distance from the assumed cdf F . Two
main concepts for robust measures analyze the sensitivity of an estimator to
infinitesimal deviations, ε → 0, and to finite (large) deviations, ε > 0, respec-
tively. Despite generality of the concept, easy interpretation and technical
difficulties often limit our choice to point-mass distributions (Dirac measures)
G = δx, x ∈ R, which simply represents an (erroneous) observation at point
x ∈ R. This simplification is also used in the following text.

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