Robust Econometrics

of contamination that can cause an infinite bias:

ε*(T) = inf{ε : B(ε,T) = ∞}.

The intuitive aim of this definition specifies the breakdown point ε* (T) as
the smallest amount of contamination that makes the estimator T useless.
Note that in most cases ε* (T) ≤ 0.5 (He and Simpson, 1993). This definition
and the upper bound however apply only in simple cases, such as location
or linear regression estimation (Davies and Gather, 2005). The most general
definition of breakdown point formalizes the idea of “useless” estimates in the
following way: an estimator is said to break down if, under contamination,
it is not random anymore, or more precisely, it can achieve only a finite
set of values (Genton and Lucas, 2003). This definition is based on the
fact that estimates are functions of observed random samples and are thus
random quantities themselves unless they fail. Although the latter definition
includes the first one, the latter one may generally depend on the underlying
model F , for example in time-series context.

2 Estimation approaches

Denote by Fn an empirical distribution function (edf) corresponding to a
sample {x_i}_iⁿ₌₁ ∈ R drawn from a model based on probability distribution F .
Most estimation methods can be defined as an extremum problem, minimiz-
ing a contrast h(z, θ)dF (z) over θ in a parameter space, or as a solution

<<   4   5   6   7   8   9   10   >>

More intriguing information