Robust Econometrics

of an equation,   g(z, θ)dF (z) = 0 in θ. The estimation for a given sam-
ple utilizes finite-sample equivalents of these integrals,   h(z, θ)dF_n (z) and

g(z, θ)dF_n(z), respectively.

Consider the pure location model Xi = μ + σεi, i = 1,..., n, with a known
scale σ and ε ~ F. The cdf of X is then F{(x — μ)/σ}. With a quadratic
contrast function h(x, θ) = (x - θ)² , the estimation problem is to minimize
ʃ(x — θ)²dF{(x — μ)/σ} with respect to θ. For known F, this leads to θ = μ
and one sees that, without loss of generality, one can assume μ = 0 and σ = 1.
For the sample {xi}n=₁ characterized by edf F_n, the location parameter μ is
estimated by

n

(x — θ)²dFn (x) = n^-1

~t

Note that for g(x, θ) = x-θ, the parameter μ is the solution to j g(x, θ)dF(x) =
0. The estimator may therefore be alternatively defined through μ = T(F) =
udF (u).

As indicated in the introduction, this standard estimator of location per-
forms unfortunately rather poorly under the sketched contamination model.
Estimating a population mean by the least squares (LS) or sample mean

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