Anon

408 The Basics of financial economeTrics

local Shift Sensitivity

The local shift sensitivity measures the effect of the removal of a mass at y
and its reintroduction at x. For continuous and differentiable IC, the local
shift sensitivity is given by the maximum absolute value of the slope of IC
at any point.

Winsor’s principle

Winsor’s principle states that all distributions are normal in the middle.

M-Estimators

M-estimators are those estimators that are obtained by minimizing a
function of the sample data. As explained in Chapter 13, ordinary least
squares estimators and maximum likelihood estimators are examples of
M-estimators. Suppose that we are given an N-sample of data X = (x 1 ,... ,
xN)′. The estimator T(x 1 ,... , xN) is called an M-estimator if it is obtained by

==∑ρ















= 

TJargm inti(,)xt

i

N

1

where ρ(xi, t) is a function that depends on the estimator and “argmint”
means to minimize the expression in the brackets with respect to the
parameters t.
ML estimators are M-estimators with ρ = –log f, where f is the probabil-
ity density. (Actually, the name M-estimators means maximum likelihood-
type estimators.) LS estimators are also M-estimators.

The Least Median of Squares Estimator

Instead of minimizing the sum of squared residuals, as in LS, to estimate the
parameter vector, Rousseuw^3 proposed minimizing the median of squared
residuals, referred to as the least median of squares (LMedS) estimator. This
estimator effectively trims the N/2 observations having the largest residuals,
and uses the maximal residual value in the remaining set as the criterion to be
minimized. It is hence equivalent to assuming that the noise proportion is 50%.

(^3) P. Rousseuw, “Least Median of Squares Regression,” Journal of the American Sta-
tistical Association 79 (1984): 871–890.