Anon

Robust Regressions 167

The empirical correlation coefficient is the empirical covariance nor-
malized with the product of the respective empirical standard deviations:

ˆ

ˆ

, ˆˆ

ρ σ ,

XY σσ

XY

XY

=

The empirical standard deviations are defined as

σσˆˆXi

i

N

Yi

i

N

N

XX

N

= YY

−

()− =

−

()−

==

∑∑

1

1

1

1

2

1

2

1

Empirical covariances and correlations are not robust as they are highly
sensitive to tails or outliers. Robust estimators of covariances and/or cor-
relations are insensitive to the tails. However, it does not make sense to
robustify correlations if dependence is not linear.
Different strategies for robust estimation of covariances exist; among
them are:

■ (^) Robust estimation of pairwise covariances
■ (^) Robust estimation of elliptic distributions
Here we discuss only the robust estimation of pairwise covariances. As
detailed in Huber,^4 the following identity holds:
cov(,)[XY var()var()]
ab
=+aX bY−−aX bY

1

4

Assume S is a robust scale functional:

Sa()Xb+=aS()X

A robust covariance is defined as

CXY

ab

(,)[=+Sa()XbYS−−()aX bY ]

1

4

22

Choose

a

SX

b

SY

==

11

() ()

(^4) Huber, Robust Statistics.