Anon

Robust Statistics 409

LMedS is unwieldy from a computational point of view because of its
nondifferentiable form. This means that a quasi-exhaustive search on all
possible parameter values needs to be done to find the global minimum.

The Least Trimmed of Squares Estimator

The least trimmed of squares (LTS) estimator offers an efficient way to find
robust estimates by minimizing the objective function given by

=∑













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= 

Jri

i

h

()

2

1

where r()^2 i is the ith smallest residual or distance when the residuals are
ordered in ascending order, that is, rr(1)^2 ≤≤(2)^2 r()N^2 and h is the number of
data points the residuals of which we want to include in the sum. This
estimator basically finds a robust estimate by identifying the N – h points
having the largest residuals as outliers, and discarding (trimming) them
from the data set. The resulting estimates are essentially LS estimates of the
trimmed data set. Note that h should be as close as possible to the number
of points in the data set that we do not consider outliers.

Robust Estimators of the Center

The mean estimates the center of a distribution but it is not resistant. Resis-
tant estimators of the center are the following:

■ (^) Trimmed mean. Suppose x(1) ≤ x(2) ≤... ≤ x(N) are the sample order sta-
tistics (that is, the sample sorted). The trimmed mean TN(δ, 1 – γ) is
defined as follows:
δ−γ= ∑
−
δγ∈=δ= γ
=+

T

UL

x

LNUN

(,1)

1

,(0,0.5) floor[] floor[]

N

NN

j

jL

U

NN

N^1

N

■ (^) Winsorized mean. The Winsorized mean XW is the mean of Winsorized
data:

=

=

≤

+≤≤

≥+

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

=

+

+

y

x

x

xx

jL

LjU

jU

XY

1

1

j

I

j

jU

N

NN

N

W

1

1

N

N