Working Paper
92-35
Divisi6n
de
Economfa
September
1992
Universidad Carlos
ill
de
Madrid
Calle Madrid,
126
28903
Getafe
(Spain)
Fax
(341)
624
9849
"
( .
'..
ON
THE
ESTIMATION
OF
THE
INFLUENCE
CURVE
Antonio
Cuevas
and
Juan
Romo·
Abstract _
We
prove the asymptotic validity of
boots
trap
confidence
bands
for
the
influence curve
from
its
usual
estimator (the sensitive curve).
The
proof
is
based
on
the
use
of Gill's
(1989)
generalized
delta
method
for
Hadamard
differentiable operators. The
scope
and
applicability
of
this
result are
also
discussed.
Key
words:
Influence curve, sensitivity curve,
boots
trap
confidence
bands,
Hadamard differentiability.
·Cuevas, Departamento
de
Matematicas,
Universidad
Aut6noma
de
Madrid,
Spain;
Romo,
Departamento
de
Estadfstica y
Econometrfa,
Universidad
Carlos
III
de
Madrid, Spain.
)
~\
)
,
j
1.
INTRODUCTION
AND
BACKGROUND
It
is well-known
that,
in many cases of practical interest,
the
estimators can
be considered
as
restrictions of functionals defined on
the
space
:F
of distri-
bution functions. In fact, this idea goes back to the origins of mathematical
statistics since
it
is
implicit in the early notion of consistency proposed by
Fisher. In precise terms, let
T
n
= T
n
(X
ll
...
, X
n
) be (for all n =
1,2,
...
) an
estimator taking values in
~,
defined on random samples X
I,
...
,X
n
from
a univariate distribution. The sequence
{T
n
}
is said to
be
generated by a
functional
T :
:F
o
C
:F
~
~,
iffor
all n and for each sample
Xl,""
X
n
,
we
have
Tn(X}, .
..
,X
n
)
=
T(F
n
),
where F
n
is the empirical distribution associ-
ated
with
Xl,'
..
,X
n
•
Many usual estimators fulfil this condition;
this
is
the
case, for instance, of
M-
and L-estimators [see,
e.
g.
Huber (1981 )]. By
:F
n
we
will represent
the
set of empirical distributions of order n in
:F,
that
is,
the
set of discrete probability measures in
:F
whose atoms have probabilities
equal to
1/n
or to a multiple of 1/n. Obviously, the domain
:F
o
of T has
to
include
:F
n
for all n E N.
In this setting, a natural idea is to use the differentiability properties
of
the
functional T in order to get statistical results for
the
sequence
{T
n
}.
The
works of von Mises (1947) and Kallianpur and Rao (1945) are pio-
neering contributions on this topic
but,
in fact,
the
use of differentiation
techniques only became really popular in
the
late sixties coinciding with
the
rapid development of
the
robustness theory. An
important
example is
the
so-called influence function,
T'(F;
x) (of a functional T
at
a distribution
F E
:F), which is nothing
but
the partial derivative of T along
the
direc-
tion
corresponding to the degenerate distribution h
x
(for each x),
that
is,
T'(F;
x) =lim(
.....
o+
[T((l-
f)F
+
fh
x
) -
T(F)]/f
[see
Hampel (1974), Hampel
et
al.
(1987)].
If
we
assume
that
the sequence
{Tn}of
estimators generated
by
T is consistent, in probability under G (for each G), to
T(
G)
then
T'(F;
x)
represents (for small values of f) the approximate value of asymptotic bias
introduced by a contamination of type
(1
-
f)F
+fh
x
at
the
distribution
F.
Some quantitative measures of robustness (gross-error sensitivity! local-shift
sensitivity! rejection points)
are also defined from
the
influence curve.
However, in order to get a deeper insight into
the
meaning of
the
influence
function, we need to impose (on
T)
further differentiability assumptions,
stronger
than
the mere existence of
T'(F;
x). This situation is similar to
that
of
the
classical analysis for functions f :
~p
--
~;
the
true
significance
2
I
of
the
gradient
'V
f (which
is
the analog of
the
influence function) arises
when
we
assume
that
f
is
differentiable since, in this case,
'V
f defines
the
best
linear
local
approximant of f.
The
general concept of differentiability, for operators or functionals,
is
inspired on
the
same idea: let Qand
'D
be normed spaces and let V : Q
---+
'D
be an operator.
We
will say
that
V is differentiable
at
G E
Q,
with respect
to a collection
S of subsets of
Q,
if
there
exists a linear and continuous map
DV(G;.)
: Q
---+
'D
(which
we
will call
the
differential
of
V
at
G)
such
that
for
~
in some neighbourhood of zero,
V(G
+~)
= V(G) +
DV(G;~)
+ R(G +
~),
where
the
remainder R satisfies
lim
R(G +
t~)
=
0,
t-O
t
uniformly in
~
E
S,
for every
SE
S.
The
most interesting particular cases correspond to the following choices
of
S: S = all singletons of
Q,
S = all compact subsets of
Q,
and S = all
bounded subsets of
Q.
They lead, respectively, to
the
concepts of Gateaux,
H
ada
mard (or compact) and Frichet differentiability.
The
application of these concepts (borrowed from
the
functional analysis)
to
statistical functionals T :
:F
---+
~,
presents an obvious hurdle:
:F
is
not a
normed space. A simple device to overcome this difficulty
is
embedding
:F
in
the
space Q =
P(F
-
H)
:
F,
H E
:F,
>..
E 'R}, endowed with
the
supremum
norm.
The statistical functionals can be often extended in a natural way
to
the
space Q (or appropriate subspaces of it). In such cases the use of
the
above
notions of differentiability
is
a very useful tool which allows
to
consider
the
influence function from a different perspective. Moreover, if
the
functional
T
is
Frechet (or Hadamard) differentiable
at
F and
the
differential can be
expressed in
the
form
DT(F;~)
=
1:
\lI(x
)d~(x),
then
it
is
not difficult to prove
[see
Boos and Serfling (1980)]
that
\11
(x
) coin-
cides with the influence curve, and the sequence
{T
n
}
of estimators generated
3
J
)
,)
)
)
I
'
•...
/
by T
is
asymptotically normal with asymptotic variance
1:
T'(F; x)2dF(x).
This is, perhaps, the most important point in connection with
the
influence
curve: under
standard
conditions the asymptotic variance can
be
expressed
in terms
of
T'(F; x).
In
particular, the estimates of
the
influence curve are
potentially useful in
the
estimation of
the
asymptotic variance
[see
Presedo
(1991
)].
The
choice between Hadamard or Frechet differential in each particular
application
is
usually guided by technical considerations.
In
general terms,
Frechet differential
is
more natural and easier to handle. Some applications
can be found in Kallianpur and Rao (1955), Boos and Serfling (1980), Clarke
(1986),
Parr
(1985) and Arcones and Gine (1992). Nevertheless,
the
com-
pact differentiation has, in principle, a broader applicability since
it
imposes
a weaker (less restrictive) condition;
it
is, in fact,
the
weakest notion of dif-
ferential which
is
still manageable in the sense offulfilling
the
chain rule. For
applications, see Fernholz (1983), Esty
et
al.
(1985) and Gill (1989).
In
this paper
we
use Hadamard differential to prove (in Section 2 below)
the
validity of bootstrap confidence bands for
the
standard estimator of
the
influence curve.
The
basic tools used in the proof are
the
results on bootstrap
of empirical processes
[see
Gine and Zinn (1990)] and
the
generalized delta
method
established
by
Gill (1989). Section 3 contains some final remarks.
2.
BOOTSTRAP
CONFIDENCE
BANDS
FOR
THE
INFLUENCE
CURVE
\Ve
consider
now
the
problem of estimating
the
influence curve
T'
(F; x)
from a random sample
Xl,'
..
,X
n
of F. Three estimators have been con-
sidered in the literature:
the
sensitivity curve,
the
empirical influence curve,
and
the
jackknife approximation
[see
Hampel
et
al.
(1987), p.
92].
The
first
one
is
perhaps
the
most popular:
it
is
defined by
_
T((l
-
~)Fn-l
+
~t5x)
-
T(F
n
_
l
)
sc ( )
n x - 11n .
Curiously enough,
the
asymptotic properties of this estimate have received
little attention; maybe the reason
is
that
the influence curve
is
often used for
descriptive aims, in order to get a general idea of
the
behavior of
the
sequence
4