STAT 583 SPRING 2008
Lecture Notes 1
Statistical Functionals
The Gâteux derivative of a statistical functional T(F) is the limit
d1T(F;G−F)=lim
↓0
T(F+(G−F)) −T(F).
If Q( ) =T(F+(G−F)) has a McLaurin expansion, we get an expansion (the von
Mises expansion) of T by noting that Q(0)=T(F), Q(1)=T(G), Q'(0+)=d1T(F;G−F)
etc., yielding
T(G)=T(F)+1
k!dkT(F;G−F)+Rm(G).
k=1
m
∑
We are usually particularly interested in G = Fn, and we write Rm(Fn)=Rmn.
T has a differential at F with respect to a norm
•
if there is a linear functional
˙
T (F;∆)
such that for all G
T(G)
−T(F)−˙
T (F;G−F)=o(G−F)
.
˙
T
is called the Frechét derivative of T.
Theorem 1: If T has a differential at F with respect to
•
, then for any G the Gâteux
derivative d
1T(F;G−F)
exists and equals
˙
T (F;G−F).
Theorem 2: Let T have a differential at F with respect to
•
. Let X1,...,Xn be observations
from F (not necessarily independent), such that n F
n−F=OP(1).
Then nR
1n→
P0.
Define the influence curve of T at F by h(F;x)
=d1(F,x−F),
where
x
is the cdf of
point mass at x.
Theorem 3: Suppose T has a linear derivative satisfying
(a) 0
<VarFh(F;X)< ∞
(b) nR
1n→
P0.
Define
(T,F)=EFh(F;X)
and
2(T,F)=VarFh(F;X)
. Then
T(F
n) ~ AsN(T(F)+(T,F), 2(T,F) / n)
.
