We proof that Hadamard differentiability in addition with usual assumptions on the loss function for M estimates implies differentiability in quadratic mean. Thus both concepts are exchangeable. --Hadamard differential,Differentiability in quadratic mean

Tinkl, Fabian

English

Research Papers in Economics

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Tinkl, Fabian
Working Paper
A note on Hadamard differentiability
and differentiability in quadratic mean
IWQW discussion paper series, No. 08/2010
Provided in cooperation with:
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Suggested citation: Tinkl, Fabian (2010) : A note on Hadamard differentiability and
differentiability in quadratic mean, IWQW discussion paper series, No. 08/2010, http://
hdl.handle.net/10419/41556_____________________________________________________________________ 
 
Friedrich-Alexander-Universität 
IWQW 
Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung 
 
 
IWQW 
 
Institut für Wirtschaftspolitik und Quantitative 
Wirtschaftsforschung 
 
 
Diskussionspapier 
Discussion Papers 
 
No. 8/2010 
 
 
A note Hadamard differentiability and differentiability 
in quadratic mean 
 
 
 
 
 
 
 
Fabian Tinkl 
University of Erlangen-Nuremberg 
 
 
 
 
 
 
 
 
ISSN 1867-6707 A note on Hadamard di￿erentiability and di￿erentiability in
quadratic mean
Working Paper, 21. Oktober 2010
Fabian Tinkl1
University of Erlangen-Nuremberg
Fabian.Tinkl@wiso.uni-erlangen.de
Keywords and phrases: Hadamard di￿erential, Di￿erentiability in quadratic mean
Abstract
We proof that Hadamard di￿erentiability in addition with usual assumptions on the
loss function for M estimates implies di￿erentiability in quadratic mean. Thus both
concepts are exchangeable.
1Correspondence author: Fabian Tinkl, Department of Statistics and Econometrics, University
of Erlangen-Nuremberg, Lange Gasse 20, D-90403 Nuremberg, E-Mail: Fabian.Tinkl@wiso.uni-
erlangen.de
11 Introduction
As it was pointed out by [Pollard, 1997] there is a connection between di￿erentia-
bility in an L2 sense, di￿erentiability in quadratic mean and local asymptotic nor-
mality. The idea of local asymptotic normality (LAN) goes back to [LeCam, 1970],
who showed that di￿erentiability in quadratic mean implies an quadratic approx-
imation for the log-likelihood function, without using the 2nd derivative. We will
show, how a di￿erent concept of di￿erentiability, namely the Hadamard di￿eren-
tials, are under mild regularity conditions equivalent to the de￿nition of di￿erentials
in quadratic mean, and therefore the results of [Pollard, 1997], [LeCam, 1970] and
[Fernholz, 1983] will coincide for M estimates. Thus it can be seen, that it is also
su￿cient for M estimates to be Hadamard di￿erentiable to enjoy the LAN property.
2 The equivalence Theorem
Throughout this article we regard an -￿nite probability space and a sampling Xi
(not necessarily iid.) from a measurable random variable X : 
 ! R, where X has
a distribution function, which is absolutely continuous w.r.t. some -￿nite measure.
De￿ne a M estimate as the solution of the following minimization problem
^ n = argmin
2
1
n
n X
i=1
(Xi;); (1)
where  is some unknown parameter, where   , and  is some metric space with
the uniform metric. Under standard conditions, that  ! () is continuous and 
is compact as well as  is uniformly integrable w.r.t. , we get that ^ n
P ! 0, where
0 = argmin
2
E[(X;)]: (2)
We suppress the dependence of  on ! 2 
 from now on, and treat  as a function
of the parameter(-vector) . If  is di￿erentiable w.r.t. to , one may take a ￿rst-
order Taylor series expansion around ^ n in neighborhood of 0, where (2) attains its
minimum:
(^ n) = (0) + (^ n   0)
T
0
(0) + Rem()(^ n   0): (3)
Suppose that Rem ! 0 in some proper sense. If Rem
L2
! 0, i.e. E[jRem( ^ n 0)j2] =
0, we call  di￿erentiable in quadratic mean (DQM). If  is Hadamard di￿erentiable,
2then Rem
P ! 0, see [Fernholz, 1983]. Under usual regularity conditions on  and
its derivative we can conclude, that
p
n(^ n  0) ! N(0;2), with 2 = var((X)),
and  is the in￿uence function. De￿ne Hadamard di￿erentials as follows (see
[van der Vaart, 1998]):
De￿nition 1 A function  : D ! W is Hadamard di￿erentiable (or compact
di￿erentiable) at  2 D if there exists 0
 2 L1(D;W) such that for any compact
set   of D,
lim
t!0
   
( + tht)   ()
t
  
0
(h)

  
W
! 0; (4)
for every ht ! h, s.t.  + tht 2 D for t > 0 and t small enough.
One may write instead of (4):
Rem1( ^ n   0) = (^ n)   (0)   (^ n   0)
T
0
(0) (5)
as the remainder term of (3). Note that 0
 indicates the Hadamard derivative. It is
shown by [Fernholz, 1983], that if
lim
!0
Rem1(   0)
j   0j
= 0;
then:
p
nRem1(   0) = oP(1). DQM can be de￿ned using (3):
De￿nition 2 () is said to be DQM if there exists a function D 2 L2(P), s.t.
lim
!0
k()   (0)   (   0)TD(0)k2
k   0k
= 0 (6)
Once again we write for (3):
Rem2( ^ n   0) = (^ n)   (0)   (^ n   0)
TD(0) (7)
We now state the result:
Theorem 1 Suppose () is Hadamard di￿erentiable w.r.t.  in a neighborhood N
of 0, supE[j()j] < 1 and supE[2()] < 1 for  2 N and ^ n is a consistent
estimate for 0, then () is also di￿erentiable in quadratic mean, and both derivative
coincide on N.
3Proof Write t = ^ n   0,  = 0 and ht ! 0 in (4), so we can rewrite (4):
lim
^ n!0
    
(^ n)   (0)   (^ n   0)T0
(0)
^    0
    
W
= 0 (8)
Because ^ n
P ! 0 is a Cauchy sequence we have for ^ n 2 N, that j^ n   0j ! 0 and
we get expression (6). We now have to show, that both remainder terms converge
in a L2 sense, as well as in probability to zero. Then from (3) we can deduce
that Rem1 = Rem2 a.s. and therefore both derivatives must coincide a.s.. From
the assumptions imposed on  we can conclude that  is uniformly integrable (see
corollary 6.22 in [Klenke, 2006]). If ^ n
P ! 0 we have that
p
nRem1(^ n 0) = oP(1)
and
p
nRem2(^ n   0)
L2
! 0. As () 2 L2(P) it follows with continuity of ()
w.r.t.  that h = j0
j 2 L2, where 0
 is the Hadamard derivative. Therefore we
have Rem1(^ n   0) 2 L2 for n large enough. We now have: Rem1 is uniformly
integrable, and Rem1
P ! 0 and thus
j
p
nRem1(^ n   0)j
2 n!1 ! j0j
2
Write g = j
p
nRem1(^ n   0)   0j2, then gn
P ! 0 and gn is uniformly integrable, as
gn = j
p
nRem1(^ n   0)j
2  4  joP(1)j
2:
Therefore gn 2 L2(P) and thus D = 0
 a.s. in N.

3 Conclusion
We proved the equivalence of Hadamard di￿erentiability and di￿erentiability in
quadratic mean for M estimates usually considered in robust statistics. This re-
sult can be used to deduce alternative assumptions imposed on M estimators to be
consistent and asymptotically normally distributed.
References
[Fernholz, 1983] Fernholz, L. (1983). Von Mises calculus for statistical functionals .
New York: Springer-Verlag.
4[Klenke, 2006] Klenke, A. (2006). Wahrscheinlichkeitstheorie . Berlin: Springer-
Verlag.
[LeCam, 1970] LeCam, L. (1970). On the assumptions used to prove asymptotic
normality of maximum likelihood estimates. The Annals of Statistics, 41, 802￿
828.
[Pollard, 1997] Pollard, D. (1997). Another look at di￿erentiability in quadratic
mean. In Festschrift for Lucien Le Cam: Research papers in probability and
statistics (pp. 305￿314). Springer-Verlag.
[van der Vaart, 1998] van der Vaart, A. (1998). Asymptotic statistics. Cambridge:
Cambridge University Press.
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