Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications in stochastic programming and asymptotic statistics.
A map
ϕ
:
D
→
E
between Banach spaces
D
and
E
is Hadamard directionally differentiable at
θ
∈
D
in the direction
h
∈
D
if there exists a map
ϕ
θ
′
:
D
→
E
such that
ϕ
(
θ
+
t
n
h
n
)
−
ϕ
(
θ
)
t
n
→
ϕ
θ
′
(
h
)
for all sequences
h
n
→
h
and
t
n
↓
0
. Note that this definition does not require continuity or linearity of the derivative with respect to the direction
h
. Although continuity follows automatically from the definition, linearity does not.
If Hadamard directional derivative exists, then Gâteaux derivative also exists and the two derivatives coincide
Hadamard derivative is readily generalized for maps between Hausdorff topological vector spaces
A version of functional delta method holds for Hadamard directionally differentiable maps. Namely, let
X
n
be a sequence of random elements in a Banach space
D
(equipped with Borel sigma-field) such that weak convergence
τ
n
(
X
n
−
μ
)
→
Z
holds for some
μ
∈
D
, some sequence of real numbers
τ
n
→
∞
and some random element
Z
∈
D
with values concentrated on a separable subset of
D
. Then for a measurable map
ϕ
:
D
→
E
that is Hadamard directionally differentiable at
μ
we have
τ
n
(
ϕ
(
X
n
)
−
ϕ
(
μ
)
)
→
ϕ
μ
′
(
Z
)
(where the weak convergence is with respect to Borel sigma-field on the Banach space
E
).
This result has applications in optimal inference for wide range of econometric models, including models with partial identification and weak instruments.
