In mathematics, the Hadamard derivative is a concept of
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
for maps between
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s. It is particularly suited for applications in
stochastic programming
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, ...
and
asymptotic statistics In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size may grow indefinitely; the properties of estimato ...
.
Definition
A map
between
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s
and
is Hadamard-directionally differentiable
at
in the direction
if there exists a map
such that
for all sequences
and
. Note that this definition does not require continuity or linearity of the derivative with respect to the direction
. Although continuity follows automatically from the definition, linearity does not.
Relation to other derivatives
* If the Hadamard directional derivative exists, then the
Gateaux derivative also exists and the two derivatives coincide.
* The Hadamard derivative is readily generalized for maps between
Hausdorff topological vector spaces.
Applications
A version of functional
delta method
In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.
History
The delta meth ...
holds for Hadamard directionally differentiable maps. Namely, let
be a sequence of random elements in a Banach space
(equipped with
Borel sigma-field) such that
weak convergence holds for some
, some sequence of real numbers
and some random element
with values concentrated on a separable subset of
. Then for a measurable map
that is Hadamard directionally differentiable at
we have
(where the weak convergence is with respect to Borel sigma-field on the Banach space
).
This result has applications in optimal inference for wide range of
econometric models, including models with
partial identification and weak
instruments.
See also
*
* - generalization of the total derivative
*
*
*
References
{{Analysis in topological vector spaces
Directional statistics
Generalizations of the derivative