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 s ...
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 vector ...
s. It is particularly suited for applications in
stochastic programming 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 estimat ...
.
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 vector ...
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
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
also exists and the two derivatives coincide.
* The Hadamard derivative is readily generalized for maps between
Hausdorff topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s.
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 method ...
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 model Econometric models are statistical models used in econometrics. An econometric model specifies the statistical relationship that is believed to hold between the various economic quantities pertaining to a particular economic phenomenon. An econometr ...
s, including models with
partial identification In statistics and econometrics, set identification (or partial identification) extends the concept of identifiability (or "point identification") in statistical models to situations where the distribution of observable variables is not informative o ...
and weak
instruments
Instrument may refer to:
Science and technology
* Flight instruments, the devices used to measure the speed, altitude, and pertinent flight angles of various kinds of aircraft
* Laboratory equipment, the measuring tools used in a scientific lab ...
.
See also
*
* - generalization of the total derivative
*
*
*
References
{{Analysis in topological vector spaces
Directional statistics
Generalizations of the derivative