In mathematics, the 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.
φ :
between Banach spaces
is Hadamard-directionally differentiable[2] at
if there exists a map
φ ( θ +
) − φ ( θ )
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.
A version of functional delta method holds for Hadamard directionally differentiable maps.
be a sequence of random elements in a Banach space
(equipped with Borel sigma-field) such that weak convergence
, 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
) − φ ( μ ) ) →
(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.