Hadamard derivative

A map $\backslash varphi\; :\; \backslash mathbb\{D\}\backslash to\; \backslash mathbb\{E\}$ between Banach spaces $\backslash mathbb\{D\}$ and $\backslash mathbb\{E\}$ is Hadamard-directionally differentiable{{Cite journal|last=Shapiro|first=Alexander|year=1991|title=Asymptotic analysis of stochastic programs|journal=Annals of Operations Research|volume=30|issue=1|pages=169–186|doi=10.1007/bf02204815|s2cid=16157084}} at $\backslash theta\; \backslash in\; \backslash mathbb\{D\}$ in the direction $h\; \backslash in\; \backslash mathbb\{D\}$ if there exists a map $\backslash varphi\_\backslash theta\text{'}:\; \backslash ,\; \backslash mathbb\{D\}\; \backslash to\; \backslash mathbb\{E\}$ such that

$$\backslash frac\{\backslash varphi(\backslash theta+t\_n\; h\_n)-\backslash varphi(\backslash theta)\}\{t\_n\}\; \backslash to\; \backslash varphi\_\backslash theta\text{'}(h)$$

for all sequences $h\_n\; \backslash to\; h$ and $t\_n\; \backslash to\; 0$.

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 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.

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 $\backslash mathbb\{D\}$ (equipped with Borel sigma-field) such that weak convergence $\backslash tau\_n\; (X\_n-\backslash mu)\; \backslash to\; Z$ holds for some $\backslash mu\; \backslash in\; \backslash mathbb\{D\}$, some sequence of real numbers $\backslash tau\_n\backslash to\; \backslash infty$ and some random element $Z\; \backslash in\; \backslash mathbb\{D\}$ with values concentrated on a separable subset of $\backslash mathbb\{D\}$. Then for a measurable map $\backslash varphi:\; \backslash mathbb\{D\}\backslash to\backslash mathbb\{E\}$ that is Hadamard directionally differentiable at $\backslash mu$ we have $\backslash tau\_n\; (\backslash varphi(X\_n)-\backslash varphi(\backslash mu))\; \backslash to\; \backslash varphi\_\backslash mu\text{'}(Z)$ (where the weak convergence is with respect to Borel sigma-field on the Banach space $\backslash mathbb\{E\}$).

This result has applications in optimal inference for wide range of econometric models, including models with partial identification and weak instruments.{{Cite arXiv |last1=Fang|first1=Zheng|last2=Santos|first2=Andres|year=2014|title=Inference on directionally differentiable functions|eprint=1404.3763|class=math.ST}}

• {{annotated link|Directional derivative}}
• {{annotated link|Fréchet derivative}} - generalization of the total derivative
• {{annotated link|Gateaux derivative}}
• {{annotated link|Generalizations of the derivative}}
• {{annotated link|Total derivative}}

{{reflist}}

{{Analysis in topological vector spaces}}

Category:Directional statistics

Category:Generalizations of the derivative