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Hadamard derivative

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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.

Contents

Definition

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.

Relation to other derivatives

  • 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
  • Applications

    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.

    References

    Hadamard derivative Wikipedia


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