Frechet differentiability check

\(\|\phi(\theta+h) - \phi(\theta) - \phi'_\theta(h)\|/\|h\| \to 0\). Frechet needs ONE linear map valid in every direction, so the numerical derivative is a single Jacobian – recomputing a directional derivative per h would return the Hadamard derivative and report even a kinked map as Frechet differentiable. Mirrors morie.fn.ksr050.

Usage

morie_frechet_check(phi, theta, h_n)

Arguments

phi

functional.

theta

base point.

h_n

list of shrinking perturbations.

Value

list: ratios, norms, ratio_shrinking, derivative.

References

Kosorok (2008), Ch. 2. van der Vaart (1998), Asymptotic Statistics, Lemma 21.3 (the quantile map is Hadamard but not Frechet). Reeds, J. A. (1976), PhD thesis, Harvard.

Examples

morie_frechet_check(function(z) z^2, 1, list(0.1, 0.05, 0.01))$ratio_shrinking
#> [1] TRUE