bibkey: “fang2016directional” authors: “Zheng Fang; Andres Santos” year: 2016 title: “Inference on Directionally Differentiable Functions” doi: “10.1093/restud/rdy049” url: “https://arxiv.org/abs/1404.3763v2” claim: “Theorem 2.1 extends the delta method to Hadamard directionally differentiable maps and states the additional condition for a stochastic expansion.” strata_touched: [] license: “citation-only” triage: “anchor”

Directional limits and exact-capacity selection

The checked primary text is arXiv:1404.3763v2, submitted 2016-01-13, with a December 2015 draft date and 63 PDF pages. The year above identifies that version. The DOI identifies the subsequent journal article; its Crossref record gives online publication on 2018-09-11. The theorem numbering below refers to the checked preprint.

Assumptions 2.1–2.2, PDF page 11, require Banach spaces, Hadamard directional differentiability tangentially to a set containing the support of the tight scaled estimator limit, and convergence of that estimator. Theorem 2.1, PDF page 12, gives the transformed limiting law. Continuous extension of the derivative to the ambient space additionally yields its stochastic expansion. The theorem does not impose iid observations as an extra premise once the estimator convergence is established. The paper attributes the preceding directional delta method to Shapiro (1991) and Dümbgen (1993).

A continuous positively homogeneous map consisting of two different linear slopes on the positive and negative half-lines is covered by this classical mechanism. Its non-Gaussian transform of a Gaussian is literature-attested context, not a new general distributional principle. Chapter 31 of the parity fluctuation volume instead proves the model-specific approximation: the exact quota correction remains in the two adjacent lattice layers; actual fixed-size posterior labels have different limiting inclusion probabilities in those layers; and the remainder is small in first absolute mean. The last property transfers the exact finite minimax center and the paired-threshold mean difference. An expansion merely in probability would not supply this mean transfer.

Those actual-data localization, posterior and exact-center arguments are repo-derived ordinary mathematics. Classical discrete-quantile and trimmed order-statistic limits also exhibit one-sided Gaussian transformations. The inspected directional theorem does not supply the compensated stationary path/pair experiment, its quota calibration, or its posterior label correction. No globally new delta method, two-piece distribution, or general selection principle is claimed. The model’s joint process statement additionally uses the already established common-data limits in Chapters 29–30.