{
  "abstract": "Higher-order influence functions, introduced in a series of articles (Robins et al., 2008, 2009a; van der Vaart, 2014; Robins et al., 2016, 2023; Liu et al., 2017), are a unified framework for constructing rate-optimal point estimates of a class of statistical functionals under various complexity-reducing assumptions on the posited statistical model that generates the observed data. Although higher-order (influence functions) estimators are theoretically appealing, they have very limited practical uptake compared to their first-order counterparts. The original higher-order estimators proposed in Robins et al. (2008) and Robins et al. (2017) involve nonparametric density estimation of multi-dimensional covariates, a highly nontrivial statistical and computational problem on its own. The density estimator is, in turn, used in the evaluation of the inverse population Gram matrix $Ω$ of a set of $k$-dimensional basis transformations of covariates. There, $k$ is allowed to be as large as $o (n^2)$. To partially address this potential shortcoming, Liu et al. (2017) restrict $k$ to $o (n)$ and instead estimate $Ω$ directly using the inverse sample Gram matrix estimator, but computed from an independent sample often obtained by sample-splitting. Liu et al. (2017) refer to this alternative estimator as the empirical higher-order estimator. Although the empirical higher-order estimator bypasses density estimation, it suffers from numerical instability due to inverting a large-dimensional sample Gram matrix. In this article, for a class of bilinear forms/functionals that often appear in substantive fields, we propose a new stabilized higher-order estimator without sample splitting, which exhibits more stable finite-sample performance compared to the empirical higher-order estimator. We also prove that this new class of higher-order estimators enjoys similar statistical guarantees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04743",
  "author_total": 4,
  "authors": [
    {
      "name": "Na Liu"
    },
    {
      "name": "Chang Li"
    },
    {
      "name": "Yujia Gu"
    },
    {
      "name": "Lin Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/39fnrac4a6mjn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04743",
  "fulltext_url": null,
  "id": "39fnrac4a6mjn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04743",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "econ.EM"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
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      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Causal Inference Techniques"
    ]
  ],
  "openalex": "W7167628544",
  "pdf_url": "https://arxiv.org/pdf/2607.04743",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/39fnrac4a6mjn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.04743",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04743"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms",
  "title_text": "Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/39fnrac4a6mjn",
  "venue": "arXiv",
  "versions": []
}