An Unbounded Gap Between Identification and Identifiable Elicitation: Central Moments
Abstract
For n≥2, two expectation equations identify the mean together with the nth central moment. How many coordinates must a report have if a one-observation loss uniquely elicits it and the report determines that moment? We prove that n coordinates are necessary and sufficient among all identifiable intermediaries. The result holds on every open convex set of positive distributions on any n+1 distinct real outcomes. It requires no regularity of the loss, report map, identification function, or link. Thus elicℐ(μn)=n while iden(μn)≤2, giving an unbounded gap between identifiable elicitation and indirect identification. This resolves the central-moment conjecture of Frongillo and Kash on the stated domains. The proof constructs a C1 manifold of supporting coefficients from the affine fibers of a convex negative Bayes risk. Translations along that manifold force an n-dimensional tangent space. Standardized moments of order at least three have the same exact complexity. The bound also extends to convex distribution classes with a full-dimensional raw-moment image.
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BibTeX
@misc{jia2026unbounded,
title = {{An Unbounded Gap Between Identification and Identifiable Elicitation: Central Moments}},
author = {Jingyi Jia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24009},
eprint = {2610.24009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Unbounded Gap Between Identification and Identifiable Elicitation: Central Moments
AU - Jingyi Jia
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.24009
UR - https://arxiv.org/abs/2610.24009
AB - For n≥2, two expectation equations identify the mean together with the nth central moment. How many coordinates must a report have if a one-observation loss uniquely elicits it and the report determines that moment? We prove that n coordinates are necessary and sufficient among all identifiable intermediaries. The result holds on every open convex set of positive distributions on any n+1 distinct real outcomes. It requires no regularity of the loss, report map, identification function, or link. Thus elic_ℐ(μ_n)=n while iden(μ_n)≤2, giving an unbounded gap between identifiable elicitation and indirect identification. This resolves the central-moment conjecture of Frongillo and Kash on the stated domains. The proof constructs a C^1 manifold of supporting coefficients from the affine fibers of a convex negative Bayes risk. Translations along that manifold force an n-dimensional tangent space. Standardized moments of order at least three have the same exact complexity. The bound also extends to convex distribution classes with a full-dimensional raw-moment image.
ER -