Shape without scale: an identifiability dichotomy for a bounded tail observed through a non-additive measurement kernel
Abstract
A latent severity has a bounded lower tail with density of shape alpha and scale L. It is observed only through a fixed Markov kernel K that is biased and non-additive. The relative conditional spread of K diverges at the endpoint. Our sample is i.i.d. from the marginal Q alone, with no anchoring covariate or instrument. We prove a dichotomy. The shape index alpha is identifiable: for every admissible choice of the class constants, any two observationally equivalent members of a lean class share alpha, determined by a near-endpoint expansion of Q. The rate, namely L and the fixed-scale exceedance p_tau, does not survive. There exist admissible shared class constants and two members of a smaller regularity class whose observed laws coincide exactly. Across the pair alpha agrees, whereas L and p_tau move. A degenerate Le Cam two-point bound excludes any uniformly consistent estimator of either, and pointwise consistency fails at one member. Only the rate needs an anchor. We conjecture that a known kernel family with known edge map identifies the rate fiber by fiber if and only if the family satisfies a fixed-scale injectivity clause, and we prove the sufficiency direction. In surrogate safety, uncalibrated conflict data give the shape of near-crash risk, not its absolute rate.
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BibTeX
@misc{qi2026shape,
title = {{Shape without scale: an identifiability dichotomy for a bounded tail observed through a non-additive measurement kernel}},
author = {Jiarui Qi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27384},
eprint = {2609.27384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27384},
}
RIS
TY - UNPB
TI - Shape without scale: an identifiability dichotomy for a bounded tail observed through a non-additive measurement kernel
AU - Jiarui Qi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.27384
UR - https://arxiv.org/abs/2609.27384
AB - A latent severity has a bounded lower tail with density of shape alpha and scale L. It is observed only through a fixed Markov kernel K that is biased and non-additive. The relative conditional spread of K diverges at the endpoint. Our sample is i.i.d. from the marginal Q alone, with no anchoring covariate or instrument. We prove a dichotomy. The shape index alpha is identifiable: for every admissible choice of the class constants, any two observationally equivalent members of a lean class share alpha, determined by a near-endpoint expansion of Q. The rate, namely L and the fixed-scale exceedance p_tau, does not survive. There exist admissible shared class constants and two members of a smaller regularity class whose observed laws coincide exactly. Across the pair alpha agrees, whereas L and p_tau move. A degenerate Le Cam two-point bound excludes any uniformly consistent estimator of either, and pointwise consistency fails at one member. Only the rate needs an anchor. We conjecture that a known kernel family with known edge map identifies the rate fiber by fiber if and only if the family satisfies a fixed-scale injectivity clause, and we prove the sufficiency direction. In surrogate safety, uncalibrated conflict data give the shape of near-crash risk, not its absolute rate.
ER -