Abstract:The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, \varphi)$, where $T$ is a tempered distribution and $\varphi$ is a rapidly decaying kernel. We develop the thesis that the kernel acts as a geometric regulariser, placing parametric statistical models in generic (transversal) position relative to degeneracy loci encoding non-identifiability, singular information, moment indeterminacy, and representation failure.
Using the transversality theorems of Whitney, Thom, and Mather, we prove a finite-dimensional weak transversality theorem: for a generic kernel in any sufficiently rich family, the kernel-induced feature map avoids degeneracy strata of sufficiently high codimension. We establish verifiable conditions -- formulated as rank conditions on the Jacobian of the joint feature map -- under which the transversality hypothesis can be checked, and verify them for location families, the log-normal, Stein discrepancies, and graphical models.
The present results apply to parametric models; extensions to semiparametric and nonparametric settings are discussed. The degeneracy classification includes representation degeneracy (Type 0) for models without closed-form densities and higher-order instabilities (Type IV) in non-chordal graphical models. Identifiability, robustness, moment determinacy, Fisher information regularity, Stein discrepancy, inferential separation, and the Behrens-Fisher problem all admit a unified geometric interpretation as transversality conditions on the feature map. This paper serves as a geometric companion to a series of papers developing the distributional framework.
| Comments: | 22 pages, no figures no tables. In the second version some sketches were replaced by proofs, an example of M-determinancy was added |
| Subjects: | Statistics Theory (math.ST); Differential Geometry (math.DG); Methodology (stat.ME) |
| MSC classes: | 62B05, 62F35, 57R45, 46F05, 53B12 |
| Cite as: | arXiv:2605.04536 [math.ST] |
| (or arXiv:2605.04536v2 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2605.04536 arXiv-issued DOI via DataCite |
From: Rodrigo Labouriau [view email]
[v1]
Wed, 6 May 2026 06:24:41 UTC (21 KB)
[v2]
Sun, 24 May 2026 19:16:59 UTC (25 KB)
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