arXiv subjects

R. Labouriau

Publications and source records attributed to R. Labouriau.

7 recordsLinked to original sources

arXiv2026

From Coefficients to Distributions: De~Moivre and the Operational View of Probability

We trace a conceptual genealogy from Abraham de Moivre's derivation of the normal curve (1733) to the modern distributional approach to statistics. De Moivre's Approximatio ad Summam Terminorum Binomii gave the first systematic derivation of the Gaussian density, its normalising constant (completed by Stirling's identification of $B = \sqrt{2\pi}$), and its tail probabilities computed to six decimal places -- more than seventy years before Gauss. His method -- extracting information from probability laws by evaluating sums against indicator probes -- is recognisably an instance of the operational viewpoint that underlies distributional statistics. We identify a four-stage chain: coefficient extraction (De Moivre) $\to$ generating functions (Euler, Laplace) $\to$ characteristic functions (Fourier, L\'evy) $\to$ distributional pairings $\langle T, \varphi \rangle$ (Schwartz). At each stage the probes become more flexible and the class of laws that can be studied grows wider. The distributional framework, in which a probability law is represented by a distribution--kernel pair $(T, \varphi) \in \mathcal{S}'(\mathbb{R}) \times \mathcal{S}(\mathbb{R})$, is the natural endpoint of this progression. We formulate and prove a distributional version of the De Moivre--Laplace theorem: the standardised binomial distribution converges to the Gaussian in $\mathcal{S}'(\mathbb{R})$, with De Moivre's original computation corresponding to the special case of indicator test functions. We also discuss the transversality framework, which provides a geometric explanation -- via infinite codimension of degeneracy strata -- for why pathologies such as moment indeterminacy, non-identifiability, and singular Fisher information are rarely encountered in parametric statistical models.

math.HO↗

arXiv2026

Inference Functionals and Observation Operators for Distributional Statistical Models

This paper generalises inference functions (Godambe, 1960) to distributional statistical models, in which each probability measure is represented by a distribution--kernel pair $(T_\theta, \varphi) \in \mathcal S'(\mathbb R) \times \mathcal S(\mathbb R)$. The generalisation is strategically motivated: the key properties of maximum likelihood estimation-consistency and asymptotic normality -derive not from maximising the likelihood but from the MLE being the root of a regular inference function. Extending inference functions to the distributional setting provides an optimality theory for models lacking classical densities or finite moments. The extension requires enlarging the notion of observation. We introduce observation operators $\mathcal O : \mathcal S'(\mathbb R) \to \mathcal Y$ mapping distributional models to an observation space, and define inference functionals as estimating equations composed with these operators. The framework encompasses classical point observations, interval-censored data, convolutional measurements, and transform-based statistics. We establish asymptotic theory (consistency, asymptotic normality, Godambe optimality) under mild conditions and derive a hierarchy of information bounds -- classical Fisher information dominates the information available through the observation operator, which in turn dominates the information captured by any inference functional -- via the H\'ajek--Le~Cam convolution theorem. The two gaps quantify distinct sources of information loss: the observation mechanism and the choice of inference functional. Examples include sinusoidal inference functions for heavy-tailed distributions, interval-censored location inference, elliptically contoured models, and nuisance parameters via the Bhapkar--Godambe projection.

math.ST↗

arXiv2026

Notes on Transversality and Statistical Degeneracies in Distributional Models

These notes provide a pedagogical introduction to the role of transversality theory in the analysis of statistical degeneracies within the framework of distributional statistical models. The classical question of when a statistical model is well-behaved - in the sense of being identifiable, having non-singular Fisher information, and admitting robust estimation - is reformulated as a question about the geometry of a kernel-induced feature map. Statistical pathologies correspond to geometric degeneracies of this map, and transversality theory provides a precise language for understanding when and why such degeneracies are non-generic. The exposition is organised in three parts. Part I surveys the statistical phenomena that motivate the geometric treatment: representation failure, non-identifiability, moment indeterminacy, singular information, nuisance parameters, and the Behrens-Fisher problem. Part II develops the necessary geometric toolkit - smooth maps, Sard's theorem, transversality, jets, stratifications, and the parametric transversality theorem - at a level accessible to students with a background in analysis and linear algebra but no prior exposure to differential topology. Part~III returns to the statistical problems of Part~I and shows how each one admits a unified geometric interpretation as a transversality condition on the feature map. These notes are a pedagogical companion to the research paper Labouriau (2026) "Transversality and Geometric Regularisation in Distributional Statistical Models" (arXiv:2605.04536 [math.ST]), expanding its arguments with motivating examples, geometric intuition, and exercises aimed at advanced Master's and PhD students with a background in mathematical statistics and measure theory. They are designed to support seminars or reading groups.

math.HO↗

arXiv2026

Transversality and Geometric Regularisation in Distributional Statistical Models

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.

math.ST↗

arXiv2026

Weak Moment Methods for Statistical Inference: with an Application to Robust Estimation

A companion paper develops a generalised framework in which a probability law is represented by a tempered distribution $T\in\mathcal{S}'$ - on the same footing as a density or characteristic function - and information is extracted by pairing $T$ with a positive Schwartz kernel $\varphi$ that acts as a measurement instrument rather than as part of the law; the resulting weak moments of all orders exist unconditionally. The present paper turns this into a methodology for statistical inference: estimation via weak moment matching, weak characteristic functions, weak cumulants, and regularised density reconstruction by Tikhonov inversion. Parametric inference proceeds directly from weak expectations, without reconstructing the density. The central result is that weak moment estimators are automatically locally robust in the sense of Hampel: their score is bounded and redescending, their influence function has a closed form, and their gross error sensitivity is finite in every identifiable parametric model - all inherited from the kernel's decay, with no ad hoc truncation. The kernel plays the role of Huber's tuning constant, but as a structural component of the model rather than a post-hoc modification. The framework is worked out for the Cauchy location model (where no classical moment estimator exists), a Student $t_3$ location-scale model, a bivariate Cauchy location model, a bivariate $t_3$ location-scale model, and a location of a moving atom (a non-dominated model). Monte Carlo comparisons show weak moment estimators matching or outperforming classical robust benchmarks under contamination; in the bivariate $t_3$ case the MLE scale estimate breaks down while the weak moment estimator converges at the parametric rate. The reconstruction route is inherently non-parametric and opens a path to weak density estimation.

stat.ME↗

arXiv2026

Distributional Statistical Models: Weak Moments, Cumulants, and a Central Limit Theorem

Many important statistical models fall outside classical moment-based methods due to the non-existence of moments or moment generating functions. We propose a generalised probabilistic framework in which a probability law is represented by a tempered distribution $T \in \mathcal{S}'$, on the same footing as a density, a distribution function, or a characteristic function. Information about the law is extracted by evaluating $T$ on test functions regularised by a given positive Schwartz kernel $\varphi \in \mathcal{S}$ -- the kernel serving as a probe, not as part of the law. Expectations are defined via the action of distributions on regularised test functions, yielding well-defined weak moments, weak characteristic functions, and weak cumulants of all orders. These extend classical quantities and retain key algebraic properties such as additivity under independence and natural affine transformation rules. The main results are: (i) a systematic algebra of weak cumulants; (ii) a weak moment problem where existence of all moments holds unconditionally and uniqueness depends on the kernel, with uniqueness results under Gaussian kernels (via Hermite completeness), positive Schwartz kernels with an exponential tail bound and square-integrable densities (via a Carleman-type criterion), and kernels with exponential decay (via Denjoy-Carleman quasi-analyticity); and (iii) a weak central limit theorem formulated as convergence of weak characteristic functions to a Gaussian limit, covering cases where the classical theorem fails. The framework is illustrated with Student's $t$, stable, and hyperbolic distributions. As a statistical consequence, the weak first moment yields a consistent estimator of the location parameter in the Cauchy model, where no classical moment-based estimator exists. A full statistical treatment is given in a companion paper.

math.PR↗

arXiv2021

Multivariate Methods for Detection of Rubbery Rot in Storage Apples by Monitoring Volatile Organic Compounds: An Example of Multivariate Generalised Mixed Models

This article is a case study illustrating the use of a multivariate statistical method for screening potential chemical markers for early detection of post-harvest disease in storage fruit. We simultaneously measure a range of volatile organic compounds (VOCs) and two measures of severity of disease infection in apples under storage: the number of apples presenting visible symptoms and the lesion area. We use multivariate generalised linear mixed models (MGLMM) for studying association patterns of those simultaneously observed responses via the covariance structure of random components. Remarkably, those MGLMMs can be used to represent patterns of association between quantities of different statistical nature. In the particular example considered in this paper, there are positive responses (concentrations of VOC, Gamma distribution based models), positive responses possibly containing observations with zero values (lesion area, Compound Poisson distribution based models) and binomially distributed responses (proportion of apples presenting infection symptoms). We represent patterns of association inferred with the MGLMMs using graphical models (a network represented by a graph), which allow us to eliminate spurious associations due to a cascade of indirect correlations between the responses.

stat.AP↗