arXiv subjects

Gilles Germain

Publications and source records attributed to Gilles Germain.

3 recordsLinked to original sources

arXiv2026

The general regularisation scheme applied to conditional density estimation

The general regularisation scheme, a versatile approach for nonparametric estimation, has been successfully applied to regression, density ratio, and score estimation. In this paper, we introduce a unified framework encompassing these settings and extend it to conditional density estimation, deriving a new estimator with rigorously established convergence rates. We implement the Landweber regularisation, which is computationally more tractable than Tikhonov regularisation in this context. Numerical experiments demonstrate that our estimator matches or outperforms the Nadaraya-Watson estimator in various scenarios, including time series models.

math.ST↗

arXiv2023

One-dimensional Stein's method with bespoke derivatives

We introduce a version of Stein's method of comparison of operators specifically tailored to the problem of bounding the Wasserstein-1 distance between continuous and discrete distributions on the real line. Our approach rests on a new family of weighted discrete derivative operators, which we call bespoke derivatives. We also propose new bounds on the derivatives of the solutions of Stein equations for Integrated Pearson random variables; this is a crucial step in Stein's method. We apply our result to several examples, including the Central Limit Theorem, Polya-Eggenberger urn models, the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator, the stationary distribution for the number of genes in the Moran model, and the stationary distribution of the Erlang-C system. Whenever our bounds can be compared with bounds from the literature, our constants are sharper.

math.PR↗

arXiv2022

A note on one-dimensional Poincaré inequalities by Stein-type integration

We study the weighted Poincaré constant $C(p,w)$ of a probability density $p$ with weight function $w$ using integration methods inspired by Stein's method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on $C(p,w)$ in terms of the so-called Stein kernel of $p$. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincaré constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectral problem. Our results rely on the properties of a pseudo inverse operator of the classical Sturm-Liouville operator. We illustrate our methods on a variety of examples: Gaussian functionals, weighted Gaussian, beta, gamma, Subbotin, and Weibull distributions.

math.PR↗