Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2204.13231

Asymptotic Inference for Infinitely Imbalanced Logistic Regression

Dorian Goldman·Bo Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we extend the work of Owen (2007) by deriving a second order expansion for the slope parameter in logistic regression, when the size of the majority class is unbounded and the minority class is finite. More precisely, we demonstrate that the second order term converges to a normal distribution and explicitly compute its variance, which surprisingly once again depends only on the mean of the minority class points and not their arrangement under mild regularity assumptions. In the case that the majority class is normally distributed, we illustrate that the variance of the the limiting slope depends exponentially on the z-score of the average of the minority class's points with respect to the majority class's distribution. We confirm our results by Monte Carlo simulations.

Explore related subjects

math.STstat.MLstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dorian Goldman, Bo Zhang. 2022-04-27. Asymptotic Inference for Infinitely Imbalanced Logistic Regression. https://arxiv.org/abs/2204.13231

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $ρ_0$, we prove in the infill asymptotics that the MLE is $n$-consistent, where $n$ denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in $ρ_0$, the analysis is highly non-standard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument $ρ$ is to $ρ_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neighborhood of $ρ_0$ asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in $ρ$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $ρ_0$.

math.ST↗
arXiv2026

Statistical models as natural transformations: meaningfulness, coherence and priors as states in Markov categories

We show that a statistical model in the sense of McCullagh, in the form given by Brøns, is a natural transformation between two functors from the category of designs to the Kleisli category Stoch of the Giry monad, provided that its components are measurable in the parameter. The condition is empty for finite models. A design-indexed quantity is a family of morphisms of Stoch defined on the parameter objects, called meaningful if it is natural. We prove that Tjur's criterion, imposed on parameter functions indexed by finite samples with multiplicities, forces the indexing by the support and then coincides with naturality over the insertions. For finite designs we show that a quantity can be corrected to a natural one within a given class of corrections if and only if a class vanishes in the first cohomology group of a Baues-Wirsching complex relative to that class, while its image in the absolute group is always zero. In the one-way layout, marginal dispersion is not meaningful, and within-group dispersion is the unique correction that leaves the merged design unchanged. A prior is a family of states on the parameter objects, called coherent over a class of design morphisms if it is natural over that class. We show that coherence at a merge confines the prior to the image of the corresponding parameter map, that coherence over the insertions is Kolmogorov consistency, and that coherence over the injections adds the exchangeability assumed by the categorical de Finetti theorem. In the finite one-way scheme, the coherent priors form polytopes of known dimension. The analogue of Jeffreys' general rule is not coherent, while the analogue for location-scale families is. Finally, we show that ridge regression is the Bayesian inversion of the Gaussian linear model with respect to a Gaussian prior, which is coherent over the insertions and never over the injections.

math.ST↗
arXiv2026

Inference for H{ü}sler-Reiss block models

Estimating the H{ü}sler-Reiss precision matrix is a fundamental problem for statistical inference in multivariate extremes. In high-dimensional settings, the number of unknown parameters grows quadratically with the dimension, making regularisation indispensable. Existing approaches regularise the estimation problem by exploiting sparsity. In this paper, we consider an alternative structural assumption, namely that the H{ü}sler-Reiss precision matrix is block-structured. To estimate such a matrix, we introduce a new regularisation framework based on a convex fusion penalty. By encouraging rows and columns to merge, this approach provides a parsimonious representation of the precision matrix, allowing for the simultaneous estimation of its coefficients and the underlying partition of the variables. The resulting convex optimisation problem is solved by an efficient algorithm combining gradient-based updates with progressive fusion steps. We establish non-asymptotic concentration bounds for the empirical weights entering the penalty and prove consistency of both block recovery and precision matrix estimation under suitable regularity conditions. Numerical experiments demonstrate that our methodology accurately recovers the latent block structure while accurately estimating the H{ü}sler-Reiss precision matrix across various configurations, illustrating the practical benefits of fusion-based regularisation for multivariate extremes. These benefits are also demonstrated by applying the proposed method to foreign exchange data.

math.ST↗