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Antonio Di Noia

Publications and source records attributed to Antonio Di Noia.

10 recordsLinked to original sources
arXiv2026

Asymptotic theory for nonparametric testing of $k$-monotonicity in discrete distributions

In shape-constrained nonparametric inference, it is often necessary to perform preliminary tests to verify whether a probability mass function (p.m.f.) satisfies qualitative constraints such as monotonicity, convexity, or in general $k$-monotonicity. In this paper, we are interested in nonparametric testing of $k$-monotonicity of a finitely supported discrete distribution. We consider a unified testing framework based on a natural statistic which is directly derived from the very definition of $k$-monotonicity. The introduced framework allows us to design a new consistent method to select the unknown knot points that are required to consistently approximate the limit distribution of several test statistics based either on the empirical measure or the shape-constrained estimators of the p.m.f. We show that the resulting tests are asymptotically valid and consistent for any fixed alternative. Additionally, for the test based solely on the empirical measure, we study the asymptotic power under contiguous alternatives and derive a quantitative separation result that provides sufficient conditions to achieve a given power. We employ this test to design an estimator for the largest parameter $k \in \mathbb N_0$ such that the p.m.f. is $j$-monotone for all $j = 0, \ldots, k$, and show that the estimator is different from the true parameter with probability which is asymptotically smaller than the nominal level of the test. Finally, we conduct an extensive simulation study to validate the theory, assess the finite-sample performance of the proposed methods, and illustrate them on several real datasets.

math.ST↗
arXiv2026

Scale adaptive and robust intrinsic dimension estimation via optimal neighbourhood identification

The Intrinsic Dimension (ID) is a key concept in unsupervised learning and feature selection, as it is a lower bound to the number of variables which are necessary to describe a system. However, in almost any real-world dataset the ID depends on the scale at which the data are analysed. Quite typically at a small scale, the ID is very large, as the data are affected by measurement errors. At large scale, the ID can also appear erroneously large, due to the curvature and the topology of the manifold containing the data. In this work, we introduce an automatic protocol to select the sweet spot, namely the correct range of scales in which the ID is meaningful and useful. This protocol is based on imposing that for distances smaller than the correct scale the density of the data is constant. In the presented framework, to estimate the density it is necessary to know the ID, therefore, this condition is imposed self-consistently. We illustrate the usefulness and robustness of this procedure to noise by benchmarks on artificial and real-world datasets.

stat.ML↗
arXiv2026

Likelihood distortion and Bayesian local robustness

Robust Bayesian analysis has been mainly devoted to detecting and measuring robustness w.r.t. the prior distribution. Many contributions in the literature aim to define suitable classes of priors which allow the computation of variations of quantities of interest while the prior changes within those classes. The literature has devoted much less attention to the robustness of Bayesian methods w.r.t. the likelihood function due to mathematical and computational complexity, and because it is often arguably considered a more objective choice compared to the prior. In this contribution, we propose a new approach to Bayesian local robustness, mainly focusing on robustness w.r.t. the likelihood function. Successively, we extend it to account for robustness w.r.t. the prior, as well as the prior and the likelihood jointly. This approach is based on the notion of distortion function introduced in the literature on risk theory. The novel robustness measure is a local sensitivity measure that turns out to be very tractable and easy to compute for several classes of distortion functions. Asymptotic properties are derived, and numerical experiments illustrate the theory and its applicability for modelling purposes.

math.ST↗
arXiv2026

Deconvolution in unlinked linear models

Unlinked regression, in which covariates and responses are observed separately without known correspondence, has recently gained increasing attention. Deconvolution, on the other hand, is a fundamental and challenging problem in nonparametric statistics with the aim of estimating the distribution of a latent random variable $Z$ based on observations contaminated by some additive noise. The complexity of this task is heavily influenced by the smoothness of the noise distribution and often leads to slow estimation rates. In this paper, we combine the recent unlinked linear regression problem with the classical deconvolution framework. Specifically, we study nonparametric deconvolution under the assumption that $Z$ is a linear function of an observable multidimensional covariate. This structural constraint allows us to introduce a nonparametric estimator of the distribution of $Z$ which achieves the parametric rate of convergence in the Wasserstein distance of order 1, where the smoothness of the noise does not affect the rate. Furthermore, we introduce nonparametric estimators for the unconditional density of $Z$ and the conditional density of $Z$ given an observed response. This allows us to study the problem of estimating the value of the latent linear predictor, whose link to the observed response is not accessible. Through several simulations, we illustrate the fast convergence rate of our deconvolution estimator and the performance of the proposed conditional estimators of the latent predictor in different simulation scenarios.

math.ST↗
arXiv2026

A general framework for adaptive nonparametric dimensionality reduction

Dimensionality reduction is a fundamental task in modern data science. Several projection methods specifically tailored to take into account the non-linearity of the data via local embeddings have been proposed. Such methods are often based on local neighbourhood structures and require tuning the number of neighbours that define this local structure, and the dimensionality of the lower-dimensional space onto which the data are projected. Such choices critically influence the quality of the resulting embedding. In this paper, we exploit a recently proposed intrinsic dimension estimator which also returns the optimal locally adaptive neighbourhood sizes according to some desirable criteria. In principle, this adaptive framework can be employed to perform an optimal hyper-parameter tuning of any dimensionality reduction algorithm that relies on local neighbourhood structures. Numerical experiments on both real-world and simulated datasets show that the proposed method can be used to significantly improve well-known projection methods when employed for various learning tasks, with improvements measurable through both quantitative metrics and the quality of low-dimensional visualizations.

stat.ML↗
arXiv2025

Estimation and goodness-of-fit testing for non-negative random variables with explicit Laplace transform

Many flexible families of positive random variables exhibit non-closed forms of the density and distribution functions and this feature is considered unappealing for modelling purposes. However, such families are often characterized by a simple expression of the corresponding Laplace transform. Relying on the Laplace transform, we propose to carry out parameter estimation and goodness-of-fit testing for a general class of non-standard laws. We suggest a novel data-driven inferential technique, providing parameter estimators and goodness-of-fit tests, whose large-sample properties are derived. The implementation of the method is specifically considered for the positive stable and Tweedie distributions. A Monte Carlo study shows good finite-sample performance of the proposed technique for such laws.

math.ST↗
arXiv2024

A family of consistent normally distributed tests for Poissonity

A family of consistent tests, derived from a characterization of the probability generating function, is proposed for assessing Poissonity against a wide class of count distributions, which includes some of the most frequently adopted alternatives to the Poisson distribution. Actually, the family of test statistics is based on the difference between the plug-in estimator of the Poisson cumulative distribution function and the empirical cumulative distribution function. The test statistics have an intuitive and simple form and are asymptotically normally distributed, allowing a straightforward implementation of the test. The finite sample properties of the test are investigated by means of an extensive simulation study. The test shows satisfactory behaviour compared to other tests with known limit distribution.

math.ST↗
arXiv2024

Bayesian size-and-shape regression modelling

Building on Dryden et al. (2021), this note presents the Bayesian estimation of a regression model for size-and-shape response variables with Gaussian landmarks. Our proposal fits into the framework of Bayesian latent variable models and allows a highly flexible modelling framework.

stat.ME↗
arXiv2024

Goodness-of-fit test for count distributions with finite second moment

A goodness-of-fit test for one-parameter count distributions with finite second moment is proposed. The test statistic is derived from the $L^1$ distance of a function of the probability generating function of the model under the null hypothesis and that of the random variable actually generating data, when the latter belongs to a suitable wide class of alternatives. The test statistic has a rather simple form and it is asymptotically normally distributed under the null hypothesis, allowing a straightforward implementation of the test. Moreover, the test is consistent for alternative distributions belonging to the class, but also for all the alternative distributions whose probability of zero is different from that under the null hypothesis. Thus, the use of the test is proposed and investigated also for alternatives not in the class. The finite-sample properties of the test are assessed by means of an extensive simulation study.

math.ST↗