arXiv subjects

Giacomo Francisci

Publications and source records attributed to Giacomo Francisci.

7 recordsLinked to original sources

arXiv2026

Central subspace data depth

Statistical data depth plays an important role in the analysis of multivariate data sets. The main outcome is a center-outward ordering of the observations that can be used both to highlight features of the underlying distribution of the data and as input to further statistical analysis. An important property of data depth is related to symmetric distributions as the point with the highest depth value, the center, coincides with the point of symmetry. However, there are applications in which it is more natural to consider symmetry with respect to a subspace of a certain dimension rather than to a point, i.e. a subspace of dimension zero. We provide a general framework to construct statistical data depths which attain maximum value in a subspace, providing a center-outward ordering from that subspace. We refer to these data depths as central subspace data depths. Moreover, if the distribution is symmetric with respect to a subspace, then the depth is maximized at that subspace. We introduce general notions of symmetry about a subspace for distributions, study the properties of central subspace data depths and provide asymptotic convergence for the corresponding sample versions. Additionally, we discuss connections with projection pursuit and dimension reduction. An application based on custom data fraud detection shows the importance of the proposed approach and strengthens its potential.

math.ST↗

arXiv2026

Estimation of Lévy-driven CARMA models under renewal sampling

Continuous-time autoregressive and moving average (CARMA) models are extensively used to model high-frequency and irregularly sampled data. We study Whittle estimation for the model parameters when the process is observed at renewal times. The driving noise is assumed to be a Lévy process allowing for more flexibility including heavy-tailed marginal distributions and jumps in the sample paths. We show that the Whittle estimator based on the integrated periodogram is consistent and asymptotically normal under very mild conditions. To obtain these results, we establish the asymptotic normality of the integrated periodogram.

math.ST↗

arXiv2024

Functional limit laws for the intensity measure of point processes and applications

Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform laws of large numbers and uniform central limit theorems over a class of bounded measurable functions for estimates of the intensity measure. Using these results, we derive the uniform asymptotic properties of half-space depth and, as corollaries, obtain the asymptotic behavior of medians and other quantiles of the standardized intensity measure. Additionally, we obtain uniform concentration upper bound for the estimator of half-space depth. As a consequence of our results, we also derive uniform consistency and uniform asymptotic normality of Lotka-Nagaev and Harris-type estimators for the Laplace transform of the point processes in a branching random walk.

math.PR↗

arXiv2024

Generalization of the simplicial depth: no vanishment outside the convex hull of the distribution support

The simplicial depth, like other relevant multivariate statistical data depth functions, vanishes right outside the convex hull of the support of the distribution with respect to which the depth is computed. This is problematic when it is required to differentiate among points outside the convex hull of the distribution support, with respect to which the depth is computed, based on their depth values. We provide the first proposal for simplicial depth which do not vanish right outside the convex hull of the distribution. The properties of the proposal and of the corresponding estimator are studied theoretically and by means of Monte Carlo simulations and analysis of datasets.

math.ST↗

arXiv2023

Branching Processes in Random Environments with Thresholds

Motivated by applications to COVID dynamics, we describe a branching process in random environments model $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(τ_j, ν_j)\}$ - specifically the values of the process at crossing times, {\it{viz.}}, $\{(Z_{τ_j}, Z_{ν_j})\}$ - along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distribution of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.

math.PR↗

arXiv2022

Analytical and statistical properties of local depth functions motivated by clustering applications

Local general depth ($LGD$) functions are used for describing the local geometric features and mode(s) in multivariate distributions. In this paper, we undertake a rigorous systematic study of $LGD$ and establish several analytical and statistical properties. First, we show that, when the underlying probability distribution is absolutely continuous with density $f(\cdot)$, the scaled version of $LGD$ (referred to as $τ$-approximation) converges, uniformly and in $L^d(\mathbb{R}^p)$ to $f(\cdot)$ when $τ$ converges to zero. Second, we establish that, as the sample size diverges to infinity the centered and scaled sample $LGD$ converge in distribution to a centered Gaussian process uniformly in the space of bounded functions on $\mathcal{H}_G$, a class of functions yielding $LGD$. Third, using the sample version of the $τ$-approximation ($S τA$) and the gradient system analysis, we develop a new clustering algorithm. The validity of this algorithm requires several results concerning the uniform finite difference approximation of the gradient system associated with $S τA$. For this reason, we establish \emph{Bernstein}-type inequality for deviations between the centered and scaled sample $LGD$, which is also of independent interest. Finally, invoking the above results, we establish consistency of the clustering algorithm. Applications of the proposed methods to mode estimation and upper level set estimation are also provided. Finite sample performance of the methodology are evaluated using numerical experiments and data analysis.

math.ST↗

arXiv2021

Statistical depth in abstract metric spaces

The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid development of technology, in particular the advent of `Big Data', we extend here that concept to general metric spaces, propose a natural depth measure and explore its properties as a statistical depth function. Working in a general metric space allows the depth to be tailored to the data at hand and to the ultimate goal of the analysis, a very desirable property given the polymorphic nature of modern data sets. This flexibility is thoroughly illustrated by several real data analyses.

stat.ME↗