arXiv subjects

Gerardo Sanz

Publications and source records attributed to Gerardo Sanz.

6 recordsLinked to original sources

arXiv2026

Estimating the tail index of Pareto-type distributions from geometric records

In this paper, we develop a novel inferential approach based on geometric records for estimating the tail index of heavy-tailed distributions. We construct a maximum likelihood estimator for the Pareto model and establish strong consistency and asymptotic normality, providing also an explicit expression for the asymptotic variance. These results are then extended to a broad class of Pareto-type distributions. The performance of the estimator is assessed via Monte Carlo simulation and compared with classical estimators from the literature. The proposed method is particularly well suited for settings where data arrive sequentially, as it yields smooth estimation trajectories. It is also especially advantageous in applications such as destructive testing, where measuring each item is costly. In this context, the estimator achieves a comparable level of estimation accuracy to Hill's estimator, but with a considerably lower number of fully measured items. An application to the analysis of the distribution of fluctuations of the Dow Jones Industrial Average (DJI) is also presented.

math.ST↗

arXiv2025

Characterisation of distributions via record-like observations

We characterise probability distributions via a martingale property associated with a natural generalisation of record values, known as $δ$-records. For an independent and identically distributed sequence $(X_n)$ with running maximum $M_n$, let $N_n$ be the number of $δ$-records (those $X_k$ with $X_k>M_{k-1}+δ$). We determine distributions for which $N_n-cM_n$ is a martingale, and show that this property uniquely determines the underlying distribution within broad classes. We show that the problem can be reformulated in terms of a delay-integrated Cauchy functional equation. A distinctive feature of this equation is that it is required to hold on a set that depends on the unknown distribution itself, which both complicates the analysis and allows for a rich variety of solutions. A complete characterisation is obtained when $δ<0$. For $δ>0$, all solutions with bounded support are identified. In the case of $δ>0$ and unbounded support, we consider both continuous and lattice distributions. In the continuous case, the characterisation reduces to a delay differential equation, which admits classical exponential-type solutions as well as broader families, including mixtures of exponential and gamma distributions. An analogous discrete analysis leads to difference equations whose solutions include mixtures of geometric and negative binomial distributions. In particular, this yields a new characterisation of the geometric distribution based on weak records.

math.PR↗

arXiv2025

Stochastic ordering, attractiveness and couplings in non-conservative particle systems

We analyse the stochastic comparison of interacting particle systems allowing for multiple arrivals, departures and non-conservative jumps of individuals between sites. That is, if $k$ individuals leave site $x$ for site $y$, a possibly different number $l$ arrive at destination. This setting includes new models, when compared to the conservative case, such as metapopulation models with deaths during migrations. It implies a sharp increase of technical complexity, given the numerous changes to consider. Known results are significantly generalised, even in the conservative case, as no particular form of the transition rates is assumed. We obtain necessary and sufficient conditions on the rates for the stochastic comparison of the processes and prove their equivalence with the existence of an order-preserving Markovian coupling. As a corollary, we get necessary and sufficient conditions for the attractiveness of the processes. A salient feature of our approach lies in the presentation of the coupling in terms of solutions to network flow problems. We illustrate the applicability of our results to a flexible family of population models described as interacting particle systems, with a range of parameters controlling births, deaths, catastrophes or migrations. We provide explicit conditions on the parameters for the stochastic comparison and attractiveness of the models, showing their usefulness in studying their limit behaviour. Additionally, we give three examples of constructing the coupling.

math.PR↗

arXiv2025

Estimating hazard rates from $δ$-records in discrete distributions

This paper focuses on nonparametric statistical inference of the hazard rate function of discrete distributions based on $δ$-record data. We derive the explicit expression of the maximum likelihood estimator and determine its exact distribution, as well as some important characteristics such as its bias and mean squared error. We then discuss the construction of confidence intervals and goodness-of-fit tests. The performance of our proposals is evaluated using simulation methods. Applications to real data are given, as well. The estimation of the hazard rate function based on usual records has been studied in the literature, although many procedures require several samples of records. In contrast, our approach relies on a single sequence of $δ$-records, simplifying the experimental design and increasing the applicability of the methods.

math.ST↗

arXiv2020

Exact and asymptotic properties of $δ$-records in the linear drift model

The study of records in the Linear Drift Model (LDM) has attracted much attention recently due to applications in several fields. In the present paper we study $δ$-records in the LDM, defined as observations which are greater than all previous observations, plus a fixed real quantity $δ$. We give analytical properties of the probability of $δ$-records and study the correlation between $δ$-record events. We also analyse the asymptotic behaviour of the number of $δ$-records among the first $n$ observations and give conditions for convergence to the Gaussian distribution. As a consequence of our results, we solve a conjecture posed in J. Stat. Mech. 2010, P10013, regarding the total number of records in a LDM with negative drift. Examples of application to particular distributions, such as Gumbel or Pareto are also provided. We illustrate our results with a real data set of summer temperatures in Spain, where the LDM is consistent with the global-warming phenomenon.

math.ST↗

arXiv2007

Asymptotic normality for the counting process of weak records and δ-records in discrete models

Let $\{X_n,n\ge1\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $δ$-record if $X_n>\max\{X_1,...,X_{n-1}\}+δ$, where $δ$ is an integer constant. We use martingale arguments to show that the counting process of $δ$-records among the first $n$ observations, suitably centered and scaled, is asymptotically normally distributed for $δ\ne0$. In particular, taking $δ=-1$ we obtain a central limit theorem for the number of weak records.

math.PR↗