arXiv subjects

José María Sarabia

Publications and source records attributed to José María Sarabia.

14 recordsLinked to original sources

arXiv2026

Beyond the Beta Lorenz Curve: A New Parametric Family for Poverty and Inequality Estimation

The estimation of inequality and poverty measures is frequently constrained by a lack of individual data. When only income shares are available, the Beta Lorenz curve introduced by Kakwani (Econometrica, 48, 1980) has become a standard tool for reconstructing income distributions. Together with the General Quadratic (GQ) Lorenz curve, it is used by the World Bank to produce official poverty estimates when microdata are unavailable. In this paper, we show that Kakwani's specification does not generally satisfy the formal requirements of a genuine Lorenz curve and introduce a new Lorenz curve rooted in the corrected parameter space. Using more than 1,700 datasets, we show that the proposed model yields valid Lorenz curves in all applications and consistently outperforms competing specifications in both accuracy and sampling precision. The GQ, by contrast, fails to provide genuine curves in 15 percent of datasets and underestimates extreme poverty severity in 98 percent of them.

stat.AP↗

arXiv2026

Revisiting "A universal model for the Lorenz curve with novel applications''

This research reviews several crucial aspects of the universal model for the Lorenz curve proposed by Sitthiyot and Holasut (2023) (hereafter, SH (2023)). A first issue concerns the mathematical definition of the proposed curves. The four functional forms introduced by SH (2023) do not satisfy the necessary and sufficient conditions for a valid Lorenz curve. We propose corrected versions of the previous curves and derive analytical expressions for some measures of inequality.

math.ST↗

arXiv2026

Multiple integral representations of the Catalan's constant

In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem that enables the representation of Catalan's constant in terms of a single integral. Finally, we provide a multiple integral representation of Catalan's constant in dimensions greater than or equal to two using the Lerch function. The results are accompanied by illustrative examples.

math.NT↗

arXiv2025

New Representations of Catalan's Constant, Apery's Constant and the Euler Numbers Obtained from the Half Hyperbolic Secant Distribution

New expressions and bounds for Catalan's and Apery's constants, derived from the half hyperbolic secant distribution, are presented. These constants are obtained by using expressions for the Lorenz curve, the Gini and Theil indices, convolutions and a mixture of distributions, among other approaches. The new expressions are presented both in terms of integral (simple and double) representation and also as an interesting series representation. Some of these features are well known, while others are new. In addition, some integral representations of Euler's numbers are obtained.

math.NT↗

arXiv2021

Inequality in the use frequency of patent technology codes

Technology codes are assigned to each patent for classification purposes and to identify the components of its novelty. Not all the technology codes are used with the same frequency - if we study the use frequency of codes in a year, we can find predominant technologies used in many patents and technology codes not so frequent as part of a patent. In this paper, we measure that inequality in the use frequency of patent technology codes. First, we analyze the total inequality in that use frequency considering the patent applications filed under the Patent Co-operation Treaty at international phase, with the European Patent Office as designated office, in the period 1977-2018, on a yearly basis. Then, we analyze the decomposition of that inequality by grouping the technology codes by productive economic activities. We show that total inequality had an initial period of growth followed by a phase of relative stabilization, and that it tends to be persistently high. We also show that total inequality was mainly driven by inequality within productive economic activities, with a low contribution of the between-activities component.

econ.GN↗

arXiv2020

The risk of death in newborn businesses during the first years in market

In this paper, we analyzed how business age and mortality are related during the first years of life, and tested the different hypotheses proposed in the literature. For that, we used data on U.S. business establishments, with 1-year resolution in the range of age of 0-5 years, in the period 1977-2016, published by the United States Census Bureau. First, we explored the adaptation of classical techniques of survival analysis (the Life Table and Peto-Turnbull methods) to the business survival analysis. Then, we considered nine parametric probabilistic models, most of them well-known in reliability analysis and in the actuarial literature, with different shapes of the hazard function, that we fitted by maximum likelihood method and compared with the Akaike information criterion. Our findings show that newborn firms seem to have a decreasing failure rate with the age during the first five years in market, with the exception of the first months of some years in which the risk can rise.

econ.GN↗

arXiv2018

The nonlinear distribution of employment across municipalities

In this paper, the nonlinear distribution of employment across Spanish municipalities is analyzed. In addition, we explore the properties of the family of generalized power law (GPL) distributions, and test its adequacy for modelling employment data. The hierarchical structure of the GPL family that includes the hierarchy of Pareto (power law) distributions is deeply studied. A new subfamily of heavy-tailed GPL distributions that is right tail equivalent to a Pareto (power law) model is derived. Our findings show on the one hand that the distribution of employment across Spanish municipalities follows a power law behavior in the upper tail and, on the other hand, the adequacy of GPL models for modelling employment data in the whole range of the distribution.

stat.ME↗

arXiv2018

Estimation of income inequality from grouped data

Grouped data in form of income shares have been conventionally used to estimate income inequality due to the lack of availability of individual records. Most prior research on economic inequality relies on lower bounds of inequality measures in order to avoid the need to impose a parametric functional form to describe the income distribution. These estimates neglect income differences within shares, introducing, therefore, a potential source of measurement error. The aim of this paper is to explore a nuanced alternative to estimate income inequality, which leads to a reliable representation of the income distribution within shares. We examine the performance of the generalized beta distribution of the second kind and related models to estimate different inequality measures and compare the accuracy of these estimates with the nonparametric lower bound in more than 5000 datasets covering 182 countries over the period 1867-2015. We deploy two different econometric strategies to estimate the parametric distributions, non-linear least squares and generalised method of moments, both implemented in R and conveniently available in the package GB2group. Despite its popularity, the nonparametric approach is outperformed even the simplest two-parameter models. Our results confirm the excellent performance of the GB2 distribution to represent income data for a heterogeneous sample of countries, which provides highly reliable estimates of several inequality measures. This strong result and the access to an easy tool to implement the estimation of this family of distributions, we believe, will incentivize its use, thus contributing to the development of reliable estimates of inequality trends.

stat.AP↗

arXiv2017

Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions

The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al, 2016), Gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.

stat.ME↗

arXiv2016

A Generalization of the Power Law Distribution with Nonlinear Exponent

The power law distribution is usually used to fit data in the upper tail of the distribution. However, commonly it is not valid to model data in all the range. In this paper, we present a new family of distributions, the so-called Generalized Power Law (GPL), which can be useful for modeling data in all the range and possess power law tails. To do that, we model the exponent of the power law using a nonlinear function which depends on data and two parameters. Then, we provide some basic properties and some specific models of that new family of distributions. After that, we study a relevant model of the family, with special emphasis on the quantile and hazard functions, and the corresponding estimation and testing methods. Finally, as an empirical evidence, we study how the debt is distributed across municipalities in Spain. We check that power law model is only valid in the upper tail; we show analytically and graphically the competence of the new model with municipal debt data in the whole range; and we compare the new distribution with other well-known distributions including the Lognormal, the Generalized Pareto, the Fisk, the Burr type XII and the Dagum models.

nlin.AO↗

arXiv2015

Risks aggregation in multivariate dependent Pareto distributions

In this paper we obtain closed expressions for the probability distribution function, when we consider aggregated risks with multivariate dependent Pareto distributions. We work with the dependent multivariate Pareto type II proposed by Arnold (1983, 2015), which is widely used in insurance and risk analysis. We begin with the individual risk model, where we obtain the probability density function (PDF), which corresponds to a second kind beta distribution. We obtain several risk measures including the VaR, TVaR and other tail measures. Then, we consider collective risk model based on dependence, where several general properties are studied. We study in detail some relevant collective models with Poisson, negative binomial and logarithmic distributions as primary distributions. In the collective Pareto-Poisson model, the PDF is a function of the Kummer confluent hypergeometric function, and in the Pareto-negative binomial is a function of the Gauss hypergeometric function. Using the data set based on one-year vehicle insurance policies taken out in 2004-2005 (Jong and Heller, 2008), we conclude that our collective dependent models outperform the classical collective models Poisson-exponential and geometric-exponential in terms of the AIC and CAIC statistics.

stat.ME↗

arXiv2014

A variation of the Dragulescu-Yakovenko income model

In the context of the Dragulescu-Yakovenko (2000) model, we show that empirical income distribution with truncated datasets, cannot be properly modeled by the one-parameter exponential distribution. However, a truncated version characterized by an exponential distribution with two parameters gives an accurate fit.

q-fin.ST↗

arXiv2013

Modelling Road Accident Blackspots Data with the Discrete Generalized Pareto distribution

This study shows how road traffic networks events, in particular road accidents on blackspots, can be modelled with simple probabilistic distributions. We considered the number of accidents and the number of deaths on Spanish blackspots in the period 2003-2007, from Spanish General Directorate of Traffic (DGT). We modelled those datasets, respectively, with the discrete generalized Pareto distribution (a discrete parametric model with three parameters) and with the discrete Lomax distribution (a discrete parametric model with two parameters, and particular case of the previous model). For that, we analyzed the basic properties of both parametric models: cumulative distribution, survival, probability mass, quantile and hazard functions, genesis and rth-order moments; applied two estimation methods of their parameters: the $μ$ and ($μ+1$) frequency method and the maximum likelihood method; and used two goodness-of-fit tests: Chi-square test and discrete Kolmogorov-Smirnov test based on bootstrap resampling. We found that those probabilistic models can be useful to describe the road accident blackspots datasets analyzed.

stat.AP↗

arXiv2013

Modelling major failures in power grids in the whole range

Empirical research with electricity transmission networks reliability data shows that the size of major failures - in terms of energy not supplied (ENS), total loss of power (TLP) or restoration time (RT) - appear to follow a power law behaviour in the upper tail of the distribution. However, this pattern - also known as Pareto distribution - is not valid in the whole range of those major events. We aimed to find a probability distribution that we could use to model them, and hypothesized that there is a two-parameter model that fits the pattern of those data well in the entire domain. We considered the major failures produced between 2002 and 2009 in the European power grid; analyzed those reliability indicators: ENS, TLP and RT; fitted six alternative models: Pareto II, Fisk, Lognormal, Pareto, Weibull and Gamma distributions, to the data by maximum likelihood; compared these models by the Bayesian information criterion; tested the goodness-of-fit of those models by a Kolmogorov-Smirnov test method based on bootstrap resampling; and validated them graphically by rank-size plots. We found that Pareto II distribution is, in the case of ENS and TLP, an adequate model to describe major events reliability data of power grids in the whole range, and in the case of RT, is the best choice of the six alternative models analyzed.

physics.soc-ph↗