arXiv subjects

Helton Saulo

Publications and source records attributed to Helton Saulo.

At least 19 recordsLinked to original sources

arXiv2026

Income inequality estimation with gamma mixtures

This paper studies the estimation of the $m$th Gini index under finite mixtures of gamma distributions. We derive closed-form expressions for the $m$th Gini index and for the expectation and bias of its non-parametric U-statistic estimator, extending previous results for both single gamma populations and gamma mixture models. We further establish the asymptotic properties of the estimator for gamma mixtures sharing a common rate parameter, including an asymptotic lower bound for the bias, asymptotic unbiasedness, strong consistency, and asymptotic normality. Although these theoretical results require a common rate parameter, a Monte Carlo study also investigates the estimator under mixtures with different rates and compares its performance with bias-corrected and parametric estimators. Finally, the proposed methodology is illustrated through the analysis of an income dataset.

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arXiv2026

Unbiased estimation of normalized scale-invariant indices under the gamma distribution

We introduce a broad class of normalized scale-invariant indices (NPRIs) generated by homogeneous functions and encompassing several well-known measures, including the Gini coefficient, generalized Gini indices, entropy-based measures, and variability indices. Explicit expressions are obtained for these indices under gamma populations. Exploiting the independence between the total sum and the associated Dirichlet proportions, we derive a simple unbiased estimator based on a U-statistic. The resulting estimator is shown to be unbiased for any NPRI when the underlying population follows a gamma distribution. Several examples are provided to illustrate the general theory. A Monte Carlo simulation study is carried out that shows the good performance of the unbiased estimator in several scenarios of index choices. We also present a simulation study that goes beyond the established theory by examining the estimator's applicability in settings characterized by a generalized gamma distribution. We evaluate the effectiveness of the NPRIs and their estimates in modeling a real-world dataset related to gross domestic product per capita in the Americas.

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arXiv2026

Quantile autoregressive moving average models for ratio-based bounded time series

This paper proposes the quantile unit-log-symmetric autoregressive moving average (QULS--ARMA) model for bounded time series on the open unit interval $(0,1)$. The model extends the unit-log-symmetric family by introducing a quantile-based reparameterization and embedding autoregressive and moving-average dynamics directly in the conditional quantile, thereby overcoming limitations of mean-based approaches and providing a coherent framework for proportion data arising from ratios of dependent positive variables. The proposed specification accommodates asymmetric behavior and heavy tails through flexible log-symmetric kernels, including the normal and Student-$t$ distributions. Parameter estimation is carried out via conditional maximum likelihood, and asymptotic properties are established. Monte Carlo simulations and an empirical application to hydroelectric energy storage proportions in Brazil assess the finite-sample performance and practical advantages of the QULS--ARMA model. The results show the good performance of the proposed estimators across a range of scenarios and kernel specifications.

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arXiv2026

Length-biased Birnbaum-Saunders quantile regression with application to water evaporation

Length-biased distributions arise naturally in environmental, reliability, and economic studies where the sampling mechanism favors larger observational units. In this paper, we propose a quantile regression model based on the length-biased Birnbaum--Saunders (QLBS) distribution. The model is constructed through a reparameterization of the length-biased Birnbaum--Saunders distribution in terms of its quantile function, thereby allowing direct interpretation of covariate effects on conditional quantiles of the response variable. We derive the log-likelihood function and the corresponding score equations, and obtain maximum likelihood estimators via numerical optimization. Asymptotic and bootstrap confidence intervals are considered. Two types of residuals are proposed for model assessment, namely the generalized Cox--Snell and randomized quantile residuals. An elaborate Monte Carlo simulation study is carried out to evaluate the performance of the maximum likelihood estimators for several sample sizes and quantile levels. The proposed methodology is illustrated with a real meteorological data set from Brazil.

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arXiv2026

A multivariate Birnbaum-Saunders autoregressive moving average model with application to air pollution concentration data

Fine particulate matter (PM$_{2.5}$) concentration data are positive, right-skewed series that arise naturally in environmental monitoring and are well described by the Birnbaum-Saunders (BS) distribution. In this paper, we propose a multivariate BS autoregressive moving average (MBSARMA) model with exogenous terms for the joint analysis of correlated positive asymmetric time series. The proposed model combines the multivariate log-linear BS framework with dynamic autoregressive moving average components on the conditional location parameter of each response. We estimate the model parameters by means of the Expectation-Maximisation (EM) algorithm. The performance of the proposed conditional likelihood estimators is evaluated by means of a Monte Carlo simulation study under several correlation levels and sample sizes. An application to weekly PM$_{2.5}$ pollution concentration data recorded at three monitoring stations in Santiago, Chile, obtained from the National Air Quality Information System of Chile (SINCA), is presented. The results show the good performance of the proposed methodology.

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arXiv2026

The arithmetic-harmonic inequality index: Theory, inference, and finite-sample analysis

We investigate the arithmetic-harmonic inequality (AHI) index, a bounded and scale-invariant measure of dispersion for positive random variables, defined through the interplay between the mean and its reciprocal. We derive analytical expressions for the AHI index within the generalized inverse Gaussian (GIG) family, encompassing the inverse Gaussian and gamma distributions as important special cases. We study the associated estimator, obtain a tractable expression for its expectation, establish its asymptotic properties, and derive explicit first-order bias approximations. A Monte Carlo study is conducted to evaluate the finite-sample performance of the estimator under various scenarios. An application to GDP per capita data for countries in the Americas illustrates the role of the AHI index within the broader Atkinson family across several values of the inequality-aversion parameter. The results show the good performance of the AHI index as a tractable and interpretable measure of economic dispersion.

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arXiv2026

Unifying the Hoover and Gini indices: Analytical, bias, and computational aspects

We propose a new family of inequality indices that bridges the Hoover index and the Gini coefficient. The measure is defined as the normalized expected absolute value of a convex combination of deviations from the mean and pairwise differences, providing a continuous interpolation between these two classical indices. We establish key theoretical properties, including scale invariance, boundedness, continuity, and compliance with the Pigou-Dalton transfer principle. Analytical representations are derived, allowing explicit evaluation under gamma distributions and leading to closed-form expressions involving incomplete gamma functions. From a statistical perspective, we study the plug-in estimator, obtaining a general expression for its expectation and explicit formulas for its bias under gamma populations. Simulation results indicate good finite-sample performance, with decreasing bias and mean squared error as the sample size increases. An empirical application to GDP per capita data illustrates the practical usefulness of the proposed index as a flexible tool for inequality analysis.

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arXiv2026

Bias analysis of a linear order-statistic inequality index estimator: Unbiasedness under gamma populations

This paper studies a class of rank-based inequality measures built from linear combinations of expected order statistics. The proposed framework unifies several well-known indices, including the classical Gini coefficient, the $m$th Gini index, the extended $m$th Gini index and particular cases of the $S$-Gini index, and also connects to spectral inequality measures through an integral representation. We investigate the finite-sample behavior of a natural U-statistic-type estimator that averages weighted order-statistic contrasts over all subsamples of fixed size and normalizes by the sample mean. A general bias decomposition is derived in terms of components that isolate the effect of random normalization on each rank level, yielding analytical expressions that can be evaluated under broad non-negative distributions via Laplace-transform methods. Under mild moment conditions, the estimator is shown to be asymptotically unbiased. Moreover, we prove exact unbiasedness under gamma populations for any sample size, extending earlier unbiasedness results for Gini-type estimators. A Monte Carlo study is performed to numerically check that the theoretical {unbiasedness} under gamma populations. Finally, a data set on GDP per capita across $34$ countries in the Americas is analyzes to illustrate the proposed methodology.

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arXiv2026

On the bias of the Hoover index estimator: Results for the gamma distribution

The Hoover index is a widely used measure of inequality with an intuitive interpretation, yet little is known about the finite-sample properties of its empirical estimator. In this paper, we derive a simple expression for the expected value of the Hoover index estimator for general non-negative populations, based on Laplace transform techniques and exponential tilting. This unified framework applies to both continuous and discrete distributions. Explicit bias expressions are obtained for gamma population, showing that the estimator is generally biased in finite samples. Numerical and simulation results illustrate the magnitude of the bias and its dependence on the underlying distribution and sample size.

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arXiv2025

On the bias of the Gini estimator: Poisson and geometric cases, a characterization of the gamma family, and unbiasedness under gamma distributions

In this paper, we derive a general representation for the expectation of the Gini coefficient estimator in terms of the Laplace transform of the underlying distribution, together with the mean and the Gini coefficient of its exponentially tilted version. This representation leads to a new characterization of the gamma family within the class of nonnegative scale families, based on a stability property under exponential tilting. As direct applications, we show that the Gini estimator is biased for both Poisson and geometric populations and provide an alternative, unified proof of its unbiasedness for gamma populations. By using the derived bias expressions, we propose plug-in bias-corrected estimators and assess their finite-sample performance through a Monte Carlo study, which demonstrates substantial improvements over the original estimator. Compared with existing approaches, our framework highlights the fundamental role of scale invariance and exponential tilting, rather than distribution-specific algebraic calculations, and complements recent results in Baydil et al. (2025) [Unbiased estimation of the gini coefficient. SPL, 222:110376] and Vila and Saulo (2025a,b) [Bias in Gini coefficient estimation for gamma mixture populations. STPA, 66:1-18; and The mth gini index estimator: Unbiasedness for gamma populations. J. Econ. Inequal].

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arXiv2025

Closed-form parameter estimation for the bivariate gamma distribution: New approaches

We propose new closed-form estimators for the parameters of McKay's bivariate gamma distribution by exploiting monotone transformations of the likelihood equations. As a special case, our framework recovers the estimators recently introduced by Zhao et al. (2022) [Zhao, J., Jang, Y.-H., and Kim, H. (2022). Closed-form and bias-corrected estimators for the bivariate gamma distribution. Journal of Multivariate Analysis, 191:105009]. Theoretical properties, including strong consistency and asymptotic normality, are established. We further introduce a second family of closed-form estimators that is explicitly built from the stochastic relationship between gamma random variables. Our second approach encompasses the estimators of Nawa and Nadarajah (2023) [Nawa, V. M. and Nadarajah, S. (2023). New closed form estimators for a bivariate gamma distribution. Statistics, 57(1):150-160]. Monte Carlo experiments are conducted to assess finite-sample performance, showing that the new estimators perform comparably to maximum likelihood estimators while avoiding iterative optimization, and improve upon the existing closed-form approach by Zhao et al. (2022) and Nawa and Nadarajah (2023). A real hydrological data set is analyzed to illustrate the proposed approaches.

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arXiv2025

A transformed-score approach to closed-form and one-step efficient estimation for the beta distribution

Power transformations of beta random variables produce unbiased estimating equations from transformed-model likelihood scores. This construction clarifies the connection between moment-type and likelihood-based estimating equations, recovers recent closed-form estimators for the beta shape parameters, and yields a new family indexed by a scalar transformation parameter $r$. For each fixed admissible $r$, we establish strong consistency and joint asymptotic normality. We also show that likelihood-guided selection over a prespecified finite grid preserves root-$n$ consistency and use one Fisher-scoring update to obtain an asymptotically efficient estimator. A Monte Carlo study with 1000 replications compares numerical maximum likelihood, two existing closed-form procedures, the proposed likelihood-selected closed-form estimator, and its one-step refinement across nine parameter configurations and four sample sizes. All procedures are evaluated on the same simulated samples within each replication. The selected closed-form estimator closely tracks maximum likelihood, while the one-step refinement is nearly indistinguishable from it for moderate sample sizes. Finally, the methods are illustrated descriptively using the proportions of municipal area devoted to crops and pasture in the 15 municipalities of Roraima, Brazil.

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arXiv2025

Two Tunable Gini-Type Measures with U-Statistic Estimation: Theory, Simulation, and an Empirical Application to GDP per Capita in the Americas

We introduce two families of inequality measures, $G_p$ and $H_q$, that converge to the classical Gini coefficient as $p,q\to\infty$. The tuning parameters $p>1$ and $q>0$ regulate the influence of disparities between observations. For each index we derive closed-form $U$-statistic plug-in estimators and establish strong consistency and asymptotic normality under mild moment conditions. A Monte Carlo study assesses finite-sample behavior across $(n,p,q)$, and an empirical illustration with GDP per capita in the Americas shows how the tuning parameters influence the measure of inequality.

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arXiv2025

On classes of distributions on the unit interval: structural properties and application to inequality data

Probability distributions defined on the unit interval are widely used in fields ranging from econometrics to reliability studies. Traditional models such as the beta and Kumaraswamy distributions are well-established due to their flexibility and tractability. In this paper, we introduce two novel families of unit-interval distributions derived via non-injective transformations of the gamma ratio. These transformations, denoted $S_r$ and $T_r$, allow the construction of new random variables with support on $(0,1)$ and admit simple closed-form expressions for their densities when the underlying variables are independent gamma distributed. Notably, for $r = 1/2$, these constructions yield sample-based estimators of the Gini and Atkinson indices, establishing a direct link with classical inequality measures. We derive the distributional laws, cumulative distribution functions, quantile functions, and raw moments, and discuss maximum likelihood estimation for the proposed models. A Monte Carlo simulation study is conducted to assess the finite sample behavior of the maximum likelihood estimators under different parameter configurations. An application to cross-country Gini index data illustrates the flexibility and practical relevance of the proposed distributions in modeling real inequality indicators.

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arXiv2025

Bias in estimating Theil, Atkinson, and dispersion indices for gamma mixture populations

This paper examines the finite-sample bias of estimators for the Theil and Atkinson indices, as well as for the variance-to-mean ratio (VMR), under the assumption that the population follows a finite mixture of gamma distributions with a common rate parameter. Using Mosimann's proportion-sum independence theorem and the structural relationship between the gamma and Dirichlet distributions, these estimators were rewritten as functions of Dirichlet vectors, which enabled the derivation of closed-form analytical expressions for their expected values. A Monte Carlo simulation study evaluates the performance of both the traditional and bias-corrected estimators across a range of mixture scenarios and sample sizes, revealing systematic bias induced by population heterogeneity and demonstrating the effectiveness of the proposed corrections, particularly in small and moderate samples. An empirical application to global per capita GDP data further illustrates the practical relevance of the methodology and confirms the suitability of gamma mixtures for representing structural economic heterogeneity.

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arXiv2025

Unbiased estimation in new Gini index extensions under gamma distributions, with application to real income data

In this paper, we introduce two flexible extensions of the classical Gini index, referred to as the extended lower and upper Gini indices. The proposed measures are based on the differences between an observation and the minimum and maximum order statistics in samples of size $m\geqslant 2$ and reduce to the classical Gini coefficient when $m=2$. Unlike conventional Gini-type measures, they provide a position-oriented assessment of inequality relative to the lower and upper tails of the distribution. We establish the consistency and asymptotic normality of the proposed estimators under mild regularity conditions. For gamma-distributed populations, we derive exact expressions for their expectations and prove their unbiasedness, thereby extending previous results of [Deltas, G. 2003. The small-sample bias of the gini coefficient: Results and implications for empirical research. Review of Economics and Statistics 85:226-234] and [Baydil, B., de la Pe\~na, V. H., Zou, H., and Yao, H. 2025. Unbiased estimation of the gini coefficient. Statistics & Probability Letters 222:110376]. The finite-sample performance of the estimators is investigated through Monte Carlo simulations, and an application to 2023 GDP per capita data from South American countries illustrates the practical usefulness of the proposed measures. The results show that the extended lower and upper Gini indices provide a richer and more informative characterization of inequality than traditional Gini-type measures.

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arXiv2025

Closed-form solutions for parameter estimation in exponential families based on maximum a posteriori equations

In this paper, we derive closed-form estimators for the parameters of certain exponential family distributions through the maximum a posteriori (MAP) equations. A Monte Carlo simulation is conducted to assess the performance of the proposed estimators. The results show that, as expected, their accuracy improves with increasing sample size, with both bias and mean squared error approaching zero. Moreover, the proposed estimators exhibit performance comparable to that of traditional MAP and maximum likelihood (ML) estimators. A notable advantage of the proposed method lies in its computational simplicity, as it eliminates the need for numerical optimization required by MAP and ML estimation.

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arXiv2025

An unbiased estimator of a novel extended Gini index for gamma distributed populations

In this paper, we introduce a novel flexible Gini index, referred to as the extended Gini index, which is defined through ordered differences between the $j$th and $k$th order statistics within subsamples of size $m$, for indices satisfying $1 \leqslant j \leqslant k \leqslant m$. We derive a closed-form expression for the expectation of the corresponding estimator under the gamma distribution and prove its unbiasedness, thereby extending prior findings by \cite{Deltas2003}, \cite{Baydil2025}, and \cite{Vila2025}. A Monte Carlo simulation illustrates the estimator's finite-sample unbiasedness. A real data set on gross domestic product (GDP) per capita is analyzed to illustrate the proposed measure.

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