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Roberto Vila

Publications and source records attributed to Roberto Vila.

At least 37 records · Page 2Linked to original sources
arXiv2026

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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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

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

A general class of continuous asymmetric distributions with positive support

In order to better fit real-world datasets, studying asymmetric distribution is of great interest. In this work, we derive several mathematical properties of a general class of asymmetric distributions with positive support which shows up as a unified framework for Extreme Value Theory asymptotic results. The new model generalizes some well-known distribution models such as Generalized Gamma, Inverse Gamma, Weibull, Fréchet, Half-normal, Modified half-normal, Rayleigh, and Erlang. To highlight the applicability of our results, the performance of the analytical models is evaluated through real-life dataset modeling.

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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

On unbiased estimators for functions of the rate parameter of the exponential distribution

In this paper, we explicitly derive unbiased estimators for various functions of the rate parameter of the exponential distribution in the absence of a location parameter, including powers of the rate parameter, the $q$th quantile, the $p$th moment, the survival function, the maximum, minimum, probability density function, mean past lifetime, moment generating function, and others. This work non-trivially complements established formulas for unbiased estimators of functions of parameters of the location-rate exponential distribution. Additionally, we establish a result demonstrating the asymptotic normality of the proposed unbiased estimators.

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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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arXiv2025

The $m$th Gini index estimator: Unbiasedness for gamma populations

This paper establishes the theoretical result that the sample $m$th Gini index is an unbiased estimator of the population $m$th Gini index, introduced by Gavilan-Ruiz (2024), for gamma-distributed populations. An illustrative Monte Carlo simulation study confirms the unbiasedness of the sample $m$th Gini index estimator in finite samples.

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arXiv2025

Bias in Gini coefficient estimation for gamma mixture populations

This paper examines the properties of the Gini coefficient estimator for gamma mixture populations and reveals the presence of bias. In contrast, we show that sampling from a gamma distribution yields an unbiased estimator, consistent with prior research (Baydil et al., 2025). We derive an explicit bias expression for the Gini coefficient in gamma mixture populations, which serves as the foundation for proposing a bias-corrected Gini estimator. We conduct a Monte Carlo simulation study to evaluate the behavior of the bias-corrected Gini estimator.

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arXiv2025

A novel unit-asymmetric distribution based on correlated Fréchet random variables

In this paper, we propose a new distribution with unitary support which can be characterized as a ratio of the type $W=X_1/(X_1+X_2)$, where $(X_1, X_2)^\top$ follows a bivariate extreme distribution with Fréchet margins, that is, $X_1$ and $X_2$ are two correlated Fréchet random variables. Some mathematical properties such as identifiability, symmetry, stochastic representation, characterization as a ratio, moments, stress-strength probability, quantiles, and the maximum likelihood method are rigorously analyzed. Two applications of the ratio distribution are discussed.

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arXiv2024

A new unit-bimodal distribution based on correlated Birnbaum-Saunders random variables

In this paper, we propose a new distribution over the unit interval which can be characterized as a ratio of the type $Z=Y/(X+Y)$ where $X$ and $Y$ are two correlated Birnbaum-Saunders random variables. The density of $Z$ may be unimodal or bimodal. Simple expressions for the cumulative distribution function, moment-generating function and moments are obtained. Moreover, the stress-strength probability between $X$ and $Y$ is calculated explicitly in the symmetric case, that is, when the respective scale parameters are equal. Two applications of the ratio distribution are discussed.

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arXiv2024

Moment-type estimators for a weighted exponential family

In this paper, we propose and study closed-form moment type estimators for a weighted exponential family. We also develop a bias-reduced version of these proposed closed-form estimators using bootstrap techniques. The estimators are evaluated using Monte Carlo simulation. This shows favourable results for the proposed bootstrap bias-reduced estimators.

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arXiv2024

Family of multivariate extended skew-elliptical distributions: Statistical properties, inference and application

In this paper we propose a family of multivariate asymmetric distributions over an arbitrary subset of set of real numbers which is defined in terms of the well-known elliptically symmetric distributions. We explore essential properties, including the characterization of the density function for various distribution types, as well as other key aspects such as identifiability, quantiles, stochastic representation, conditional and marginal distributions, moments, Kullback-Leibler Divergence, and parameter estimation. A Monte Carlo simulation study is performed for examining the performance of the developed parameter estimation method. Finally, the proposed models are used to analyze socioeconomic data.

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