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Helton Saulo

Publications and source records attributed to Helton Saulo.

At least 37 records · Page 2Linked to original sources
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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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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arXiv2023

Parametric quantile autoregressive conditional duration models with application to intraday value-at-risk

The modeling of high-frequency data that qualify financial asset transactions has been an area of relevant interest among statisticians and econometricians -- above all, the analysis of time series of financial durations. Autoregressive conditional duration (ACD) models have been the main tool for modeling financial transaction data, where duration is usually defined as the time interval between two successive events. These models are usually specified in terms of a time-varying mean (or median) conditional duration. In this paper, a new extension of ACD models is proposed which is built on the basis of log-symmetric distributions reparametrized by their quantile. The proposed quantile log-symmetric conditional duration autoregressive model allows us to model different percentiles instead of the traditionally used conditional mean (or median) duration. We carry out an in-depth study of theoretical properties and practical issues, such as parameter estimation using maximum likelihood method and diagnostic analysis based on residuals. A detailed Monte Carlo simulation study is also carried out to evaluate the performance of the proposed models and estimation method in retrieving the true parameter values as well as to evaluate a form of residuals. Finally, the proposed class of models is applied to a price duration data set and then used to derive a semi-parametric intraday value-at-risk (IVaR) model.

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arXiv2023

Bivariate autoregressive conditional models: A new method for jointly modeling duration and number of transactions of irregularly spaced financial data

In this paper, a new approach to bivariate modeling of autoregressive conditional duration (ACD) models is proposed. Specifically, we consider the joint modeling of durations and the number of transactions made during the spell. The proposed bivariate ACD model is based on log-symmetric distributions, which are useful for modeling strictly positive, asymmetric and light- and heavy-tailed data, such as transaction-level high-frequency financial data. A Monte Carlo simulation is performed for the assessment of the estimation method and the evaluation of a form of residuals. A real financial transactions data set is analyzed in order to illustrate the proposed method.

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arXiv2023

Scale-mixture Birnbaum-Saunders quantile regression models applied to personal accident insurance data

The modeling of personal accident insurance data has been a topic of extreme relevance in the insurance literature. This kind of data often exhibits positive skewness and heavy tails. In this work, we propose a new quantile regression model based on the scale-mixture Birnbaum-Saunders distribution for modeling personal accident insurance data. The maximum likelihood estimates of the model parameters are obtained via the EM algorithm. Two Monte Carlo simulation studies are performed using the R software. The first study aims to analyze the performances of the EM algorithm to obtain the maximum likelihood estimates, and the randomized quantile and generalized Cox-Snell residuals. In the second simulation study, the size and power of the the Wald, likelihood ratio, score and gradient tests are evaluated. The two simulation studies are conducted considering different quantiles of interest and sample sizes. Finally, a real insurance data set is analyzed to illustrate the proposed approach.

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arXiv2023

An upper bound and a characterization for Gini's mean difference based on correlated random variables

In this paper, we obtain an upper bound for the Gini mean difference based on mean, variance and correlation for the case when the variables are correlated. We also derive some closed-form expressions for the Gini mean difference when the random variables have an absolutely continuous joint distribution. We then examine some particular examples based on elliptically contoured distributions, and specifically multivariate normal and Student-$t$ distributions.

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arXiv2023

Bivariate distributions on the unit square: Theoretical properties and applications

We introduce the bivariate unit-log-symmetric model based on the bivariate log-symmetric distribution (BLS) defined in [Vila et al., 2022, Bivariate Log-symmetric Models: Theoretical Properties and Parameter Estimation. Avaliable at arXiv:2211.13839] as a flexible family of bivariate distributions over the unit square. We then study its mathematical properties such as stochastic representations, quantiles, conditional distributions, independence of the marginal distributions and moments. Maximum likelihood estimation method is discussed and examined through Monte Carlo simulation. Finally, the proposed model is used to analyze soccer data.

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arXiv2022

Bivariate log-symmetric models: distributional properties, parameter estimation and an application to fatigue data analysis

The bivariate Gaussian distribution has been a key model for many developments in statistics. However, many real-world phenomena generate data that follow asymmetric distributions, and consequently bivariate normal model is inappropriate in such situations. Bidimensional log-symmetric models have attractive properties and can be considered as good alternatives in these cases. In this paper, we discuss bivariate log-symmetric distributions and their characterizations. We establish several distributional properties and obtain the maximum likelihood estimators of the model parameters. A Monte Carlo simulation study is performed for examining the performance of the developed parameter estimation method. A real data set is finally analyzed to illustrate the proposed model and the associated inferential method.

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arXiv2022

Unit-log-symmetric models: Characterization, statistical properties and its use in analyzing internet access data

We present here a unit-log-symmetric model based on the bivariate log-symmetric distribution. It is a flexible family of distributions over the interval $(0, 1)$. We then discuss its mathematical properties such as stochastic representation, symmetry, modality, moments, quantile function, entropy and maximum likelihood estimators, paying particular attention to the special cases of unit log-normal, unit-log-Student-$t$ and unit-log-Laplace distributions. Finally, some empirical results and practical illustrations are presented.

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arXiv2022

Parametric quantile regression for income data

Univariate normal regression models are statistical tools widely applied in many areas of economics. Nevertheless, income data have asymmetric behavior and are best modeled by non-normal distributions. The modeling of income plays an important role in determining workers' earnings, as well as being an important research topic in labor economics. Thus, the objective of this work is to propose parametric quantile regression models based on two important asymmetric income distributions, namely, Dagum and Singh-Maddala distributions. The proposed quantile models are based on reparameterizations of the original distributions by inserting a quantile parameter. We present the reparameterizations, some properties of the distributions, and the quantile regression models with their inferential aspects. We proceed with Monte Carlo simulation studies, considering the maximum likelihood estimation performance evaluation and an analysis of the empirical distribution of two residuals. The Monte Carlo results show that both models meet the expected outcomes. We apply the proposed quantile regression models to a household income data set provided by the National Institute of Statistics of Chile. We showed that both proposed models had a good performance both in terms of model fitting. Thus, we conclude that results were favorable to the use of Singh-Maddala and Dagum quantile regression models for positive asymmetric data, such as income data.

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arXiv2022

Symmetric generalized Heckman models

The sample selection bias problem arises when a variable of interest is correlated with a latent variable, and involves situations in which the response variable had part of its observations censored. Heckman (1976) proposed a sample selection model based on the bivariate normal distribution that fits both the variable of interest and the latent variable. Recently, this assumption of normality has been relaxed by more flexible models such as the Student-t distribution (Marchenko and Genton, 2012; Lachos et al., 2021). The aim of this work is to propose generalized Heckman sample selection models based on symmetric distributions (Fang et al., 1990). This is a new class of sample selection models, in which variables are added to the dispersion and correlation parameters. A Monte Carlo simulation study is performed to assess the behavior of the parameter estimation method. Two real data sets are analyzed to illustrate the proposed approach.

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