Generalized Beta Distribution Research Articles

Novel closed-form point estimators for the beta distribution

In this paper, we propose and investigate novel closed-form point estimators for the beta distribution. The estimators of the first type are a modified version of Pearson's method of moments. The underlying idea is to involve the sufficient statistics, i.e., log-moments in the moment estimation equations and solve the mixed type of moment equations simultaneously. The estimators of the second type are based on an approximation to Fisher's likelihood principle. The idea is to solve two score equations derived from the log-likelihood function of generalized beta distributions. Both two resulted estimators are in closed forms, strongly consistent and asymptotically normal. In addition, through theoretical analyses and extensive simulations, the proposed estimators are shown to perform very close to the maximum likelihood estimators in both small and large samples, and they significantly outperform the method of moment estimators.

Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

Beyond Zipf’s law: Exploring the discrete generalized beta distribution in open-source repositories

Rank-size distributions, such as Zipf’s Law, have been instrumental in providing insights into the emergence of hierarchies across diverse systems, from linguistic corpuses to urban structures. However, the application of Zipf’s Law reveals limitations, particularly in its focus on distribution tails, sometimes overlooking a large proportion of the data which might play a pivotal role in system dynamics. Yet, fitting rank-size distributions other than a straight line on the log–log scale requires caution. In this study, we re-evaluate the utility of rank-size distributions by contrasting the traditional Zipf’s Law with the Discrete Generalized Beta Distribution (DGBD). We show the need of cautious fitting techniques for rank distributions, including the use of binning to prevent overfitting to data tails. Through both analytical derivation and empirical validation on commit data of open-source repositories, we show that DGBD consistently improves over Zipf distribution for concave rank distributions of large datasets (N≥100). This approach contributes to the advancement of methodologies for analyzing hierarchical systems.

A new class of composite GBII regression models with varying threshold for modeling heavy-tailed data

The four-parameter generalized beta distribution of the second kind (GBII) has been proposed for modeling insurance losses with heavy-tailed features. The aim of this paper is to present a parametric composite GBII regression modeling by splicing two GBII distributions using mode matching method. It is designed for simultaneous modeling of small and large claims and capturing the policyholder heterogeneity by introducing the covariates into the scale parameter. The threshold that splits two GBII distributions is allowed to vary across individuals policyholders based on their risk features. The proposed regression modeling also contains a wide range of insurance loss distributions as the head and the tail respectively and provides the close-formed expressions for parameter estimation and model prediction. A simulation study is conducted to show the accuracy of the proposed estimation method and the flexibility of the regressions. Some illustrations of the applicability of the new class of distributions and regressions are provided with a Danish fire losses data set and a Chinese medical insurance claims data set, comparing with the results of competing models from the literature.

Gbdmr: identifying differentially methylated CpG regions in the human genome via generalized beta regressions

BackgroundDNA methylation is a biochemical process in which a methyl group is added to the cytosine-phosphate-guanine (CpG) site on DNA molecules without altering the DNA sequence. Multiple CpG sites in a certain genome region can be differentially methylated across phenotypes. Identifying these differentially methylated CpG regions (DMRs) associated with the phenotypes contributes to disease prediction and precision medicine development.ResultsWe propose a novel DMR detection algorithm, gbdmr. In contrast to existing methods under a linear regression framework, gbdmr assumes that DNA methylation levels follow a generalized beta distribution. We compare gbdmr to alternative approaches via simulations and real data analyses, including dmrff, a new DMR detection approach that shows promising performance among competitors, and the traditional EWAS that focuses on single CpG sites. Our simulations demonstrate that gbdmr is superior to the other two when the correlation between neighboring CpG sites is strong, while dmrff shows a higher power when the correlation is weak. We provide an explanation of these phenomena from a theoretical perspective. We further applied the three methods to multiple real DNA methylation datasets. One is from a birth cohort study undertaken on the Isle of Wight, United Kingdom, and the other two are from the Gene Expression Omnibus database repository. Overall, gbdmr identifies more DMR CpGs linked to phenotypes than dmrff, and the simulated results support the findings.ConclusionsGbdmr is an innovative method for detecting DMRs based on generalized beta regression. It demonstrated notable advantages over dmrff and traditional EWAS, particularly when adjacent CpGs exhibited moderate to strong correlations. Our real data analyses and simulated findings highlight the reliability of gbdmr as a robust DMR detection tool. The gbdmr approach is accessible and implemented by R on GitHub: https://github.com/chengzhouwu/gbdmr.

Information entropy of the generalized beta distribution

The output distribution of statistical variables characterizes the level of uncertainty of a complex system, the state of which is given by probabilistic changes in internal relationships. Information entropy is an effective tool for analysing the distribution of data in complex systems. Subfamilies of generalized beta distributions of the first and second types includes many simple distributions as special cases. These subfamilies are often used to model the output state of the system by distributions of random variables. There are conditions in the paper, that are imposed on the parameters of the generalized beta distribution to obtain relationships with simpler types. The paper contains an expression for calculating the information entropy, the generalized beta distribution, which includes distributions of the first and second kind as special cases. Expressions of information entropies for the most famous subfamilies of generalized beta distribution also are given.

Volatility Forecasting Using Quasi-Score-Driven Models with an Application to the Coronavirus Pandemic Period

Abstract We study the statistical and volatility forecasting performances of the recent quasi-score-driven EGARCH (exponential generalized autoregressive conditional heteroscedasticity) models. We compare the quasi-score-driven EGARCH models with GARCH, asymmetric power ARCH (A-PARCH), and all relevant score-driven EGARCH models of the literature. For score-driven and quasi-score-driven EGARCH, we use the following seven score-driven probability distributions: Student’s t-distribution; general error distribution (GED); generalized t-distribution (Gen-t); skewed generalized t-distribution (Skew-Gen-t); exponential generalized beta distribution of the second kind (EGB2); normal-inverse Gaussian distribution (NIG); Meixner distribution (MXN). We use all combinations of those distributions for (i) the probability distribution of the dependent variable, and (ii) the probability distribution which defines the quasi-score function updating term of the quasi-score-driven filters. We use daily data for the Standard & Poor’s 500 (S&P 500) index. We find that both in-sample and out-of-sample, quasi-score-driven EGARCH is superior to GARCH, A-PARCH, and score-driven EGARCH. We report in-sample results for the period of January 2000 to December 2020, providing evidence in favor of the quasi-score-driven EGARCH model for the last two decades. We report out-of-sample volatility forecasting results for a period within the coronavirus disease 2019 (COVID-19) pandemic, providing evidence in favor of the quasi-score-driven EGARCH model for a crisis period.

Estimating Income Distributions From Grouped Data: A Minimum Quantile Distance Approach

This paper focuses on the estimation of income distribution from grouped data in the form of quantiles. We propose a novel application of the minimum quantile distance (MQD) approach and compare its performance with the maximum likelihood (ML) technique. The estimation methods are applied using three parametric distributions: the generalized beta distribution of the second kind (GB2), the Dagum distribution, and the Singh–Maddala distribution. We provide the density-quantile functions for these distributions, along with reproducible R code. A simulation study is conducted to evaluate the performance of the MQD and ML methods. The proposed methods are then applied to data from 30 European countries, utilizing the aforementioned parametric distributions. To validate the accuracy of the estimates, we compare them with estimates obtained from more detailed and informative microdata sets. The findings confirm the excellent performance of the considered parametric distributions in estimating income distribution. Additionally, the MQD approach is identified as a straightforward and reliable method for this purpose. Notably, the MQD method displays superior robustness in comparison to the ML technique when it comes to selecting suitable starting values for the underlying computation algorithm, specifically when dealing with the GB2 distribution.

EXCEL-oriented procedures for determining the entropy of a distribution function and its relative parametric sensitivity (elasticity) under the conditions of two-sided restrictions on the range of values of a continuous random variable

The goal of the work. Development of an EXCEL-oriented calculator for calculating entropy and its elasticity for distribution functions under the condition of a limited domain of definition of a continuous random variable. Subject of study. Probability density functions and their entropy under the condition of bilateral restrictions on the domain of possible values of random variables. Research methods. Algorithmic and numerical analysis of procedures for obtaining numerical values of entropy of density functions of continuous random variables under the condition of two-way restrictions on the area of its definition. The obtained results. The work proposes an EXCEL-oriented calculator for calculating entropy and its elasticity for distribution functions under the condition of a limited area of definition of a continuous random variable. All distribution functions used in the work are divided into three categories depending on the form in which entropy and its elasticity are presented. The first category includes distribution functions for which entropy and its elasticity can be determined analytically, namely: Simpson distribution, Right-angled and negatively skew version distribution, Right-angled and positively skew version distribution, Sine Wave Distribution, Ark – Sine Distribution Type I, Ark – Sine Distribution Type IІ, Ordinary Cosine Distribution, Raised Cosine Distribution, U-shaped parabolic distribution Type I, Semi-elliptical distribution, Ark – Sine Distribution Type II. Another category includes distribution functions for which entropy can be defined in the analytical group, its elasticity can be defined in the tabular part, namely: Truncated Normal Distribution, Kumaraswamy Distribution, Beta Distribution, Generalized Beta Distribution, Uniformly increasing distribution density, Ark-Sine Distribution Type III. The third category includes distribution functions for which entropy and its elasticity can be determined in tabular form. Basic information about the structure of the proposed calculator and examples of its application are provided. The method of determining the confidence interval of the point estimate of entropy is described. An example of using the calculator proposed in this message for planning an experiment related to entropy data analysis is given.

Opportune warning of COVID-19 in a Mexican health care worker cohort: Discrete beta distribution entropy of smartwatch physiological records

We present a statistical study of heart rate, step cadence, and sleep stage registers of health care workers in the Hospital General de México “Dr. Eduardo Liceaga” (HGM), monitored continuously and non-invasively during the COVID-19 contingency from May to October 2020, using the Fitbit Charge 3® Smartwatch device. The HGM-COVID cohort consisted of 115 participants assigned to areas of COVID-19 exposure. We introduce a novel biomarker for an opportune signal for the likelihood of SARS-CoV-2 infection based on the Shannon Entropy of the Discrete Generalized Beta Distribution fit of rank ordered smartwatch registers. Our statistical test indicated infection for 94% of patients confirmed by positive polymer chain reaction (PCR+) test, 47% before the test, and 47% in coincidence. These results required innovative data preprocessing for the definition of a new biomarker index. The statistical method parameters are data-driven, confidence estimates were calibrated based on sensitivity tests using appropriately derived surrogate data as a benchmark. Our surrogate tests can also provide a benchmark for comparing results from other anomaly detection methods (ADMs). Biomarker comparison of the negative Immunoglobulin G Antibody (IgG-) subgroup with the PCR+ subgroup showed a statistically significant difference (p < 0.01, effect size = 1.44). The distribution of the uninfected population had a lower median and less dispersion than the PCR+ population. A retrospective study of our results confirmed that the biomarker index provides an early warning of the likelihood of COVID-19, even several days before the onset of symptoms or the PCR+ test request. The method can be calibrated for the analysis of different SARS-CoV-2 strains, the effect of vaccination, and previous infections. Furthermore, our biomarker screening could be implemented to provide general health profiles for other population sectors based on physiological signals from smartwatch wearable devices.

Frequency Analysis of Extreme Events Using the Univariate Beta Family Probability Distributions

This manuscript presents three families of distributions, namely the Beta, Beta Prime and Beta Exponential families of distributions. From all the distributions of these families, 14 statistical distributions of three, four and five parameters are presented that have applicability in the analysis of extreme phenomena in hydrology. These families of distributions were analyzed regarding the improvement of the existing legislation for the determination of extreme events, specifically the elaboration of a norm regarding frequency analysis in hydrology. To estimate the parameters of the analyzed distributions, the method of ordinary moments and the method of linear moments were used; the latter conforms to the current trend for estimating the parameters of statistical distributions. The main purpose of the manuscript was to identify other distributions from these three families with applicability in flood frequency analysis compared to the distributions already used in the literature from these families, such as the Log–logistic distribution, the Dagum distribution and the Kumaraswamy distribution. The manuscript does not exclude the applicability of other distributions from other families in the frequency analysis of extreme values, especially since these families were also analyzed within the research carried out in the Faculty of Hydrotechnics and presented in other materials. All the necessary elements for their use are presented, including the probability density functions, the complementary cumulative distribution functions, the quantile functions and the exact and approximate relations for estimating parameters. A flood frequency analysis case study was carried out for the Prigor RiverRiver, to numerically present the proposed distributions. The performance of this distributions were evaluated using the relative mean error, the relative absolute error and the L-skewness–L-kurtosis diagram. The best fit distributions are the Kumaraswamy, the Generalized Beta Exponential and the Generalized Beta distributions, which presented a stability related to both the length of the data and the presence of outliers.

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

To simulate non-Gaussian stochastic processes based on the first four moments, various simulation methods are presented, in which the determination of the transformation model and the calculation of the correlation coefficients between non-Gaussian stochastic processes and Gaussian stochastic processes are the critical procedures in these simulation methods. However, some existing simulation methods are limited to specific ranges. Furthermore, their practical applications are affected negatively due to the expensive cost of determining the transformation model and the correlation coefficients between non-Gaussian and Gaussian stochastic processes. Therefore, an accurate and efficient simulation method of a non-Gaussian stochastic process with a broader range is proposed in this article. Since the simulation of non-Gaussian processes and the Nataf transformation of non-Gaussian variables have some similar characteristics, a new combined distribution is proposed based on the unified Hermite polynomial model (UHPM) and the generalized beta distribution (GBD). Then, the combined distribution is employed in the simulation of non-Gaussian stochastic processes, in which the transformation model is deduced by the combined distribution. The correlation coefficient transformation function (CCTF) between the Gaussian and non-Gaussian stochastic processes can be evaluated through the interpolation method. Furthermore, numerical examples are presented to show the accuracy and effectiveness of the proposed simulation method for non-Gaussian stochastic processes.

Rethinking Generalized Beta family of distributions

We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the "traditional" GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the "traditional" GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S\&P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings.

Inverse Exponentiated Odd Lomax Exponential Distribution: Properties and Applications

Background: New family of distributions has important functions in generalization of distributions by modifying some existing distributions for getting more flexible irrespective to applied and practical view point. The Inverse Exponentiated Odd Lomax Exponential Distribution (IEOLE) having four parameters is suggested. Proposed model is based on T-X family of distribution which is the extended form of beta-generated distribution. Based on the LSE, MLE, and CVM methods, the parameters of the proposed distribution are estimated. Different model validation criteria and model comparisons are done by considering other existing models.&#x0D; Materials and methods: IEOLE is compound distribution derived by using theoretical concept of Odd Lomax Exponential distribution and T-X family of distribution. The parameters of the proposed distribution are estimated through the least square, Cramer–Von Mises and maximum likelihood methods. The applicability of the proposed model is evaluated using R programming on two real-life time data sets.&#x0D; Results: The statistical properties and different characteristics like the hazard rate function, the cumulative distribution function, quantile function, skewness, and kurtosis of the proposed model are discussed. Box plots, TTT plot, density fits etc. shows that the proposed model fits better to considered two real data sets. Different model validation criteria such as AIC, BIC, and CAIC are obtained and compared with some existing well-known probability distributions.&#x0D; Conclusion: This study presents new probability model called Inverse Exponentiated Odd Lomax Exponential distribution. The density curves of the model show that it is unimodal having right skewed. The suggested model has proven to be versatile for modeling real-world data due to its increasing-decreasing, right-skewed form. Also, the hazard rate function (HRF) graph is decreasing, decreasing-increasing or inverted bathtub shaped according to the value of the model constants. Different validation criteria show that the suggested model fits data well and the goodness-of-fit shows that proposed model has lesser test statistic value and higher p- value with respect to some existing models.

Designing acceptance single sampling plans: An optimization‐based approach under generalized beta distribution

AbstractThe availability of information that suppliers possess about the production process, as well as about the technical and economic consequences for customers, encourages the development and application of acceptance sampling plans that follow economic criteria, such as the Bayesian ones proposed in the literature. The combination of prior knowledge described by the prior distribution and empirical knowledge based on the sample leads to the decision to accept or reject the lot under inspection. The main purpose of this study was to derive acceptance sampling plans for attributes based on a prior generalized beta distribution following the economic criterion to minimize the expected total cost of quality. Specifically, a procedure is proposed to define the optimal sampling plan based on the technical characteristics of the production process and the costs inherent in the quality of the product. After the methodological aspects are described in detail, an extensive simulation study is reported that demonstrates how the optimal plan changes according to the main parameters, providing guidance for practitioners.

Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution

Abstract In this paper, statistical and volatility forecasting performances of the non-path-dependent score-driven multi-regime Markov-switching (MS) exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models are explored. Three contributions to the existing literature are provided. First, we use all relevant score-driven distributions from the literature - namely, the Student’s t-distribution, general error distribution (GED), skewed generalized t-distribution (Skew-Gen-t), exponential generalized beta distribution of the second kind (EGB2), and normal-inverse Gaussian (NIG) distribution. We then introduce the score-driven Meixner (MXN) distribution-based EGARCH model to the literature on score-driven models. Second, proving the sufficient conditions of the asymptotic properties of the maximum likelihood (ML) estimator for non-path-dependent score-driven MS-EGARCH models is an unsolved problem. We provide a partial solution to that problem by proving necessary conditions for the asymptotic theory of the ML estimator. Third, to the best of our knowledge, this work includes the largest number of international stock indices from the G20 countries in the literature, covering the period of 2000–2022. We provide a discussion on the major events which caused common or non-common switching to the high-volatility regime for the G20 countries. The statistical performance and volatility forecasting results support the adoption of score-driven MS-EGARCH for the G20 countries.

Validating citation models by proxy indices

• We compare three-dimensional model of scientific impact with another citation models. • We test their ability to reproduce the values of selected bibliometric indices. • We indicate which indices can be predicted with high accuracy. There are many approaches to the modelling of citation vectors of individual authors. Models may serve different purposes, but usually they are evaluated with regards to how well they align to citation distributions in large networks of papers. Here we compare a few leading models in terms of their ability to correctly reproduce the values of selected bibliometric indices of individual authors. Our recently-proposed three-dimensional model of scientific impact serves this purpose equally well as the discrete generalised beta distribution and the log-normal models, but has fewer parameters which additionally are all easy to interpret. We also indicate which indices can be predicted with high accuracy and which are more difficult to model.

A rank-size approach to analyse soccer competitions and teams: the case of the Italian football league “Serie A"

In this paper, we present a data-analysis rank-size approach to assess the features of soccer competitions and competitors. We investigate the championships rankings and the teams’ final scores in the most relevant Italian league, the “Serie A”, between 1930 and 2020. We use the final rankings and the teams’ scores to explore the presence of rank-size regimes in the various yearly championships. Besides, we analyse the teams one by one, ranking their performance over the years and using the rank-size law’s parameters to compare their performances across the tournaments. We chose to do so via the Discrete Generalised Beta Distribution, a three-parameter rank-size function. We offer a cluster analysis of the rank-size law parameters based on a k-means algorithm to provide additional insights and capture similarities and deviations among championships and teams. Concluding, we propose a measure of competitiveness within championships and per team. The best fit results are statistically outstanding, and the cluster analysis presents two main clusters capturing teams’ performances and years in which they have competed in the “Serie A”. The competitiveness analysis shows that the teams at the bottom of the championships ranking have obtained decreasing scores in recent years.

Identifying intergenerational patterns of correlated methylation sites

DNA methylation can be transmitted through generations. This paper proposes a clustering method to identify the intergenerational patterns from parents to their offspring. Motivated by the potential of correlation between DNA methylation sites, we use the multivariate generalized beta distribution to model the blockwise correlation structure among the sites. A stochastic EM algorithm is implemented to estimate the parameters, and BIC is applied to determine the optimal number of clusters. Simulations demonstrate the feasibility of the proposed method. We further applied the approach to cluster DNA methylation data generated from a cohort study on asthma and allergic conditions.

The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality

When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.