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The Pareto–Feller distribution has been widely used across various disciplines to model "heavy-tailed" phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown. [ABSTRACT FROM AUTHOR]

This paper introduces a novel non-homogeneous stochastic diffusion process, useful for modeling both decreasing and increasing trend data. The model is based on a generalized Gamma-like curve. We derive the probabilistic characteristics of the proposed process, including a closed-form unique solution to the stochastic differential equation, the transition probability density function, and both conditional and unconditional trend functions. The process parameters are estimated using the maximum likelihood (ML) method with discrete sampling paths. A small Monte Carlo experiment is conducted to evaluate the finite sample behavior of the trend function. The practical utility of the proposed process is demonstrated by fitting it to two real-world data sets, one exhibiting a decreasing trend and the other an increasing trend. [ABSTRACT FROM AUTHOR]

This paper introduces a novel non-homogeneous stochastic diffusion process, useful for modeling both decreasing and increasing trend data. The model is based on a generalized Gamma-like curve. We derive the probabilistic characteristics of the proposed process, including a closed-form unique solution to the stochastic differential equation, the transition probability density function, and both conditional and unconditional trend functions. The process parameters are estimated using the maximum likelihood (ML) method with discrete sampling paths. A small Monte Carlo experiment is conducted to evaluate the finite sample behavior of the trend function. The practical utility of the proposed process is demonstrated by fitting it to two real-world data sets, one exhibiting a decreasing trend and the other an increasing trend.

We introduce and discuss a system of one-dimensional kinetic equations describing the influence of higher education in the social stratification of a multi-agent society. The system is obtained by coupling a model for knowledge formation with a kinetic description of the social climbing in which the parameters characterizing the elementary interactions leading to the formation of a social elite are assumed to depend on the degree of knowledge/education of the agents. In addition, we discuss the case in which the education level of an individual is function of the position occupied in the social ranking. With this last assumption, we obtain a fully coupled model in which knowledge and social status influence each other. In the last part, we provide several numerical experiments highlighting the role of education in reducing social inequalities and in promoting social mobility. [ABSTRACT FROM AUTHOR]

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown.

Defining properly the time distribution of a steady‐state process is crucial to its management. A common practice is to assume that process‐time is normally distributed for repetitive processes (process has constant work‐content), and exponentially for non‐repetitive processes (memoryless; no characteristic work‐content). This dichotomous distinction ignores the majority of processes, residing in between the two extreme scenarios, the semi‐repetitive ones. These processes own a characteristic duration time (as reflected in the mode), yet part of work‐content ("process identity") randomly varies between cycles. In this paper, we develop a unified platform to model process‐time, comprising all three types of processes. The effects of work‐content instability on shape characteristics of process‐time distribution are studied, and process repetitiveness measure, possibly to be used to monitor work‐content instability, is defined. The generalized gamma distribution is employed to approximate the (unknown) distribution of process‐time sample mean, becoming exact for the two extreme scenarios (repetitive and non‐repetitive processes). [ABSTRACT FROM AUTHOR]

This article considers one-off actions for a football coach who is asking for a specific outcome from a play. This will be in the form of a minimum gain in yards, usually in order to gain a first down. Using a random utility model approach we propose the play to be called is the one which maximizes the probability of the desired outcome. We specifically focus on pass plays, which requires the modeling of outcomes in terms of yards gained, for which we use the family of generalized gamma distributions. The data and results relate to the Fall 2021 Presbyterian College football team, in which we leverage specific information pertaining to the offensive playbook. [ABSTRACT FROM AUTHOR]

Background: The primary objective of this study was to analyze the long-term survival of 48,067 chemical warfare survivors who suffered from pulmonary, cutaneous, and ocular lesions in the decades following the Iran-Iraq war. Methods: The data for this study were obtained from the Veterans and Martyr Affair Foundation (VMAF) database. The survivors were divided into two groups based on whether they were evacuated/admitted (EA) to a hospital or not evacuated/admitted (NEA) to a hospital. The proportional hazard (PH) assumption for age categories, gender, exposure statuses, and eye severity was not satisfied. Therefore, we used a Generalized Gamma (GG) distribution with an Accelerated Failure Time (AFT) model for analysis. Results: The study included a total of 48,067 observations, and among them, 4342 (9.03 %) died during the study period. The mean (SD) age of the survivors was 55.99 (7.9) years. The mortality rate increased with age, and higher rates were observed in males. Survival probabilities differed significantly among age categories, provinces, lung severity, and eye severity based on log-rank tests (p-value

This paper considers a generalization of the Rényi theorem to the case of a structural distribution with a scale parameter. In terms of the zeta metric, some estimates of the convergence rate in the generalized Rényi theorem are obtained when the structural mixed Poisson distribution of the summation index is a scale mixture of the generalized gamma distribution. Estimates of the convergence rate for the structural digamma distribution are given as a special case. The paper extends the results previously obtained for the generalized gamma distribution. [ABSTRACT FROM AUTHOR]

In the paper, quasi-exponentiated normal distributions are introduced for any real power (exponent) no less than two. With natural exponents, the quasi-exponentiated normal distributions coincide with the distributions of the corresponding powers of normal random variables with zero mean. Their representability as scale mixtures of normal and exponential distributions is proved. The mixing distributions are written out in the closed form. Two approaches to the construction of asymmetric quasi-exponentiated normal distributions are described. A limit theorem is proved for sums of a random number of independent random variables in which the asymmetric quasi-exponentiated normal distribution is the limit law. [ABSTRACT FROM AUTHOR]

This article considers the Bayes analysis of generalized gamma accelerated failure time model and its two components Weibull and gamma when the given observations are left truncated and right censored. In order to perform the analysis, the paper proposes the use of an improved version of the Metropolis-Hastings algorithm, namely, the Metropolis-adjusted Langevin algorithm. Besides, the paper also checks the model compatibility and compares the considered models with its components using the Bayes factor computed on the basis of a recent methodology. A numerical illustration is provided based on a simulated as well as a real dataset. The real dataset consists of individuals infected with human immunodeficiency virus who are at the risk of acquired immunodeficiency syndrome and subsequent deaths. The numerical illustration is further extended to check the effect of different therapies including the highly active antiretroviral therapy on the lifetime of individuals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behaviors of moment for normalized extreme of the generalized gamma distribution. Under optimal norming constants, we establish higher-order expansion of moment for the maximum. The expansion is used to deduce the rate of convergence of the moment for normalized partial maximum to the moment of the associating extreme value limit. Numerical simulations are given to sustain the results of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. Based on the previously proved asymptotic normality of the estimators for the characteristic index and the shape and scale parameters of the digamma distribution constructed from a fixed-size sample, we obtain a statement about the convergence of these estimators to the scale mixtures of the normal law in the case of a random sample size. Using this result, asymptotic confidence intervals for the estimated parameters are constructed. A number of examples of the limit laws for sample sizes with special forms of negative binomial distributions are given. The results of this paper can be widely used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support. [ABSTRACT FROM AUTHOR]

For comparing largest order statistics from independent heterogeneous non-negative scale variables in the mean residual life order, a new framework is introduced here. This framework can be viewed as a generalization of the well-known multiple-outlier scale model, and it additionally includes the situation in which all the random variables are heterogeneous. We also find some sufficient conditions for comparing the largest order statistics, one with complete heterogeneous scale parameters and another with homogeneous scale parameters, in the mean residual life order. As examples of the obtained results, generalized gamma, generalized beta of the second kind, power-generalized Weibull, and half-normal distributions are all presented. The findings of this work generalize and also reinforce some of the existing results in this direction. [ABSTRACT FROM AUTHOR]

A wide range of signs are acquired from the human body called biomedical signs or biosignals, and they can be at the cell level, organ level, or sub-atomic level. Electroencephalogram is the electrical activity from the cerebrum, the electrocardiogram is the electrical activity from the heart, electrical action from the muscle sound signals referred to as electromyogram, the electroretinogram from the eye, and so on. Studying these signals can be so helpful for doctors, and it can help them examine and predict and cure many diseases. However, these signals are often affected by various types of noise. It is important to denoise the signals to get accurate information from them. The denoising process is solved by proposing an entirely novel family of flexible score functions for blind source separation, based on a family of generalized Gamma densities. To blindly extract the independent source signals, we resort to the popular fast independent component analysis (FastICA) approach; to adaptively estimate the parameters of such score functions, we use an efficient method based on maximum likelihood. The results obtained using generalized Gamma densities in our technique are better than those obtained by other distribution functions. [ABSTRACT FROM AUTHOR]

Advances in biologic technology enable researchers to obtain a huge amount of genetic and genomic data, whose dimensions are often quite high on both phenotypes and variants. Testing their association with multiple phenotypes has been a hot topic in recent years. Traditional single phenotype multiple variant analysis has to be adjusted for multiple testing and thus suffers from substantial power loss due to ignorance of correlation across phenotypes. Similarity-based method, which uses the trace of product of two similarity matrices as a test statistic, has emerged as a useful tool to handle this problem. However, it loses power when the correlation strength within multiple phenotypes is middle or strong, for some signals represented by the eigenvalues of phenotypic similarity matrix are masked by others. We propose a divided-and-combined omnibus test to handle this drawback of the similarity-based method. Based on the divided-and-combined strategy, we first divide signals into two groups in a series of cut points according to eigenvalues of the phenotypic similarity matrix and combine analysis results via the Cauchy-combined method to reach a final statistic. Extensive simulations and application to a pig data demonstrate that the proposed statistic is much more powerful and robust than the original test under most of the considered scenarios, and sometimes the power increase can be more than 0.6. Divided-and-combined omnibus test facilitates genetic association analysis with high-dimensional data and achieves much higher power than the existing similarity based method. In fact, divided-and-combined omnibus test can be used whenever the association analysis between two multivariate variables needs to be conducted. [ABSTRACT FROM AUTHOR]

This paper presents a new distribution, the product of the mixture between Maxwell-Boltzmann and a particular case of the generalized gamma distributions. The resulting distribution, called the Scale Mixture Maxwell-Boltzmann, presents greater kurtosis than the recently introduced slash Maxwell-Boltzmann distribution. We obtained closed-form expressions for its probability density and cumulative distribution functions. We studied some of its properties and moments, as well as its skewness and kurtosis coefficients. Parameters were estimated by the moments and maximum likelihood methods, via the Expectation-Maximization algorithm for the latter case. A simulation study was performed to illustrate the parameter recovery. The results of an application to a real data set indicate that the new model performs very well in the presence of outliers compared with other alternatives in the literature. [ABSTRACT FROM AUTHOR]

In other to improve the ability of decisionmakers to prepare for and deal with the unforeseen circumstances resulting from climate change as consequences of precipitation fluctuations, extreme and torrential rainfall. It is important to provide a more complete understanding of the range and likelihood of rainfall patterns a location could receive using a probabilistic model whose parameters might complement or even replace such common measures as the mean, median, variance, minimum, maximum and quartile values as major descriptors of rainfall at such location. Daily precipitation totals can be approximated by the gamma distribution as it is bounded on the left at zero and positively skewed indicating an extended tail to the right which suit the distribution of daily rainfall and accommodate the lower limit of zero which constrains rainfall values. This paper presents the comparison between Maximum Likelihood Estimation (MLE) of closed & open form solutions and Method of Moment Estimation (MME) of location and scaling parameters of the twoparameter gamma distribution, the parameters were estimated using MME and MLE with their performance adjudged and the result obtained showed that the closedform solution of the MLE outperformed the open form solution and MME by comparing their estimates for the scaling parameter. [ABSTRACT FROM AUTHOR]

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed. [ABSTRACT FROM AUTHOR]

Abstract In fast-fading scattering environments such as high-speed rail and low-altitude communications, mobile communication systems need to quickly and robustly estimate and equalize fast-fading, time-varying channels. Under these circumstances, noncoherent multiple-input multiple-output (MIMO) has received attention in recent years, since it is less influenced by factors such as phase fluctuations and has fewer requirements for channel estimation and synchronization. Spatial multiplexing MIMO-MFSK based on energy detection is different from conventional noncoherent MIMO in that it can achieve higher spatial multiplexing gain in the independent distribution of channel fading statistics. At the receiver, partial real channel state information (CSI) is available, which can improve the capacity while ensuring the reliability of the link. With partial real CSI replacing instantaneous CSI, the system performance is inferior to conventional coherent MIMO under the influence of noise. As a result, it is necessary to analyze its theoretical detection performance. By energy detection, the noise of the MIMO-MFSK system conforms to the generalized gamma distribution. On the basis of this distribution, the optimal decision rule of the system and the symbol error rate (SER) formula are derived. Additionally, we investigate the signal-dependent noise problem of minimum Euclidean distance detection. Numerical results show that the SER formula fits well with the simulation results under the condition of a high signal-to-noise ratio.

This paper considers a generalization of the Rényi theorem to the case of a structural distribution with a scale parameter. In terms of the zeta metric, some estimates of the convergence rate in the generalized Rényi theorem are obtained when the structural mixed Poisson distribution of the summation index is a scale mixture of the generalized gamma distribution. Estimates of the convergence rate for the structural digamma distribution are given as a special case. The paper extends the results previously obtained for the generalized gamma distribution.

In the paper, quasi-exponentiated normal distributions are introduced for any real power (exponent) no less than two. With natural exponents, the quasi-exponentiated normal distributions coincide with the distributions of the corresponding powers of normal random variables with zero mean. Their representability as scale mixtures of normal and exponential distributions is proved. The mixing distributions are written out in the closed form. Two approaches to the construction of asymmetric quasi-exponentiated normal distributions are described. A limit theorem is proved for sums of a random number of independent random variables in which the asymmetric quasi-exponentiated normal distribution is the limit law.

The gamma distribution is one of the most widely used statistical distribution in fitting various real life data set in the field of the reliability to fit failure data, because it is sufficiently flexible to deal with decreasing, constant, and increasing failure rates. In this article, the stock investment valuation model has been presented on the assumption of Gamma distribution. The purpose is to analysis profitability of investment in the property development sector in Malaysia based on the investment rate of return on investment. The flow of the investment cash and the accumulation of the share units over the investment period have been presented. it demonstrated the computing of MIRR by the case of a company which experienceed distributing a treasury share dividend, which increases the small portion of investor’s share units instead of receiving the dividend in cash. As the MIRR might exhibit negative values, the Gamma distribution of the transformed MIRR for certain investment periods is developed and the method of moment is applied to estimate the parameters. The study revealed that the transformed MIRR is well fitted with Gamma distribution for four years to eight years investment periods. It is shown that this approach has a better performance return on investment in Malaysia property sector from 2008 to 2018.

Theoretical properties of a new survival distribution based upon the renowned distribution has been proposed, such as the probability density function, the cumulative distribution function, the ordinary moment, the moment generating function, the incomplete moments, the survival function, the hazard rate function, the mode, and the asymptotic behavior. This new distribution is called the new mixture of generalized gamma distribution. In addition, the maximum likelihood estimator of an unknown parameter has been studied. Finally, to illustrate usefulness of the proposed distribution, the model has been applied in describing actual data sets. [ABSTRACT FROM AUTHOR]

The area of the cells of Voronoi tessellations has been modelled through different probability distributions among which the most promising are the generalized gamma and tapered Pareto distributions. In particular the latter has been used to model times and distances between successive earthquakes besides area and perimeter of cells generated by earthquake epicenters. In the framework of nonextensive statistical mechanics applied in geophysics, variables like seismic moment, inter-event time or Euclidean distance between successive earthquakes or length of faults in a given region have been studied through the so-called q-exponential distributions obtained by maximizing the Tsallis entropy under suitable conditions. These distributions take also the name of generalized Pareto distributions in the context of the limit distributions of excesses. In this work we consider the spatial distribution of a set of earthquakes and its temporal variations by modelling the area of Voronoi cells generated by the epicenters through a generalized Pareto distribution. Following the Bayesian paradigm we analyze the recent seismicity of the central Italy and we compare the posterior marginal likelihood of the aforementioned distributions in shifting time windows. We point out that the best fitting distribution varies over time and the trend of all three distributions converges to that of the exponential distribution in the preparatory phase for the mainshock.

The paper considers a new digamma distribution generalizing the distributions from the gamma and beta classes. The presentation of the digamma distribution as a fractional-scale mixture of gamma distributions is proved. Explicit forms of the moments and density of the considered distribution are given. A method for statistical estimation of unknown parameters based on logarithmic cumulants is described. A number of numerical examples of estimating concentration parameters from model samples are given. [ABSTRACT FROM AUTHOR]

In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. Based on the previously proved asymptotic normality of the estimators for the characteristic index and the shape and scale parameters of the digamma distribution constructed from a fixed-size sample, we obtain a statement about the convergence of these estimators to the scale mixtures of the normal law in the case of a random sample size. Using this result, asymptotic confidence intervals for the estimated parameters are constructed. A number of examples of the limit laws for sample sizes with special forms of negative binomial distributions are given. The results of this paper can be widely used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

In this paper, we provide a new bivariate distribution obtained from a Kibble-type bivariate gamma distribution. The stochastic representation was obtained by the sum of a Kibble-type bivariate random vector and a bivariate random vector builded by two independent gamma random variables. In addition, the resulting bivariate density considers an infinite series of products of two confluent hypergeometric functions. In particular, we derive the probability and cumulative distribution functions, the moment generation and characteristic functions, the Hazard, Bonferroni and Lorenz functions, and an approximation for the differential entropy and mutual information index. Numerical examples showed the behavior of exact and approximated expressions. [ABSTRACT FROM AUTHOR]

In fast-fading scattering environments such as high-speed rail and low-altitude communications, mobile communication systems need to quickly and robustly estimate and equalize fast-fading, time-varying channels. Under these circumstances, noncoherent multiple-input multiple-output (MIMO) has received attention in recent years, since it is less influenced by factors such as phase fluctuations and has fewer requirements for channel estimation and synchronization. Spatial multiplexing MIMO-MFSK based on energy detection is different from conventional noncoherent MIMO in that it can achieve higher spatial multiplexing gain in the independent distribution of channel fading statistics. At the receiver, partial real channel state information (CSI) is available, which can improve the capacity while ensuring the reliability of the link. With partial real CSI replacing instantaneous CSI, the system performance is inferior to conventional coherent MIMO under the influence of noise. As a result, it is necessary to analyze its theoretical detection performance. By energy detection, the noise of the MIMO-MFSK system conforms to the generalized gamma distribution. On the basis of this distribution, the optimal decision rule of the system and the symbol error rate (SER) formula are derived. Additionally, we investigate the signal-dependent noise problem of minimum Euclidean distance detection. Numerical results show that the SER formula fits well with the simulation results under the condition of a high signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

Parametric family of statistical distributions are of great importance for several applications. In particular, we propose to investigate the generalized Gamma mixture model (gΓMM) for modeling and classifying medical imaging (Chest x-ray and CT-scans). The main advantage of this mixture over some existing Gaussian models is that it allows additional flexibility in shape modeling, which is crucial for classification systems. In order to capture accurately the intrinsic nature of medical images, we propose to derive some efficient measures based on Fisher, Kullback-Leibler and Bhattacharyya distances for the mixtures of generalized Gamma distributions. Indeed, the main idea is to investigate these distances effectively via the statistical model parameters in order to make our proposed scheme particularly appropriate for image classification problem. The proposed approach involves the extraction of robust texture descriptors, the learning of mixture model gΓMM via the expectation-maximization (EM) and Newton-Raphson algorithms, and the classification of images using the derived mixtures-based distances. We evaluate our model against the challenging problem of early diagnosis of pneumonia diseases. Experimental results on different datasets show the merits of our developed framework compared with the other methods.

Abstract: In this paper, we introduce a new life time distribution . This distribution based on the reciprocal of Generalized Gamma (GG) random variable . This new distribution is called the Inverse Generalized Gamma (IGG) Distribution in which some of the inverse distributions are special cases. The important benefit of this distribution is ability to fit skewed data that cannot be fitted accurately by many other ungeneralized life time distributions. This distribution has many applications in pollution data ,engineering ,Biological fields and reliability. Some theoretical properties of the distribution has been studied such as: moments, mode, median and other properties. It is concluded that the distribution is skew with heavy tail and the skewness increased when the shape parameters increased but the scale parameter has no effect on the skewness and kurtosis.

The precision of Rayleigh distribution, as the simplest fading model in Non-Line-of-Sight (NLOS) channels, is low in high-resolution radars and long-distance communication receivers. Many currently-available statistical models with a higher precision, including Nakagami-m, Weibull and generalized hybrid Gamma models, are used to describe the radar clutter and the reflected signals in communication receivers. Although the mentioned models in NLOS channels have more accurate matching with the actual fading, a variety of models and the lack of a comprehensive model in different fading channels make it difficult to select an appropriate model. In this paper, the generalized Gamma model is analyzed and evaluated to demonstrate that it adapts to other fading models. Moreover, it also matches with long-term fading, as well as combined long and short-term fading models. Using a practical sample, it is stated that the generalized Gamma model is also adaptable to the channels to which none of the other available closed-form models are suitable. The simulations and the real data results, approved by Kullback-Leibler and Kolmogorov-Smirnov criteria, prove the claim.

This paper presents a new distribution, the product of the mixture between Maxwell-Boltzmann and a particular case of the generalized gamma distributions. The resulting distribution, called the Scale Mixture Maxwell-Boltzmann, presents greater kurtosis than the recently introduced slash Maxwell-Boltzmann distribution. We obtained closed-form expressions for its probability density and cumulative distribution functions. We studied some of its properties and moments, as well as its skewness and kurtosis coefficients. Parameters were estimated by the moments and maximum likelihood methods, via the Expectation-Maximization algorithm for the latter case. A simulation study was performed to illustrate the parameter recovery. The results of an application to a real data set indicate that the new model performs very well in the presence of outliers compared with other alternatives in the literature.

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed.

Advancement in technology brings a revolutionary change in the quality of the final product or items. Most of the manufacturing processes produce a large number of conforming items along with a few non-conforming items. For real-time monitoring of these highly efficient processes, monitoring of time-between-events is a well-known approach adopted in the literature of statistical process control. Usually, it is assumed that the time-between-events follows an exponential or gamma distribution. However, the generalized gamma distribution is the most popular choice for modelling skewed data. In this article, we consider a two-sided monitoring scheme based on the generalized gamma distribution. Two-sided monitoring schemes for skewed distributions often encounter bias in its run length properties. Therefore, we address this problem with modified control limits in a more general distributional setup. A Monte Carlo simulation-based study is designed, and results showed encouraging performance properties. A couple of practical applications in connection to monitoring renewable energy and coal mine explosions have been discussed. [ABSTRACT FROM AUTHOR]

Recently, the overall reference prior has been proposed for parameters of the generalized gamma distribution. As an alternative, here we develop a matching prior that provides the same coverage probability asymptotically as a Bayesian credible interval with the corresponding frequentist counterpart, and subsequently show that this matching prior yields proper posteriors. In addition, we find that the overall reference prior is not a first-order matching prior. Simulation studies show that the derived matching priors perform better than the overall reference prior in meeting the target coverage probabilities, and meets the target coverage probabilities well even for a small sample size. [ABSTRACT FROM AUTHOR]

Many spatial statistics methods require neighbourhood structures such as the one determined by a Voronoi tessellation, so understanding statistical properties of Voronoi cells is crucial. While distributions of cell properties when data locations follow an unbounded homogeneous Poisson process have been studied, little attention has been given to how these properties change when a boundary is imposed. This is important when geographical data are gathered within a restricted study area, such as a national boundary or a coastline. We study the effects of imposing a boundary on the cell properties of a Poisson Voronoi tessellation. The area, perimeter and number of edges of individual cells with and without boundary conditions are investigated by simulation. Distributions of these properties differ substantially when boundaries are imposed, and these differences are affected by proximity to the boundary. We also investigate how changes in such properties when boundaries are imposed vary over two-dimensional space. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new bivariate distribution obtained from a Kibble-type bivariate gamma distribution. The stochastic representation was obtained by the sum of a Kibble-type bivariate random vector and a bivariate random vector builded by two independent gamma random variables. In addition, the resulting bivariate density considers an infinite series of products of two confluent hypergeometric functions. In particular, we derive the probability and cumulative distribution functions, the moment generation and characteristic functions, the Hazard, Bonferroni and Lorenz functions, and an approximation for the differential entropy and mutual information index. Numerical examples showed the behavior of exact and approximated expressions.

In this paper, we discuss the parameter estimation for the generalized gamma distribution based on left‐truncated and right‐censored data. A stochastic version of the expectation‐maximization (EM) algorithm is proposed as an alternative method to compute approximate maximum likelihood estimates. Two different methods to obtain reliable initial estimates of the parameters required for the iterative algorithms are also proposed. Interval estimation based on a parametric bootstrap method is discussed. The proposed methodologies are illustrated with a numerical example. Then, a Monte Carlo simulation study is used to evaluate the performance of the proposed estimation procedures and to compare with the direct optimization method and the conventional EM algorithm. Based on the simulation results, we show that the proposed stochastic EM algorithm is a useful alternative estimation method for the model fitting of the generalized gamma distribution. [ABSTRACT FROM AUTHOR]

The generalized gamma distribution (GGD) is a popular distribution because it is extremely flexible. Due to the density function structure of GGD, estimating the parameters of the GGD family by statistical point estimation techniques is a complicated task. In other words, for the parameter estimation, the maximizing likelihood function of GGD is a problematic case. Hence, alternative approaches can be used to obtain estimators of GGD parameters. This paper proposes an alternative parameter estimation method for GGD by using the heuristic optimization approaches such as Genetic Algorithms (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), and Simulated Annealing (SA). A comparison between different modern heuristic optimization methods applied to maximize the likelihood function for parameter estimation is presented in this paper. The paper also investigates both the performance of heuristic methods and estimation of GGD parameters. Simulations show that heuristic approaches provide quite accurate estimates. In most of the cases, DE has better performance than other heuristics in terms of bias values of parameter estimations. Besides, the usefulness of an alternative parameter estimation method for GGD using the heuristic optimization approach is illustrated with a real data set. [ABSTRACT FROM AUTHOR]

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.

Statistical modeling of synthetic aperture radar (SAR) data is a crucial step in SAR image processing. In this context, the generalized gamma (GGamma) distribution, which generalizes many common distributions, has been applied to model SAR image statistics. Parameter estimation remains, however, a challenging step that conditions the quality of model fitting to data and, thus, of the required processing. In this article, we propose a novel parameter estimation method for GGamma distribution in the log-domain, named as the maximum likelihood and logarithmic cumulants (ML–LC) method. The ML–LC method constructs a novel scale-independent shape parameter estimator in the log-domain based on the Mellin transform and maximum-likelihood estimation and estimates the distribution parameters based on the multistart local search, gradient descent, and bisection methods, rather than solving the system of highly nonlinear equations in the traditional estimations. The ML–LC method is able to estimate the GGamma distribution parameters more accurately. To assess the performance of our estimation method, we perform the goodness-of-fit test on simulated data and real SAR images. In addition, we apply the ML–LC method in some SAR image processing tasks covering image segmentation, classification, and change detection. The results obtained confirm the interest of the proposed ML–LC method. [ABSTRACT FROM AUTHOR]

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker–Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a social elite. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing. [ABSTRACT FROM AUTHOR]

The performance of reliability inference strongly depends on the modeling of the product's lifetime distribution. Many products have complex lifetime distributions whose optimal settings are not easily found. Practitioners prefer to use simpler lifetime distribution to facilitate the data modeling process while knowing the true distribution. Therefore, the effects of model mis-specification on the product's lifetime prediction is an interesting research area. This article presents some results on the behavior of the relative bias (RB) and relative variability (RV) of pth quantile of the accelerated lifetime (ALT) experiment when the generalized Gamma (GG3) distribution is incorrectly specified as Lognormal or Weibull distribution. Both complete and censored ALT models are analyzed. At first, the analytical expressions for the expected log-likelihood function of the misspecified model with respect to the true model is derived. Consequently, the best parameter for the incorrect model is obtained directly via a numerical optimization to achieve a higher accuracy model than the wrong one for the end-goal task. The results demonstrate that the tail quantiles are significantly overestimated (underestimated) when data are wrongly fitted by Lognormal (Weibull) distribution. Moreover, the variability of the tail quantiles is significantly enlarged when the model is incorrectly specified as Lognormal or Weibull distribution. Precisely, the effect on the tail quantiles is more significant when the sample size and censoring ratio are not large enough. for this article are available online. [ABSTRACT FROM AUTHOR]

This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techniques to estimate its parameters. Many authors con-sidered estimating the parameters of the generalized gamma distribution in a Bayesian framework using Jeffrey’s priors. Others used different loss functions and the least squares approach. This study uses Bayesian techniques to estimate the three-parameter generalized gamma distribution by using conjugate priors. The random Metropolis algorithm is used to simulate the Bayesian estimates of the three parameters. Then these estimates are compared to the maximum like-lihood estimates using the mean error through simulation. It has been shown in this paper that the obtained estimates using this approach is more accurate than the traditional methods of estimation such as the Maximum likelihood method. The same approach is then used to estimate the parameters of mixtures of the generalized gamma parameters using conjugate priors. [ABSTRACT FROM AUTHOR]