Search

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.

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.

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

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 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.

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.

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.

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.

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.

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.

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.

Due to the inaccuracy of modeling the correlated noise (CN) with Laplacian distribution,the rate-distortion performance of DVC is relatively low.CN modeling via generalized Gamma (GGA)distribution was proposed for pix-el-domain DVC.Statistical features of the CN modeling were analyzed.It was found that Laplacian distribution peak was lower than actual CN distribution.GGA distribution was exploited to fit CN and the GGA distribution parameters were estimated online.The simulation results indicate that the proposed model can more accurately reflect the statistics fea-tures of CN,improve rate distortion performance,and reduce computation complexity.

Due to the inaccuracy of modeling the correlated noise (CN) with Laplacian distribution,the rate-distortion performance of DVC is relatively low.CN modeling via generalized Gamma (GGA)distribution was proposed for pix-el-domain DVC.Statistical features of the CN modeling were analyzed.It was found that Laplacian distribution peak was lower than actual CN distribution.GGA distribution was exploited to fit CN and the GGA distribution parameters were estimated online.The simulation results indicate that the proposed model can more accurately reflect the statistics fea-tures of CN,improve rate distortion performance,and reduce computation complexity.

Currently, much research attention has focused on generalizations of known mathematical objects in order to obtain adequate models describing real phenomena. An important role in the applied theory of probability and mathematical statistics is the gamma class of distributions, which has proven to be a convenient and effective tool for modeling many real processes. The gamma class is quite wide and includes distributions that have useful properties such as, for example, infinite divisibility and stability, which makes it possible to use distributions from this class as asymptotic approximations in various limit theorems. One of the most important tasks of applied statistics is to obtain estimates of the parameters of the model distribution from the available real data. In this paper, we consider the gamma-exponential distribution, which is a generalization of the distributions from the gamma class. Estimators for some parameters of this distribution are given, and the asymptotic normality of these estimators is proven. When obtaining the estimates, a modified method of moments was used, based on logarithmic moments calculated on the basis of the Mellin transform for the generalized gamma distribution. On the basis of the results obtained, asymptotic confidence intervals for the estimated parameters are constructed. The results of this work can be used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are proved. Unlike the conventional analytical approach, here the presentation follows the lines of a kind of arithmetical approach in the space of random variables or vectors. Here the operation of scale mixing in the space of distributions is replaced with the operation of multiplication in the space of random vectors/variables under the assumption that the multipliers are independent. By doing so, the reasoning becomes much simpler, the proofs become shorter and some general features of the distributions under consideration become more vivid. The first part of the paper concerns the univariate case. Some known results are discussed and simple alternative proofs for some of them are presented as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions. The results obtained here open the way to deal with natural multivariate analogs of EP distributions. In the second part of the paper, we discuss the conventionally defined multivariate EP distributions and introduce the notion of projective EP (PEP) distributions. The properties of multivariate EP and PEP distributions are considered as well as limit theorems establishing the conditions for the convergence of multivariate statistics constructed from samples with random sizes (including random sums of random vectors) to multivariate elliptically contoured EP and projective EP laws. The results obtained here give additional theoretical grounds for the applicability of EP and PEP distributions as asymptotic approximations for the statistical regularities observed in data in many fields.

Mathematical models are proposed for statistical regularities of maximum daily precipitation within a wet period and total precipitation volume per wet period. The proposed models are based on the generalized negative binomial (GNB) distribution of the duration of a wet period. The GNB distribution is a mixed Poisson distribution, the mixing distribution being generalized gamma (GG). The GNB distribution demonstrates excellent fit with real data of durations of wet periods measured in days. By means of limit theorems for statistics constructed from samples with random sizes having the GNB distribution, asymptotic approximations are proposed for the distributions of maximum daily precipitation volume within a wet period and total precipitation volume for a wet period. It is shown that the exponent power parameter in the mixing GG distribution matches slow global climate trends. The bounds for the accuracy of the proposed approximations are presented. Several tests for daily precipitation, total precipitation volume and precipitation intensities to be abnormally extremal are proposed and compared to the traditional PoT-method. The results of the application of this test to real data are presented.

Background: The goal of this study is to extend the applications of parametric survival models so that they include cases in which accelerated failure time (AFT) assumption is not satisfied, and examine parametric and semiparametric models under different proportional hazards (PH) and AFT assumptions. Methods: The data for 12,531 women diagnosed with breast cancer in British Columbia, Canada, during 1990-1999 were divided into eight groups according to patients′ ages and stage of disease, and each group was assumed to have different AFT and PH assumptions. For parametric models, we fitted the saturated generalized gamma (GG) distribution, and compared this with the conventional AFT model. Using a likelihood ratio statistic, both models were compared to the simpler forms including the Weibull and lognormal. For semiparametric models, either Cox′s PH model or stratified Cox model was fitted according to the PH assumption and tested using Schoenfeld residuals. The GG family was compared to the log-logistic model using Akaike information criterion (AIC) and Baysian information criterion (BIC). Results: When PH and AFT assumptions were satisfied, semiparametric and parametric models both provided valid descriptions of breast cancer patient survival. When PH assumption was not satisfied but AFT condition held, the parametric models performed better than the stratified Cox model. When neither the PH nor the AFT assumptions were met, the log normal distribution provided a reasonable fit. Conclusions: When both the PH and AFT assumptions are satisfied, the parametric and semiparametric models provide complementary information. When PH assumption is not satisfied, the parametric models should be considered, whether the AFT assumption is met or not.

With the increase of resolution, effective characterization of synthetic aperture radar (SAR) image becomes one of the most critical problems in many earth observation applications. Inspired by deep learning and probability mixture models, a generalized Gamma deep belief network (g Γ-DBN) is proposed for SAR image statistical modeling and land-cover classification in this work. Specifically, a generalized Gamma-Bernoulli restricted Boltzmann machine (gΓB-RBM) is proposed to capture high-order statistical characterizes from SAR images after introducing the generalized Gamma distribution. After stacking the g Γ B-RBM and several standard binary RBMs in a hierarchical manner, a gΓ-DBN is constructed to learn high-level representation of different SAR land-covers. Finally, a discriminative neural network is constructed by adding an additional predict layer for different land-covers over the constructed deep structure. Performance of the proposed approach is evaluated via several experiments on some high-resolution SAR image patch sets and two large-scale scenes which are captured by ALOS PALSAR-2 and COSMO-SkyMed satellites respectively.

In this paper, the confidence intervals for the generalized gamma distribution parameters are derived based on the Bayesian approach using the informative and non-informative priors and the classical approach, via the Asymptotic Maximum likelihood estimation, based on the generalized order statistics. For measuring the performance of the Bayesian approach comparing to the classical approach, the confidence intervals of the unknown parameters have been studied, via Monte Carlo simulations and some real data. The simulation results indicated that the confidence intervals based on the Bayesian approach compete and outperform those based on the classical approach.

This paper proposes a moment-based approximation to the distribution of quadratic forms in gamma random variables. Quadratic forms in order statistics from an exponential population are considered as well. Actually,several test statistics can be expressed in terms of the latter. The density approximants are expressible as the product of a gamma type distributed base density function and a polynomial adjustment. Several illustrative examples are provided.

The kernel approach has been applied using the adaptive kernel density estimation, to inference on the generalized gamma distribution parameters, based on the generalized order statistics (GOS). For measuring the performance of this approach comparing to the Asymptotic Maximum likelihood estimation, the confidence intervals of the unknown parameters have been studied, via Monte Carlo simulations, based on their covering rates, standard errors and the average lengths. The simulation results indicated that the confidence intervals based on the kernel approach compete and outperform the classical ones. Finally, a numerical example is given to illustrate the proposed approaches developed in this paper.

The simple linear iterative clustering (SLIC) method is a recently proposed popular superpixel algorithm. However, this method may generate bad superpixels for synthetic aperture radar (SAR) images due to effects of speckle and the large dynamic range of pixel intensity. In this paper, an improved SLIC algorithm for SAR images is proposed. This algorithm exploits the likelihood information of SAR image pixel clusters. Specifically, a local clustering scheme combining intensity similarity with spatial proximity is proposed. Additionally, for post-processing, a local edge-evolving scheme that combines spatial context and likelihood information is introduced as an alternative to the connected components algorithm. To estimate the likelihood information of SAR image clusters, we incorporated a generalized gamma distribution (GГD). Finally, the superiority of the proposed algorithm was validated using both simulated and real-world SAR images.

Introduction: This paper used Generalized Gamma (GG) distribution to find the predictive factors of overall survival (OS) after haematopoietic stem cell transplant (HSCT) in acute myeloid leukemia patients. Methods: Discrimination among the exponential, Weibull, GG, log-logistic, and lognormal distributions was done using maximum likelihood and Akaike information criteria. Results: The 5-year OS in 301 patients was 65% (95%CI: 60.7-69.3). Peak mortality hazard occurred at months 6-7 after HSCT then, it was U Shape. The data was fitted by GG distribution better than other distributions. Univariate analysis using GG distribution showed a positive association between OS with dose of infused WBC (P=0.018), CD3 (p=0.001), no relapse (P