Exponentiated Gamma Distribution Research Articles

The Inverted Exponentiated Gamma Distribution: A Heavy-Tailed Model with Upside Down Bathtub Shaped Hazard Rate

In this article, we proposed and studied the inverted exponentiated gamma distribution (IEGD). IEGD is obtained by considering the inverse transformation of exponentiated gamma variate. This distribution has been motivated by the extensive use of the exponentiated gamma model in many applied areas and also due to the fact that this new generalization provides more flexibility to analyze real data with upside down bathtub (UBT) hazard rate. The shape of the distribution has been traced mathematically and found that the proposed model is compatible with UBT hazard rate models. The tail area property is also presented based on the idea of Marshall and Olkin (2007) and it is concluded that the new model belongs to the family of heavy-tailed distributions. Some other characteristics such as reliability, hazard, the quantile function, skewness and kurtosis, stochastic ordering, stress-strength reliability and order statistics have been explicitly derived. The classical and Bayesian estimation procedures have been discussed to estimate the unknown parameter of IEGD. The performances of classical and Bayes estimators are studied in terms of average mean square error (MSE) by conducting Monte Carlo simulations. Finally, a real data set with UBT type hazard rate is analyzed for the illustrative purpose of the study.

Zero-Truncated Poisson Exponentiated Gamma Distribution: Application and Estimation Methods

We define a new two-parameter lifetime model called the zero-truncated Poisson exponentiated gamma distribution. The hazard function of the new distribution has monotonic and non-monotonic shapes, which allows to fit more dispersed data. The proposed distribution has the advantage of having only two parameters and consequently provides an easier way for estimating the model parameters differently from other distributions with three or more parameters. We obtain some of its structural properties. Several methods of estimation of the model parameters are presented. A simulation with ten different models of estimators was performed in order to compare the performance and accuracy of such proposed estimators. An application to real data was included.

A study on methods for estimating the PDF and the CDF in the exponentiated gamma distribution

The exponentiated gamma distribution is a two parameters lifetime distribution with monotone and non-monotone failure rates. In this article, some estimation methods of the probability density function and the cumulative distribution function of the exponentiated gamma distribution such as uniformly minimum variance unbiased (UMVU), maximum likelihood (ML), least squares, weighted least squares and Minimum distance estimators are studied and their performances through numerical simulations are compared. By the mean integrated squared error (MISE), the UMVU and ML are approximately equivalent and more efficient than the others based on simulation studies when the sample size is more than 40. A real data set is analyzed for illustrative purposes.

Entropy-Based Parameter Estimation for the Four-Parameter Exponential Gamma Distribution

Two methods based on the principle of maximum entropy (POME), the ordinary entropy method (ENT) and the parameter space expansion method (PSEM), are developed for estimating the parameters of a four-parameter exponential gamma distribution. Using six data sets for annual precipitation at the Weihe River basin in China, the PSEM was applied for estimating parameters for the four-parameter exponential gamma distribution and was compared to the methods of moments (MOM) and of maximum likelihood estimation (MLE). It is shown that PSEM enables the four-parameter exponential distribution to fit the data well, and can further improve the estimation.

A simulation-based approach to predicting influence in social media communities: A case of U.S. border security

ABSTRACTPredicting influence in social media (SM) communities has a strong implication for cybersecurity and public policy setting. However, the rapidly growing volume and large variety of SM have made the prediction difficult. Unfortunately, research that combines the power of simulation, SM networks, and SM community features to predict influence is not widely available. In this research, we developed and validated a simulation-based approach to predicting influence in SM communities. The approach uses a power-law distribution to simulate user interaction and leverages statistical distributions to model SM posting and to predict influence of opinion leaders. We applied the approach to analyzing 1,323,940 messages posted by 380,498 users on Twitter about the U.S. border security and immigration issues. Three models for predicting behavioral responses were developed based on exponential distribution, Weibull distribution, and gamma distribution. Evaluation results show that the simulation-based approach a...

Beta Exponentiated Gamma Distribution: Some Properties and Estimation

The exponentiated gamma (EG) distribution is one of the important families of distributions in lifetime tests. In this paper, a new generalized version of this distribution which is called the beta exponentiated gamma (BEG) distribution has been introduced. The new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the BEG distribution has been provided. We derived the rth moment and moment generating function for this distribution. Moreover, we discussed the maximum likelihood estimation of this distribution under a simulation study.

The McDonald exponentiated gamma distribution and its statistical properties

In this paper, we propose a five-parameter lifetime model called the McDonald exponentiated gamma distribution to extend beta exponentiated gamma, Kumaraswamy exponentiated gamma and exponentiated gamma, among several other models. We provide a comprehensive mathematical treatment of this distribution. We derive the moment generating function and the rth moment. We discuss estimation of the parameters by maximum likelihood and provide the information matrix.AMS Subject ClassificationPrimary 62N05; secondary 90B25

Transmuted exponentiated gamma distribution: a generalization of the exponentiated gamma probability distribution

The exponentiated gamma (EG) distribution is one of the important families of distributions in lifetime tests. In this paper, a new generalized version of this distribution which is called Transmuted exponentiated gamma (TEG) distribution is introduced. A new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the TEG distribution is provided. We derive the rth moment and moment generating function this distribution. Moreover, we discuss the maximum likelihood estimation of this distribution.

Classical and Bayesian estimation of parameters on the generalized exponentiated gamma distribution

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

Bayesian analysis of exponentiated gamma distribution under type II censored samples

The paper is concerned with posterior analysis of exponentiated gamma distribution for type II censored samples. The expressions for Bayes estimators and associated risks have been derived under different priors. The entropy and quadratic loss functions have been assumed for estimation. The posterior predictive distributions have been obtained and corresponding intervals have been constructed. The study aims to find out a suitable estimator of the parameter of the distribution. The findings of the study suggest that the performance of estimators under gamma prior using entropy loss function is the best.

Moments of truncated normal/independent distributions

In this work we have considered the problem of finding the moments of a doubly truncated member of the class of normal/independent distributions. We obtained a general result and then use it to derive the moments in the case of doubly truncated versions of Pearson type VII distribution, slash distribution, contaminated normal distribution, double exponential distribution and variance gamma distribution. We also give an application of some actuarial data.

Exponentiated Gamma Distribution: Different Methods of Estimations

The exponentiated gamma (EG) distribution and Fisher information matrices for complete, Type I, and Type II censored observations are obtained. Asymptotic variances of the different estimators are derived. Also, we consider different estimators and compare their performance through Monte Carlo simulations.

Two algorithms for classification on quality control

In this article, no probability density and no prior probability are assumed for the good and defective classes, except possibly two means. A sequence of unidentified input lifetimes is used to estimate the percentages of two classes. After the classifier system is put into use, the new coming unidentified input lifetime can improve the percentages. A Monte Carlo simulation study with exponential or Weibull or gamma distribution on life is presented to demonstrate the classification. The study can be widely used in quality control.

Higher Order Moments and Recurrence Relations of Order Statistics from the Exponentiated Gamma Distribution

Higher Order Moments and Recurrence Relations of Order Statistics from the Exponentiated Gamma Distribution

Numerical solution to an integral equation in geometric process

In this article, an integral equation satisfied by the expected number of events occurred by time t in a geometric process is considered; this equation is a generalization of the renewal equation. A numerical solution based on the trapezoidal integration rule to the integral equation is introduced. To explain the method used in the article, four numerical examples with the exponential distribution, gamma distribution, Weibull distribution and lognormal distribution, respectively, are studied.

The Exponentiated Gamma Distribution with Application to Drought Data

Gamma distributions are some of the most popular models for hydrological processes. In this paper, a very flexible family which contains the gamma distrihution as a particular case is introduced. This distribution generalizes the standard gamma distribution in the same way the exponentiated exponential distribution (Gupta et al., 1998; Roy, 1998) generalizes the standard exponential distribution. We refer to this new distribution as the exponentiated gamma distribution. A comprehensive treatment of the math­ ematical properties is provided by deriving expressions for various particular cases and bounds, analytical shapes of the corresponding probability density function and the hazard rate function, moment generating function, the nth moment, skewness, kurtosis, mean deviation , Shannon entropy, and the asymptotic distribution of the extreme order statistics. Estimation procedure by the method of maximum likelihood and an expression for the Fisher information matrix are also provided. Finally, an application to drought data from the State of Nebraska is illustrated.

Study on the general model of hydrological frequency analysis

On the basis of exponential Gamma distribution, this paper presents a model consisting of 11 kinds of distribution curves for the first time. The model contains several frequency curves used commonly in China and other countries, for example, Pearson type-3 distribution, Kritsky and Menkel (USSR) distribution, Logarithmic normal distribution, and so on. Through dealing with parameters, the model may be applicable to the hydrological extreme values of different climate regions, and has very strong flexibility. In this paper, a real case study using flood datum of 240 different hydrological stations in China is submitted and the calculating results are satisfactory.

Probability Distribution of Surface Elevation in Surf and Swash Zones

Three irregular wave tests were conducted on a 1:16 smooth impermeable slope to investigate the detailed cross-shore variations of the probability distributions and statistics of the free surface elevations and middepth horizontal velocities in the shoaling, surf, and swash zones. The exponential gamma distribution with the measured mean, standard deviation, and skewness is shown to be capable of describing the measured probability distributions in a unified manner, although the agreement becomes worse in the lower swash zone. The probability distribution of the free surface elevation whose lower limit is imposed by the beach face in the swash zone becomes exponential with the skewness s= 2 and the standard deviation σ=h¯ with h¯= mean water depth. These upper limits of s and σ/h¯ in the lower swash zone are in qualitative agreement with the data in the region of h¯≳ 0.4 cm in these small-scale tests. The cross-shore variations of the mean (undertow) and standard deviation of the middepth horizontal velocity measured in the shoaling and surf zones can be predicted fairly accurately using the simple relationships derived using linear long-wave theory together with the measured values of h¯ and σ. The adopted distribution and these simple relationships will need to be verified using extensive data on sand beaches of various profiles.

Order statistics from the type i generalized logistic distribution

In this paper, we first obtain three generalized forms of the standard logistic distribution. These are derived by compounding an extreme-value distribution of the doubly exponential type, a reduced log-Weibull distribution and an exponential-gamma distribution, respectively, with a gamma distribution. The usefulness of these three types of generalized logistic distributions in some estimation problems, and also in some robustness studies, is pointed out. Finally, we study the order statistics from the Type I generalized logistic distribution in detail. We derive exact and explicit expressions for the means, variances and covariances of order statistics in terms of the gama function and its successive derivatives. Percentage points and modes of order statistics are also discussed