Generalized Exponential Distribution Research Articles

Bayesian estimation of the mixture of generalized exponential distribution: a versatile lifetime model in industrial processes

Constructing a flexible parametric classes of probability distributions is most popular approach in Bayesian analysis for the last few decades. This study is planned in the same direction for two components’ mixture of generalized exponential (GE) probability distribution by considering heterogeneous population from industry. We have considered censored sample environment due to its popularity in reliability theory. In addition, we have worked out expressions for the maximum likelihood estimates along with their variances and constructed components of the information matrix. To examine the performance of these estimators, we have evaluated their properties for different sample sizes, censoring rates, proportions of the component of mixture, and a variety of loss functions (LFs). The Bayes estimates are evaluated under squared error, entropy, squared logarithmic, and precautionary LFs. Hazard rate of GE distribution graphically and numerically compared with mixture of other life-time distributions. To highlight the practical significance, we have included an illustrative application example based on a real-life data.

Bayesian Approach on the Generalized Exponential Distribution in the Presence of Outliers

Abstract In this paper, the maximum likelihood and the Bayes estimators are derived for sample from the Generalized-Exponential distribution in the presence of k outliers. These estimators are obtained using Newton-Raphson method and Lindley's approximation (L-approximation). The proposed Bayes estimators are obtained under symmetric and asymmetric loss functions. These estimators are compared empirically using Monte Carlo simulation, when all the parameters are unknown.

Discriminating Among the Log-Normal, Weibull, and Generalized Exponential Distributions

We consider model selection and discrimination among three important lifetime distributions. These three distributions have been used quite effectively to analyze lifetime data. We study the probability of correct selection using the maximized likelihood method, as it has been used in the literature. We further compute the asymptotic probability of correct selection, and compare the theoretical, and simulation results for different sample sizes, and for different model parameters. The results have been extended for Type-I censored data also. The theoretical, and simulation results match quite well. Two real data sets have been analyzed for illustrative purposes. We also suggest a method to determine the minimum sample size required to discriminate among the three distributions for a given probability of correct selection, and a user specified protection level.

Tail Extrapolation in MLSE Receivers Using Nonparametric Channel Model Estimation

This paper presents a method for determining the probability of rare events, in particular for probability density function (pdf) and bit error rate (BER) estimation. The derivation of the method is based on the presumption that the pdf is a member of a family of distributions very often named as the generalized exponential (GE) class of distributions. Based on high reliability estimations obtained in short simulation/measurement times, the low probably events are estimated accurately by extrapolation. The suggested method can be applied to some distributions that are different from GE distributions, such as noncentral chi-square distributions, to extrapolate to low probability events, with some extrapolation error. It can also be applied to BER estimation. The method is in particular helpful for estimating channels suffering from both severe signal distortion causing undesired intersymbol interference (ISI) of several symbols, and from severe noise. Such conditions prevail, for example, in metro and long haul high-speed optical fiber communication systems. So the method may be implemented in particular in maximum-likelihood sequence estimation (MLSE) optical receivers using nonparametric channel model estimation. A special use of the extrapolation method is explained for practical systems using trellis branch metrics derived from the estimated pdf to decode the transmitted sequence of symbols.

The beta generalized exponential distribution

A new distribution called the beta generalized exponential distribution is proposed. It includes the beta exponential and generalized exponential (GE) distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. The density function can be expressed as a mixture of generalized exponential densities. This is important to obtain some mathematical properties of the new distribution in terms of the corresponding properties of the GE distribution. We derive the moment generating function (mgf) and the moments, thus generalizing some results in the literature. Expressions for the density, mgf and moments of the order statistics are also obtained. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We observe in one application to a real skewed data set that this model is quite flexible and can be used effectively in analyzing positive data in place of the beta exponential and GE distributions.

Bayes Estimator of Generalized-Exponential Parameters under Linex Loss Function Using Lindley's Approximation

In this paper, we have obtained the Bayes Estimator of Generalized-Exponential scale and shape parameter using Lindley's approximation (L-approximation) under asymmetric loss functions. The proposed estimators have been compared with the corresponding MLE for their risks based on simulated samples from the Generalized-Exponential distribution.

Performance Evaluation and Resource Management of Hierarchical MACRO-/MICRO Cellular Networks using MOSEL-2

The paper presents a performance evaluation and resource management of hierarchical MACRO-/MICRO cellular networks using the new Modeling and Evaluation Language (MOSEL-2). MOSEL-2 with new constructs has the ability to find the performance and reliability modeling and evaluation of systems with exponential and non-exponential distributions. A MACRO/MICRO cell structure is solved numerically and mathematically in this paper to handle the handoff calls. Additionally, a simulation program is written to validate these results. In order to reduce the loss probability, a guard channels are introduced at the MICRO cell and channel reservation at the MACRO cell. Additionally, the concept of queuing is introduced where there is a possibility for the handoff calls from both MACRO and MICRO layers to be queued when all the resources are occupied. MOSEL-2 is used to find the numerical solution for this problem with both exponential and general exponential (GE) distribution. The performance analysis show the efficiency of the proposed scheme to manage the handoff calls and the ability of the suggested scheme to reduce the blocking probability of handover calls and the loss probability as the main objective is to block the new connection rather than terminating the ongoing connection as well as balancing the load all over the whole network. It is shown in this paper that there are a set of important factors that affect the performance, such as: reservation policy, channel allocation, handover ratio, capacity of the queue and the variation of the inter-arrival times. These factors are discussed via some important performance measures, such as: new call blocking probability, blocking probability of handover calls, loss probability, utilization and the average delay of the queue.

Order Statistics from the Generalized Exponential Distribution and Applications

Recently, a new distribution, called generalized exponential distribution (GED), has been introduced and studied quite extensively by authors. The GED can be used as an alternative to gamma and Weibull distributions in many situations. In this article, we use the moments of order statistics from the GED derived by Raqab and Ahsanullah (2001) and Raqab (2004) to develop the correlation goodness -of-fit test for the GED. In addition, we calculate the power of the test based on some other alternative distributions. Further, we construct approximate confidence intervals for the location and scale parameters of the GED. Finally, we apply the procedures developed in the paper to real data set.

Analysis of Incomplete, Censored Data in Competing Risks Models With Generalized Exponential Distributions

This paper presents estimates of the parameters included in a competing risks model in the presence of incomplete & censored data. We consider the case when the competing risks have generalized exponential distributions. The maximum likelihood procedure is used to derive point, and asymptotic confidence interval estimations of the unknown parameters. The relative risks due to each cause of failure are investigated. A set of real data is used to test the hypothesis that the causes of failure follow exponential distributions against that they follow generalized exponential distributions

From Plant Traits to Plant Communities: A Statistical Mechanistic Approach to Biodiversity

We developed a quantitative method, analogous to those used in statistical mechanics, to predict how biodiversity will vary across environments, which plant species from a species pool will be found in which relative abundances in a given environment, and which plant traits determine community assembly. This provides a scaling from plant traits to ecological communities while bypassing the complications of population dynamics. Our method treats community development as a sorting process involving species that are ecologically equivalent except with respect to particular functional traits, which leads to a constrained random assembly of species; the relative abundance of each species adheres to a general exponential distribution as a function of its traits. Using data for eight functional traits of 30 herbaceous species and community-aggregated values of these traits in 12 sites along a 42-year chronosequence of secondary succession, we predicted 94% of the variance in the relative abundances.

Bayesian inference for the generalized exponential distribution

The two-parameter generalized exponential (GE) distribution was introduced by Gupta and Kundu [Gupta, R.D. and Kundu, D., 1999, Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41(2), 173–188.]. It was observed that the GE can be used in situations where a skewed distribution for a nonnegative random variable is needed. In this article, the Bayesian estimation and prediction for the GE distribution, using informative priors, have been considered. Importance sampling is used to estimate the parameters, as well as the reliability function, and the Gibbs and Metropolis samplers data sets are used to predict the behavior of further observations from the distribution. Two data sets are used to illustrate the Bayesian procedure.

On the comparison of Fisher information of the Weibull and GE distributions

In this paper, we consider the Fisher information matrices of the generalized exponential (GE) and Weibull distributions for complete and Type-I censored observations. Fisher information matrix can be used to compute asymptotic variances of the different estimators. Although both distributions may provide similar data fit but the corresponding Fisher information matrices can be quite different. Moreover, the percentage loss of information due to truncation of the Weibull distribution is much more than the GE distribution. We compute the total information of the Weibull and GE distributions for different parameter ranges. We compare the asymptotic variances of the median estimators and the average asymptotic variances of all the percentile estimators for complete and Type-I censored observations. One data analysis has been preformed for illustrative purposes. When two fitted distributions are very close to each other and very difficult to discriminate otherwise, the Fisher information or the above mentioned asymptotic variances may be used for discrimination purposes.

Empirical Bayes Inference for Generalized Exponential Distribution Based on Records

Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. Bayes and empirical Bayes estimators for the unknown parameter of the generalized exponential distribution are derived based on record statistics. These estimates are obtained based on squared error and LINEX loss functions. Prediction bounds for future lower record values are obtained by using Bayes and empirical Bayes techniques. Numerical example is given to illustrate the results.

Modeling the Egyptian Stock Market Volatility Pre- and Post Circuit Breaker

Circuit breakers (price limits and trading halts) are regulatory instruments aiming to reduce severe price volatility and provide markets with a cooling off period. The paper investigated empirically, using daily returns of two Egyptian Stock Market indices the Hermes Financial Index (HFI) and the Egyptian Financial Group Index (EFGI) during the period January 1993 - December 2001, the impact of regulatory policies on conditional volatility estimation. The paper examined four models GARCH, EGARCH, GJR, and APARCH under variety of density functions (Gaussian normal distribution, Student-t distribution, Skewed Student-t and Generalized Exponential Distribution (GED)). The empirical evidence provided in this paper confirms Mecagni and Sourial (1999) findings that the symmetric price limits on individual shares failed to dampen volatility in the market. Furthermore, regulatory and/or structural shifts in the market results in different conditional volatility model structure and using asymmetric models for conditional volatility estimation rather than symmetric models provide better results.

Generalized exponential distribution: Moments of order statistics

In this paper, we consider the generalized exponential distribution (GED) with shape parameter α. We establish several recurrence relations satisfied by the single and the product moments for order statistics from the GED. The relationships can be written in terms of polygamma and hypergeometric functions and used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes.

Discriminating between Weibull and generalized exponential distributions

Recently the two-parameter generalized exponential (GE) distribution was introduced by the authors. It is observed that a GE distribution can be considered for situations where a skewed distribution for a non-negative random variable is needed. The ratio of the maximized likelihoods (RML) is used in discriminating between Weibull and GE distributions. Asymptotic distributions of the logarithm of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between two overlapping families of distributions for a user specified probability of correct selection and tolerance limit.

The Galactic Disk Mass Budget. II. Brown Dwarf Mass Function and Density

In this paper, we extend the calculations conducted previously in the stellar regime to determine the brown dwarf IMF in the Galactic disk. We perform Monte Carlo calculations taking into account the brown dwarf formation rate, spatial distribution and binary fraction. Comparison with existing surveys seems to exclude a power-law MF as steep as the one determined in the stellar regime below 1 $\msol$ and tends to favor a more flatish behaviour. Comparison with methane-dwarf detections tends to favor an eventually decreasing form like the lognormal or the more general exponential distributions determined in the previous paper. We calculate predicting brown dwarf counts in near-infrared color diagrams and brown dwarf discovery functions. These calculations yield the presently most accurate determination of the brown dwarf census in the Galactic disk. The brown dwarf number density is comparable to the stellar one, $n_{BD}\simeq n_\star\simeq 0.1$ pc$^{-3}$. The corresponding brown dwarf mass density, however, represents only about 10% of the stellar contribution, i.e. $\rho_{BD}\simle 5.0\times 10^{-3} \mvol$. Adding up the local stellar density determined previously yields the density of star-like objects, stars and brown dwarfs, in the solar neighborhood $\rho_\odot \approx 5.0\times 10^{-2} \mvol$.

Non-additive entropies and quantum statistics

The properties of general non-additive entropic forms and their emergence in quantum statistical mechanics are discussed. As particular examples, we analyze the entropies associated with general exponential distributions as well as those related with independent particles satisfying Fermi, Bose and fractional exclusion statistics.

Characterization of non‐homogeneous polymers with size‐exclusion chromatography by implementing an on‐line viscometer

AbstractThe viscosity distribution of a polymer sample can be obtained by using an on‐line viscometer as a detector in size‐exclusion chromatography. This newly defined viscosity distribution is closely related to the molecular weight distribution and expresses weight fraction times intrinsic viscosity of species i as a function of the corresponding molecular weight times intrinsic viscosity (wi[ηi] vs. Mi[ηi]). The intrinsic viscosity ([η]) and number‐average molecular weight (M̄n) can be obtained directly from a viscosity distribution. If the Mark‐Houwink exponent a is known (or approximately known) for non‐homogeneous polymer the M̄w/M̄n can be estimated from the viscosity distribution when the molecular weight distribution is approximated with a known distribution function. These estimates are independent of any other detector and are valid even for non‐homogeneous polymer samples. The relation between the moments of the viscosity distribution and the M̄w/M̄n is presented for two widely used distribution functions, the Log‐Normal and the Generalized Exponential Distributions. Polymer characterization based on the viscosity distribution is shown to be a robust technique. It is particularly attractive in characterizing non‐homogeneous polymers since it is solely obtained from on‐line viscometer.

Robust Estimation, Nonnormalities, and Generalized Exponential Distributions

The detection and treatment of outliers in data has represented an important area of research. The general approach that has been adopted involves observing that outliers contribute to the “fatness” in the tails of the error distribution and using an appropriate model of the error term to take this into account. This article introduces a general dass of distributions to capture nonnormalities in the data based on the generalized exponential family. This family of distributions provides great flexibility in modeling not only Symmetrie fat-tailed distributions, but also skewed and possibly even multimodal distributions. The approach taken is to use subordinate distributions within the generalized exponential family to model the empirical distribution. This family represents a generalization of the unimodal exponential family, which contains as subordinates the normal, gamma, beta, Student t, and so forth. The parameters of the generalized exponential distribution can be estimated using maximum likelihood, weighted least squares, or minimum chi-squared methods, depending on the nature of the data. Special attention is given to analyzing the distributional properties of two subordinates of this family; namely, the generalized Student t and generalized lognormal distributions. The flexibility of these distributions is illustrated by applying them to two empirical problems and comparing the results with previous methods.