Generalized Exponential Distribution Research Articles

Stochastic analysis of a complex system under preemptive resume repair policy using Gumbel-Hougaard family copula

The present paper focuses on the study of complex system consisting of three units super priority, priority and ordinary unit under preemptive resume repair policy. Three units connected with an automatic changeover switch and human operator is handling the system. Together with the failure in super priority, priority and ordinary unit, the other important failures like deliberate failure and switch failure have been considered for study of mathematical model. All failure rates are assumed constant and follow exponential distribution however, the repair rate for partially failed state follow general distribution and complete failed states are repaired using Gumbel-Hougaard family copula (a joint probability distribution of general distribution and negative exponential distribution). The system is analysed using supplementary variable technique and Laplace transform. The various popular measures of reliability such as availability, reliability, mean time to failure (MTTF) and profit function has been calculated for particular values of failure and repair rates. Sensitivity analysis for different traditional measures has also been done. Computed results have been highlighted by tables and graphs.

Interval Estimation of a P(X<sub>1</sub> < X<sub>2</sub>) Model for Variables having General Inverse Exponential Form Distributions with Unknown Parameters

In this article the interval estimation of a P(X₁ < X₂) model is discussed when X₁ and X₂are non-negative independent random variables, having general inverse exponential form distributions with different unknown parameters. Different interval estimators are derived, by applying different approaches. A simulation study is performed to compare the estimators obtained. The comparison is carried out on basis of average length, average coverage, and tail errors. The results are illustrated, using inverse Weibull distribution as an example of the general inverse exponential form distribution.

Characterizations of Kumaraswamy-Laplace, McDonald Inverse Weibull and New Generalized Exponential Distributions

Nassar (2016) considers an interesting univariate continuous distribution called Kumaraswamy-Laplace which has different forms on two subintervals. He studies certain properties and applications of this distribution. Shahbaz et al. (2016) consider another interesting distribution called McDonald Inverse Weibull distribution. They present some basic properties of their distribution and study the estimations of the parameters as well as discussing its application via an illustrative example. What is lacking in both papers, in our opinion, is the characterizations of these two interesting distributions. the present work is intended to complete, in some way, the works of Nassar and Shahbaz et al. via establishing certain characterizations of these distributions in four directions. We also introduce several New generalized Exponential distributions and present their characterizations as well.

Interval Estimation of Stress-Strength Reliability for a General Exponential Form Distribution with Different Unknown Parameters

This paper deals with interval estimation of the stress-strength reliability, when the stress and strength variables follow a general exponential form distribution. The distribution parameters of both the stress and the strength are assumed to be unknown. Interval estimation for reliability is discussed, using different approaches. The results obtained are applicable to many well known distributions. For illustration of the general results obtained a simulation study is performed with application on Weibull distribution. Numerical comparison of the interval estimators is carried out based on average length, probability coverage, and tail errors.

Ordering Results for Order Statistics from Two Heterogeneous Marshall-Olkin Generalized Exponential Distributions

Adding parameters to a known distribution is a useful way of constructing flexible families of distributions. Marshall and Olkin (Biometrika, 84, 641–652, 1997) introduced a general method of adding a shape parameter to a family of distributions. In this paper, based on the Marshall-Olkin extension of a specified distribution, we introduce a new models referred to as Marshal-Olkin generalized exponential (MOGE) models, which include as a special case the well-known generalized exponential distribution. Next, we establish some stochastic comparisons between the corresponding order statistics based on majorization, weak majorization and p-larger theory. The results established here extend some well-known results in the literature about the generalized exponential distribution.

Generalized Exponential Truncated Negative Binomial Distribution

SYNOPTIC ABSTRACTNadarajah, Jayakumar, and Ristić (2013) introduced a family of life time models using truncated negative binomial distribution and derived some properties of the family of distributions. It is a generalization of Marshall–Olkin family of distributions. In this article, we introduce Generalized Exponential Truncated Negative Binomial (GETNB) distribution using the same approach. The GETNB distribution is a generalization of the Marshall–Olkin Generalized Exponential (MOGE) distribution studied in Ristić and Kundu (2015) and Exponential Truncated Negative Binomial (ETNB) distribution studied in Nadarajah et al. (2013). The shape properties of density function and the hazard rate of GETNB distribution are discussed. The expressions for moments, order statistics, and entropy are obtained. The parameters of GETNB distribution are estimated using the method of maximum likelihood. An application to a real data set is also presented.

Model selection for the segmentation of multiparameter exponential family distributions

We consider the segmentation problem of univariate distributions from the exponential fam- ily with multiple parameters. In segmentation, the choice of the number of segments remains a dicult issue due to the discrete nature of the change-points. In this general exponential family distribution framework, we propose a penalized log-likelihood estimator where the penalty is inspired by papers of L. Birg e and P. Massart. The resulting estimator is proved to satisfy an oracle inequality. We then further study the particular case of categorical variables by comparing the values of the key constants when derived from the specication of our general approach and when obtained by working directly with the characteristics of this distribution. Finally, a simulation study is conducted to assess the performance of our criterion for the exponential distribution, and an application on real data modeled by the categorical distribution is provided.

Slashed generalized exponential distribution

ABSTRACTIn this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient between a GE random variable and a beta-distributed random variable with one unknown parameter. The resulting distribution is a distribution with greater kurtosis than the GE distribution. Probability properties of the distribution such as moments and asymmetry and kurtosis are studied. Likewise, statistical properties are investigated using the method of moments and the maximum likelihood approach. Two real data analyses are reported illustrating better performance of the new model over the GE model.

Texture modelling with nested high-order Markov–Gibbs random fields

Currently, Markov–Gibbs random field (MGRF) image models which include high-order interactions are almost always built by modelling responses of a stack of local linear filters. Actual interaction structure is specified implicitly by the filter coefficients. In contrast, we learn an explicit high-order MGRF structure by considering the learning process in terms of general exponential family distributions nested over base models, so that potentials added later can build on previous ones. We relatively rapidly add new features by skipping over the costly optimisation of parameters.We introduce the use of local binary patterns as features in MGRF texture models, and generalise them by learning offsets to the surrounding pixels. These prove effective as high-order features, and are fast to compute. Several schemes for selecting high-order features by composition or search of a small subclass are compared. Additionally we present a simple modification of the maximum likelihood as a texture modelling-specific objective function which aims to improve generalisation by local windowing of statistics.The proposed method was experimentally evaluated by learning high-order MGRF models for a broad selection of complex textures and then performing texture synthesis, and succeeded on much of the continuum from stochastic through irregularly structured to near-regular textures. Learning interaction structure is very beneficial for textures with large-scale structure, although those with complex irregular structure still provide difficulties. The texture models were also quantitatively evaluated on two tasks and found to be competitive with other works: grading of synthesised textures by a panel of observers; and comparison against several recent MGRF models by evaluation on a constrained inpainting task.

Maximum Rainfall Model Based on Archival Pluviographic Records – Case Study for Legnica (Poland)

The scope of this study is to develop a probabilistic model for maximum rainfall in Legnica, based on a 50-year series of pluviographic records. The present authors uses Frechet, Gamma, Generalized Exponential Distribution (GED), Gumbel, Log-Normal and Weibull distributions to describe the measurement data. Distributions parameters are estimate using maximum likelihood method. Coincidence of the analyzed theoretical distributions with measured data are inspected using the Anderson-Darling test, while the best fitting distribution is chosen by Bayesian information criterion (BIC) of Schwartz as well as by the relative residual mean square error. Among others distributions Frechet, Gamma, GED, and Weibull distributions fulfill the compliance criterion for each of the 20 analyzed rainfall durations. BIC criterion indicates for a GED, but differences between GED, Gamma and Weibull is minor. Only RRMSE analysis revealed that in comparison to other distribution GED best describes the measurement rainfall data. At first glance maximum rainfall model was well described by the generalized exponential distribution. However, there is a substantial inconvenience to use it for engineering purposes. Generalization of the shape parameter α depended on the rainfall duration, by averaging and then recalculating remaining parameters λ and γ brought relatively simpler form of model.

Generalized Exponential Distribution in Flood Frequency Analysis for Polish Rivers.

Many distributions have been used in flood frequency analysis (FFA) for fitting the flood extremes data. However, as shown in the paper, the scatter of Polish data plotted on the moment ratio diagram shows that there is still room for a new model. In the paper, we study the usefulness of the generalized exponential (GE) distribution in flood frequency analysis for Polish Rivers. We investigate the fit of GE distribution to the Polish data of the maximum flows in comparison with the inverse Gaussian (IG) distribution, which in our previous studies showed the best fitting among several models commonly used in FFA. Since the use of a discrimination procedure without the knowledge of its performance for the considered probability density functions may lead to erroneous conclusions, we compare the probability of correct selection for the GE and IG distributions along with the analysis of the asymptotic model error in respect to the upper quantile values. As an application, both GE and IG distributions are alternatively assumed for describing the annual peak flows for several gauging stations of Polish Rivers. To find the best fitting model, four discrimination procedures are used. In turn, they are based on the maximized logarithm of the likelihood function (K procedure), on the density function of the scale transformation maximal invariant (QK procedure), on the Kolmogorov-Smirnov statistics (KS procedure) and the fourth procedure based on the differences between the ML estimate of 1% quantile and its value assessed by the method of moments and linear moments, in sequence (R procedure). Due to the uncertainty of choosing the best model, the method of aggregation is applied to estimate of the maximum flow quantiles.

Potential of a Spheroid with Generalized Exponential Density Distribution

AbstractThe internal potential of an inhomogeneous layered spheroid with a small ellipticity and general exponential density distribution is derived in an analytical form. The results are presented in the form of standard Gauss hypergeometric function and validated numerically. The computing time when using this formula is noticeably smaller than the time required by numerical integration.

Analysis of Generalized Exponential Distribution Under Adaptive Type-II Progressive Hybrid Censored Competing Risks Data

This paper presents estimates of the parameters based on adaptive type-II progressive hybrid censoring scheme (AT-II PHCS) in the presence of the competing risks model. We consider the competing risks have generalized exponential distributions (GED). The maximum likelihood method is used to derive point and asymptotic confidence intervals for the unknown parameters. The relative risks due to each cause of failure are investigated. A real data set is used to illustrate the theoretical results and to test the hypothesis that the causes of failure follow the generalized exponential distributions against the exponential distribution (ED). Keywords: Competing Risks; Adaptive Type-II Progressive Hybrid Censoring; Generalized Exponential Distribution; Maximum Likelihood Estimation.

Reliability estimation and ratio distribution in a general exponential distribution

We shall consider the estimation for the parameter and the right tail probability in a general exponential distribution. We also shall consider the estimation of the reliability P(X < Y ) and the skewness trends of the density function of the ratio X=(X+Y) for two independent general exponential variables each having different shape parameters and known scale parameter. We then shall consider the estimation of the failure rate average and the hazard function for a general exponential variable having the density function with the unknown shape and known scale parameters, and for a bivariate density induced by the general exponential density.

Three-parameter generalized exponential distribution in earthquake recurrence interval estimation

The purpose of this article is to study the three-parameter (scale, shape, and location) generalized exponential (GE) distribution and examine its suitability in probabilistic earthquake recurrence modeling. The GE distribution shares many physical properties of the gamma and Weibull distributions. This distribution, unlike the exponential distribution, overcomes the burden of memoryless property. For shape parameter β> 1, the GE distribution offers increasing hazard function, which is in accordance with the elastic rebound theory of earthquake generation. In the present study, we consider a real, complete, and homogeneous earthquake catalog of 20 events with magnitude above 7.0 (Yadav et al. in Pure Appl Geophys 167:1331–1342, 2010) from northeast India and its adjacent regions (20°–32°N and 87°–100°E) to analyze earthquake inter-occurrence time from the GE distribution. We apply the modified maximum likelihood estimation method to estimate model parameters. We then perform a number of goodness-of-fit tests to evaluate the suitability of the GE model to other competitive models, such as the gamma and Weibull models. It is observed that for the present data set, the GE distribution has a better and more economical representation than the gamma and Weibull distributions. Finally, a few conditional probability curves (hazard curves) are presented to demonstrate the significance of the GE distribution in probabilistic assessment of earthquake hazards.

Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor Under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring

Accelerated life testing (ALT) and partially accelerated life testing (PALT) are frequently used in modern reliability engineering. ALT and PALT are run to obtain information on the life of the products and materials in a shorter time and at lower cost. The experimental units are subject to stress conditions that are more severe than those encountered in normal use condition to induce early failures. ALT or PALT can be carried out using constant, step, progressive, cyclic and random stress loadings. This paper considers the problem of estimating the generalized exponential (GE) distribution parameters and the acceleration factor under constant-stress PALT model. The main objective is to derive the maximum likelihood estimators (MLEs) of the parameters of the GE distribution and the acceleration factor when the data are type-II censored from constant-stress PALT. Also, the performance of the MLEs is investigated numerically for different sample sizes and different parameter values using the mean square error. In addition, the approximate confidence intervals of the model parameters are constructed. Moreover, the likelihood ratio bounds (LRB) method is used to obtain confidence bounds of the model parameters when the sample size is small. For illustration, a simulation study is conducted. It is observed that the simulation results support the theoretical findings.

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.

An efficient analytical method for performance evaluation of transfer lines with unreliable machines and finite transfer-delay buffers

Buffers are widely adopted in transfer lines to reduce the fluctuations caused by the imbalances of systems or machine failures. This paper presents an efficient analytical method to evaluate the performance of transfer lines with unreliable machines and finite transfer-delay buffers. Firstly, the buffers with transfer delays are transformed equivalently into a series of perfect machines and buffers without transfer delays. Correspondingly, the initial transfer line is replaced by an equivalent transfer line with more machines and zero-transfer-delay buffers. Since in the equivalent transfer line the orders of magnitude of machines’ reliability parameters (mean times between failures and mean times to repair) are not at the same level, an advanced decomposition method is introduced to analyse the equivalent transfer line, using the general-exponential distributions instead of the exponential distributions to approximate the repair time distributions of the fictitious machines. Finally, extensive simulation and numerical cases are carried out to verify the performance of the developed method.

Bayesian Inference of Generalized Exponential Distribution Based on Lower Record Values

SYNOPTIC ABSTRACT This article addresses the problem of frequentist and Bayesian estimation of the parameters of the generalized exponential distribution (GED) using lower record values. The maximum likelihood estimates (MLE) and the Bayes estimates based on lower records are derived for the unknown parameters of the GED. We consider the Bayes estimators of the unknown parameters under the assumption of gamma priors on both the shape and the scale parameters. The Bayes estimators cannot be obtained in explicit forms. The Bayesian estimation of the parameters of the GED has been studied with respect to both symmetric and asymmetric loss functions. We have also derived the Bayes interval of this distribution and discussed the Bayesian prediction intervals of the future record values based on the observed record values. Monte Carlo simulations are performed to compare the performances of the proposed methods, and one dataset has been analyzed for illustrative purposes.

Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms

The generalized order statistics which introduced by [1] are studied in the present paper. The Gompertz distribution is widely used to describe the distribution of adult deaths, and some related models used in the economic applications [2]. Previous works concentrated on formulating approximate relationships to characterize it [3-5]. The main aim of this paper is to obtain the distribution of single, two, and all generalized order statistics from Gompertz distribution with some special cases. In addition the conditional distribution of two generalized order statistics from the same distribution is obtained. The Gompertz distribution has a continuous probability density function with location parameter a and shape parameter b, , where x restricted by the interval . The nth moment generated function of the Gompertz distributed random variable X is given on the form: where, is the generalized integro-exponential function [6]. In this paper we shall obtain joint distribution, distribution of product of two generalized order statistics from the Gompertz distribution, and then derive some useful formulas of these distributions as special cases.