Generalized Exponential Research Articles

Robust Estimation methods of Generalized Exponential Distribution with Outliers

This paper discussed robust estimation for point estimation of the shape and scale parameters for generalized exponential (GE) distribution using a complete dataset in the presence of various percentages of outliers. In the case of outliers, it is known that classical methods such as maximum likelihood estimation (MLE), least square (LS) and maximum product spacing (MPS) in case of outliers cannot reach the best estimator. To confirm this fact, these classical methods were applied to the data of this study and compared with non-classical estimation methods. The non-classical (Robust) methods such as least absolute deviations (LAD), and M-estimation (using M. Huber (MH) weight and M. Bisquare (MB) weight) had been introduced to obtain the best estimation method for the parameters of the GE distribution. The comparison was done numerically by using the Monte Carlo simulation study. The two real datasets application confirmed that the M-estimation method is very much suitable for estimating the GE parameters. We concluded that the M-estimation method using Huber object function is a suitable estimation method in estimating the parameters of the GE distribution for a complete dataset in the presence of various percentages of outliers.

A Sensitivity Analysis of a New NHPP Software Reliability Model with the Generalized Exponential Fault Detection Rate Function Considering the Uncertainty of Operating Environments

Early in the development of the software reliability model, it was assumed that the operating environment of the software was used in the same or similar field environment as the test environment. However, they may also be used in many different locations that may differ from the environment in which they were developed and tested. And also, in general, software systems are known to degrade performance and increase failure rate as their service life increases. Therefore, efforts have been made to expand the exponential distribution assuming a constant failure rate. In this paper, we propose a new software reliability model with the generalized exponential (GE) fault detection rate function considering the uncertainty of operation environments based on a non-homogeneous Poisson process (NHPP). We examine the goodness-of-fit of a new NHPP software reliability model and other existing models based on dataset. As a result of comparison of goodness-of-fit, it can be confirmed that the new proposed model is significantly better by showing that all the criteria of the new proposed model are the lowest compared to other models. In addition, the results for the sensitivity analysis show that the parameters of a new NHPP software reliability model affect the mean value function.

Molecular Structure of Atomic Nucleus

In this work, we extend our work on the Heisenberg model of the neutron formulated as a dwarf hydrogen-like atom under the influence of the More General Exponential Screened Coulomb Potential (MGESCP) to show that an atomic nucleus may possess a molecular structure made up of atoms bonding together by a potential used to describe the strong force associated with a generalised Yukawa MGESCP potential. We show that the neutrons and protons are arranged into narrow lattices therefore they may fold to form three-dimensional shells by bonding similar to hydrogen bonding. In particular, the nucleons may form stable structures such as that of fullerenes in which the vertices are occupied by the nucleons which are simply just protons. For example, a nucleus with a total number of 60 nucleons may arrange itself into the topological structure of a buckminsterfullerene. We also apply Schrödinger wave equation with central field approximation to describe the quantum dynamics of nuclei of atomic atoms that now possess the physical structure of a dwarf molecular ion.

Generalized Halanay Inequalities with Applications to Generalized Exponential Stability and Boundedness of Time-Delay Systems

In this paper, a class of generalized Halanay inequalities is investigated. Under some assumptions, the generalized exponential stability and boundedness are discussed by means of integral inequalities. We do not require the boundedness of the time-varying delays and coefficients. In addition, our results improve some previous works. At last, three examples and simulations are given to illustrate the effectiveness of the main results.

Correcting Satellite Precipitation Data and Assimilating Satellite-Derived Soil Moisture Data to Generate Ensemble Hydrological Forecasts within the HBV Rainfall-Runoff Model

An implementation of bias correction and data assimilation using the ensemble Kalman filter (EnKF) as a procedure, dynamically coupled with the conceptual rainfall-runoff Hydrologiska Byråns Vattenbalansavdelning (HBV) model, was assessed for the hydrological modeling of seasonal hydrographs. The enhanced HBV model generated ensemble hydrographs and an average stream-flow simulation. The proposed approach was developed to examine the possibility of using data (e.g., precipitation and soil moisture) from the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) Satellite Application Facility for Support to Operational Hydrology and Water Management (H-SAF), and to explore its usefulness in improving model updating and forecasting. Data from the Sola mountain catchment in southern Poland between 1 January 2008 and 31 July 2014 were used to calibrate the HBV model, while data from 1 August 2014 to 30 April 2015 were used for validation. A bias correction algorithm for a distribution-derived transformation method was developed by exploring generalized exponential (GE) theoretical distributions, along with gamma (GA) and Weibull (WE) distributions for the different data used in this study. When using the ensemble Kalman filter, the stochastically-generated ensemble of the model states generally induced bias in the estimation of non-linear hydrologic processes, thus influencing the accuracy of the Kalman analysis. In order to reduce the bias produced by the assimilation procedure, a post-processing bias correction (BC) procedure was coupled with the ensemble Kalman filter (EnKF), resulting in an ensemble Kalman filter with bias correction (EnKF-BC). The EnKF-BC, dynamically coupled with the HBV model for the assimilation of the satellite soil moisture observations, improved the accuracy of the simulated hydrographs significantly in the summer season, whereas, a positive effect from bias corrected (BC) satellite precipitation, as forcing data, was observed in the winter. Ensemble forecasts generated from the assimilation procedure are shown to be less uncertain. In future studies, the EnKF-BC algorithm proposed in the current study could be applied to a diverse array of practical forecasting problems (e.g., an operational assimilation of snowpack and snow water equivalent in forecasting models).

Bayesian Survival Analysis of Type I General Exponential Distributions

This article aims at generalizing two distribution by means of, exponentiated exponential and Weibull distribution. The researchers have employed three and four parameters life model called the Type I General Exponential exponentiated exponential distribution and Type I General Exponential Weibull distribution. Survival and hazard rate functions were provided for these two models. To fit these models into survival and hazard rate functions, we adopted the Bayesian approach. For illustration, a real survival data set has been employed. Application is carried out by R and Stan. Finally,a comparison between these two models is made by using loo package to find the best model and simulation.

A characterization of generalized exponential polynomials in terms of decomposable functions

Let G be a topological commutative semigroup with unit. We prove that a continuous function $${f\colon {G} \to \mathbb{C}}$$ is a generalized exponential polynomial if and only if there is an $${n \geq 2}$$ such that $${f(x_{1} + {\cdots} +x_{n})}$$ is decomposable; that is, if $${f(x_{1} + {\cdots} +x_{n}) = \sum_{i=1}^{k} u_{i} \cdot v_{i}}$$ , where the function ui only depends on the variables belonging to a set $${\emptyset \neq E_{i} \subsetneq \{x_{1},\ldots, x_{n}\}}$$ , and vi only depends on the variables belonging to $${\{x_{1},\ldots,x_{n}\} \setminus E_{i} (i = 1,\ldots,k)}$$ .

A Study of the Sequential Test for the Parameter of the Odd Generalized Exponential Gompertz Distribution

A Study of the Sequential Test for the Parameter of the Odd Generalized Exponential Gompertz Distribution

A Quantum Dynamics of Heisenberg Model of the Neutron Associated with Beta Decay

In this work we re-examine a model of the nucleons that involve the weak interaction which was once considered by Heisenberg; that is a neutron may have the structure of a dwarf hydrogen-like atom. We formulate a quantum dynamics for the Heisenberg model of the neutron associated with interaction that involves the beta decay in terms of a mixed Coulomb-Yukawa potential and the More General Exponential Screened Coulomb Potential (MGESCP), which has been studied and applied to various fields of physics. We show that all the components that form the MGESCP potential can be derived from a general system of linear first order partial differential equations similar to Dirac relativistic equation in quantum mechanics. There are many interesting features that emerge from the MGESCP potential, such as the MGESCP potential can be reduced to the potential that has been proposed to describe the interaction between the quarks for strong force in particle physics, and the energy spectrum of the bound states of the dwarf hydrogen-like atom is continuous with respect to distance. This result leads to an unexpected implication that a proton and an electron may also interact strongly at short distances. We also show that the Yukawa potential when restrained can generate and determine the mathematical structures of fundamental particles associated with the strong and weak fields.

Bayesian and Maximum Likelihood Estimation for the Weibull Generalized Exponential Distribution Parameters Using Progressive Censoring Schemes

In this paper we consider the estimation of the Weibull Generalized Exponential Distribution (WGED) Parameters with Progressive Censoring Schemes. In order to obtain the optimal censoring scheme for WGED, more than one method of estimation was used to reach a better scheme with the best method of estimation. The maximum likelihood method and the method of Bayesian estimation for (square error and Linex) loss function have been used. Monte carlo simulation is used for comparison between the two methods of estimation under censoring schemes. To show how the schemes work in practice; we analyze a strength data for single carbon fibers as a case of real data.

General Stability and Exponential Growth for a Class of Semi-linear Wave Equations with Logarithmic Source and Memory Terms

In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation $$\begin{aligned} u_{tt}-\Delta u + u + (g\,*\, \Delta u)(t)+ h(u_{t})u_{t}+|u|^{2}u=u\log |u|^{k}, \end{aligned}$$ in an open bounded domain $$\Omega \subseteq \mathbb {R}^3$$ whith $$h(s)=k_{0}+k_{1}|s|^{m-1}.$$ We prove a general stability of solutions which improves and extends some previous studies such as the one by Hu et al. (Appl Math Optim, https://doi.org/10.1007/s00245-017-9423-3 ) in the case $$g=0$$ and in presence of linear frictional damping $$u_{t}$$ when the cubic term $$|u|^2u$$ is replaced with u. In the case $$k_{1}=0,$$ we also prove that the solutions will grow up as an exponential function. Our result shows that the memory kernel g dose not need to satisfy some restrictive conditions to cause the unboundedness of solutions.

New Odd Generalized Exponential - Exponential Distribution: Its Properties and Application

New Odd Generalized Exponential - Exponential Distribution: Its Properties and Application

Semiparametric generalized exponential frailty model for clustered survival data

In this paper, we propose a novel and mathematically tractable frailty model for clustered survival data by assuming a generalized exponential (GE) distribution for the latent frailty effect. Both parametric and semiparametric versions of the GE frailty model are studied with main focus for the semiparametric case, where an EM-algorithm is proposed. Our EM-based estimation for the GE frailty model is simpler, faster and immune to a flat likelihood issue affecting, for example, the semiparametric gamma model, as illustrated in this paper through simulated and real data. We also show that the GE model is at least competitive with respect to the gamma frailty model under misspecification. A broad analysis is developed, with simulation results explored via Monte Carlo replications, to evaluate and compare models. A real application using a clustered kidney catheter data is considered to demonstrate the potential for practice of the GE frailty model.

Generalized fiducial inference for generalized exponential distribution

ABSTRACTThis article mainly considers interval estimation of the scale and shape parameters of the generalized exponential (GE) distribution. We adopt the generalized fiducial method to construct a kind of new confidence intervals for the parameters of interest and compare them with the frequentist and Bayesian methods. In addition, we give the comparison of the point estimation based on the frequentist, generalized fiducial and Bayesian methods. Simulation results show that a new procedure based on generalized fiducial inference is more applicable than the non-fiducial methods for the point and interval estimation of the GE distribution. Finally, two lifetime data sets are used to illustrate the application of our new procedure.

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.

Parametric regression model for response time in clinical trials – a bayesian approach

In this paper an attempt has been made to model the censored survival data using Bayesian regressions with Markov Chain Monte Carlo (MCMC) methods. Bayesian LogNormal (LN) regression model are found to be providing better fit than the other Bayesian regression models namely Exponential (E), Generalized Exponential (GE), Webull (W), LogLogistic (LL) and Gamma (G).

Control charts for generalized exponential distribution percentiles

ABSTRACTThe generalized exponential (GE) distribution, which was introduced by Mudholkar and Srivastava in 1993, has been studied for various applications of lifetime modelings. In this article, five control charts, that comprise the Shewhart-type chart and four parametric bootstrap charts based on maximum likelihood estimation method, the moment estimation method, probability plot method, and least-square error method for the GE percentiles, are investigated. An extensive Monte Carlo simulation study is conducted to compare the performance among all five control charts in terms of average run length. Finally, an example is given for illustration.

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.

Analysis of Volatility Spill Over between Oil Price and Exchange Rate in India: GARCH Approach

The present paper endeavours to analyse the volatility spill over between crude oil price and exchange rate for India using daily data for time period June 2003 to March 2016. To examine the impact of oil price on exchange rate of Indian Rupee against U.S. Dollar, Generalised Autoregressive Conditional Heteroskedasticity (GARCH) and Exponential Generalised Autoregressive Conditional Heteroskedasticity (EGARCH) models have been employed. The analysis reveals that an increase in oil price return leads to depreciation of Indian Currency with respect to U.S. Dollar. The study establishes that positive and negative oil price shocks have similar effects, in terms of magnitude, on exchange rate volatility and also have permanent effect on exchange rate volatility.

A Recent Study of Excited Energy Levels of Diatomics for Modified more General Exponential Screened Coulomb Potential: Extended Quantum Mechanics

A Recent Study of Excited Energy Levels of Diatomics for Modified more General Exponential Screened Coulomb Potential: Extended Quantum Mechanics