Generalized Exponential Research Articles

Trip Assignment Modeling for Transportation in Denpasar City Using PTV Visum Software

Mapping and macro transportation modelling are methods used to obtain an overview and analysis of transportation conditions at the macro level, encompassing all road segments in urban areas. Critical constraints on road segments can be identified through this process. Macro transportation modelling serves as a tool for studying the implementation of problem-solving policies, allowing the anticipated outcomes to be analysed through simulations in transportation modelling. This research has a specific goal: to understand the travel patterns of the population in the city of Denpasar. Calibration and validation processes in the PTV Visum modelling are conducted to test the validity of the created macrosimulation model. The trip assignment results are generated through PTV Visum modeling. The research methodology involves quantitative analysis using household interview surveys, traffic counting, correlational studies, and macrosimulation modelling. Calibration is performed by calculating the beta coefficient value (with specific calculated values), time matrices, and distance matrices to measure travel impedance functions. Validation is conducted by evaluating the Generalised Exponential Horn (GEH) value, which is less than 5, to ensure the alignment between travel output in the model and field survey data from traffic counting. From the analysis of a sample of 2,725 households, a total of 2,165,876 daily trips were obtained. Types of trips include movements from internal to internal zones (2,115,602 trips), from internal to external zones (28,799 trips), from external to internal zones (19,319 trips), and from external to external zones (2,159 trips) per day. Roads with the highest traffic volume in a single day include Hasanudin Street (42,524 pcu/hour), Diponogoro Street (42,282 pcu/hour), Sudirman Street (31,062 pcu/hour), Sutoyo Street (30,180 pcu/hour), and Sumatra Street (23,515 pcu/hour).

Minimum density power divergence estimation for the generalized exponential distribution

Statistical modeling of rainfall data is an active research area in agro-meteorology. The most common models fitted to such datasets are exponential, gamma, log-normal, and Weibull distributions. As an alternative to some of these models, the generalized exponential (GE) distribution was proposed by Gupta and Kundu (2001a). Rainfall (specifically for short periods) datasets often include outliers, and thus, a proper robust parameter estimation procedure is necessary. Here, we use the popular minimum density power divergence estimation (MDPDE) procedure developed by Basu et al. (1998) for estimating the GE parameters. We derive the analytical expressions for the estimating equations and asymptotic distributions. We analytically compare MDPDE with maximum likelihood estimation in terms of robustness, through an influence function analysis. Besides, we study the asymptotic relative efficiency of MDPDE analytically for different parameter settings. We apply the proposed technique to some simulated datasets and two rainfall datasets from Texas, United States. The results indicate superior performance of MDPDE compared to the other existing estimation techniques in most of the scenarios.

A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for α∈R+\{1}. Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for α∈R+\{1}, we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices.

Classical and Bayesian Inference for the Kavya–Manoharan Generalized Exponential Distribution under Generalized Progressively Hybrid Censored Data

This manuscript focuses on the statistical inference of the Kavya–Manoharan generalized exponential distribution under the generalized type-I progressive hybrid censoring sample (GTI-PHCS). Different classical approaches of estimation, such as maximum likelihood, the maximum product of spacing, least squares (LS), weighted LS, and percentiles under GTI-PHCS, are investigated. Based on the squared error and linear exponential loss functions, the Bayes estimates for the unknown parameters utilizing separate gamma priors under GTI-PHCS have been derived. Point and interval estimates of unknown parameters are developed. We carry out a simulation using the Monte Carlo algorithm to show the performance of the inferential procedures. Finally, real-world data collection is examined for illustration purposes.

Double Acceptance Sampling Plan for Odd Generalized Exponential Log-logistic Distribution Based on Truncated Life Test

According to the results of industrial research, product failure time is correlated with fatigue weakness, which is typically produced by repeated stress variations. A double acceptance sampling strategy was presented for shortened life tests where the lifespan of test products follows an odd generalized exponential log-logistic distribution (OGELLD), according to the findings of this study. The minimum sample sizes for the first and second samples are calculated using a producer’s risk of 0.05 to ensure that the actual median life is greater than the specified life at the chosen consumer confidence level. Based on various ratios of genuine median life to stipulated life, we analyzed operational features; we observed that reduced producer risk at the defined level was associated with the lowest median ratios to the specified level. Finally, an illustration is offered to help in the grasp of the suggested framework.

Memory‐type control charts for censored reliability data

AbstractControl charts are commonly used to monitor a process to detect undesirable changes. The main goal of this work is to propose exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts to track a process by utilizing type‐I censored generalized exponential (GE) distributed data. In particular, the censored data are replaced with the conditional expected value (CEV). A comparison between CUSUM and CUSUM ignoring unobserved covariates (CUSUM‐IUC) charts is also a part of this study. The GE distribution is considered due to its application in reliability analysis. The performance of the charts is evaluated by using the average run length along with the standard deviation of the run length. Furthermore, the study also examines the effect of smoothing parameters, censoring rate, and shifts on the proposed methodologies. The results indicate that the EWMA chart performs better than other schemes (CUSUM and CUSUM‐IUC) for shifts in the shape parameter of the GE distribution. Finally, the proposed schemes are also applied to a real‐life data set.

Solutions of the N-dimensional Klein-Gordon Equation with Ultra Generalized Exponential–Hyperbolic Potential to Predict the Mass Spectra of Heavy Mesons

Abstract: We solved the N-dimensional Klein-Gordon equation analytically using the Nikiforov-Uvarov method to obtain the energy eigenvalues and the corresponding wave function in terms of Laguerre polynomials with the ultra generalized exponential–hyperbolic potential. The present results are applied for calculating the mass spectra of heavy mesons, such as charmonium (cc ̅) and bottomonium (bb ̅), for different quantum states. The present work provides excellent results in comparison with experimental data with a maximum error of 0.0059GeV and the works of other researchers. Keywords: Ultra generalized exponential–hyperbolic potential, Klein-Gordon equation, Heavy mesons, Nikiforov-Uvarov method.

Multicomponent Stress-Strength Reliability estimation based on Unit Generalized Exponential Distribution

Estimation of stress-strength reliability is considered based on frequentist and Bayesian methods of estimation when both stress and strength variables follow unit generalized exponential distributions. In frequentist method we consider maximum likelihood, least squares, weighted least squares and maximum product spacing methods to estimate system reliability when a common scale parameter is unknown. Further asymptotic and bootstrap intervals of system reliability are obtained. Next, we discuss Bayesian procedure under different loss functions using gamma and weighted Lindley priors for model parameters to estimate system reliability. Subsequently, highest posterior density credible intervals are also obtained. Besides, uniformly minimum variance unbiased estimator of system reliability is obtained when the common scale parameter is known. Extensive Monte-Carlo simulation studies are conducted to evaluate the performance of proposed estimates with respect to various criteria. Finally, to show the applicability of the proposed methodologies in a real-life scenario, an engineering data set is analyzed.

A new extension of the two-parameter bathtub hazard shaped distribution

The need of new life time distributions that can be used to fit real data sets is crucial in lifetime data analysis. This article uses the two parameter bathtub (TPBT) and the generalized exponential (GE) distributions to propose a new family of lifetime distributions, named the odd generalized exponential two-parameter bathtub shaped distribution (OGE-TPBT). Statistical properties of the proposed distribution are discussed. The maximum likelihood and Bayesian procedures are used to estimate the model’s parameters and some of its reliability measures. For Bayes method, we use three approaches of the approximate Bayesian computation (ABC) method. Simulation study is provided to investigate the properties of the methods applied. Based on some well know diagnostic tests, we find out that the simulation data provided in this paper is appropriate. To discuss the possible improvements of the new distribution compared to the original two distributions (GE and TPBT) and its applicability, a real-life data set is analyzed. Based on the comparison results, we found out that the OGE-TPBT fits the data better than both the GE and TPBT distributions. Also, we used the same real data set to compare the three approaches of the ABC. Based on the comparisons results of these three approaches, we recommend the naive ABC to approximate Bayes estimations in the situation for which there is no analytic solution.

Bayesian Estimation of the Parameters of the Odd Generalized Exponentiated - Inverse Exponential Distribution (OGE -IED)

The Odd Generalized Exponentiated-Inverse Exponential Distribution, a three parameter distribution, is a hybrid of the Generalized Exponential distribution. Each of the parameters were assigned a gamma prior independently resulting to a posterior distribution that is mathematically intractable impossible to obtain marginal posterior distribution for two of the parameters, and a likelihood function that is not known traditionally to R or other statistical software. Resort was made to STAN in order to obtain Bayesian estimates - leveraging on STAN’s provision for user-defined distribution functions. Two datasets were used; remission times (in months) of bladder cancer patients and COVID-19 Survey data in Andalusia, Spain. In the end, the Maximum Likelihood estimates maximized the likelihood more than the Bayesian estimates - though with a slight margin of not more than 0.77. On the other hand, the Bayesian estimates proved to be more stable yielding very negligible standard errors compared to the Maximum Likelihood estimates.

Stationary GE-Process and its Application in Analyzing Gold Price Data

In this paper we introduce a new discrete time and continuous state space stationary process {Xn;n = 1,2,…}, such that Xn follows a two-parameter generalized exponential (GE) distribution. Joint distribution functions, characterization and some dependency properties of this new process have been investigated. The GE-process has three unknown parameters, two shape parameters and one scale parameter, and due to this reason it is more flexible than the existing exponential process. In presence of the scale parameter, if the two shape parameters are equal, then the maximum likelihood estimators of the unknown parameters can be obtained by solving one non-linear equation and if the two shape parameters are arbitrary, then the maximum likelihood estimators can be obtained by solving a two dimensional optimization problem. Two synthetic data sets, and one real gold-price data set have been analyzed to see the performance of the proposed model in practice. Finally some generalizations have been indicated.

Control charts for monitoring mean of generalized exponential distribution with type‐I censoring

AbstractMonitoring censored data is a challenging task as the traditional charts reported poor performance in the case of censored data. This article presents exponentially weighted moving average control charts for monitoring type‐I censored data assuming generalized exponential (GE) distribution. In particular, we replace the censored observation with the conditional expected value (CEV), conditional median (CM), and conditional standard deviation (CSD) to assess the performance of the charts. To evaluate the performance of the memory‐type charts, this study considers different shifts, smoothing weights, and censoring rates in order to compare their effectiveness. Besides a comprehensive study of average run length (ARL) using Monte Carlo simulations, special attention is paid to the standard deviation of the run length (SDRL), because it can be substantially larger for skewed distributions. Furthermore, the application of the proposed charts is illustrated by using two real data sets. Different situations are identified where CEV, CM, and CSD are preferred over each other.

The mixed Novikov–Kazamaki type condition for the uniform integrability of the general stochastic exponential

We generalize the well known mixed Novikov–Kazamaki condition and introduce a new type sufficient condition for the uniform integrability of the stochastic exponential with jumps.

Estimation of the Daily Recovery Cases in Egypt for COVID-19 Using Power Odd Generalized Exponential Lomax Distribution.

Covid-19 has become an important topic this days, because of its bad effect in many fields such as Economics, industrial and commerce. In this paper, Covid-19 will be studied statistically point of view depending on the recovery cases in the Arab Republic of Egypt in the interval of (20 March to 20 August 2020). A power odd generalized exponential Lomax distribution has been considered. Some mathematical properties of the distribution are studied. The method of maximum likelihood and maximum product of spacings are used for estimating the model parameters. Also 95% asymptotic confidence intervals for the estimates of the parameters are derived. A simulation study was conducted to evaluate the numerical behavior of the estimates. The proposed methods are utilized to find estimates of the parameters of power odd generalized exponential Lomax distribution for the recovery cases of corona virus in Egypt.

The Investigation of Approximate Solutions of Deformed Klein–Gordon and Schrödinger Equations Under Modified More General Exponential Screened Coulomb Potential Plus Yukawa Potential in NCQM Symmetries

The Investigation of Approximate Solutions of Deformed Klein–Gordon and Schrödinger Equations Under Modified More General Exponential Screened Coulomb Potential Plus Yukawa Potential in NCQM Symmetries

Two-Sided Group Chain Sampling Plans Based on Truncated Life Test for Generalized Exponential Distribution

Acceptance sampling is an important technique in quality assurance; its main goal is to achieve the most accurate decision in accepting lot using minimum resources. In practice, this often translates into minimizing the required sample sizes for the inspection, while satisfying the maximum allowable risks by consumer and producer. Numerous sampling plans have been developed over the past decades, the most recent being the incorporation of grouping to enable simultaneous inspection in the two-sided chain sampling which considers information from preceding and succeeding samples. This combination offers improved decision accuracy with reduced inspection resources. To-date, two-sided group chain sampling plan (TSGCh) for characteristic based on truncated lifetime has only been explored for Pareto distribution of the 2^nd kind. This article introduces TSGCh sampling plan for products with lifetime that follows generalized exponential distribution. It focuses on minimizing consumer's risk and operates with three acceptance criteria. The equations that derived from the set conditions involving generalized exponential and binomial distributions are mathematically solved to develop this sampling plan. Its performance is measured on the probability of lot acceptance and number of minimum groups. A comparison with the established new two-sided group chain (NTSGCh) indicates that the proposed TSGCh sampling plan performs better in terms of sample size requirement and consumers' protection. Thus, this new acceptance sampling plan can reduce the inspection time, resources, and costs via smaller sample size (number of groups), while providing the desired consumers' protection.

The Weibull Generalized Exponential Distribution with Censored Sample: Estimation and Application on Real Data

This paper is concerned with the estimation of the Weibull generalized exponential distribution (WGED) parameters based on the adaptive Type-II progressive (ATIIP) censored sample. Maximum likelihood estimation (MLE), maximum product spacing (MPS), and Bayesian estimation based on Markov chain Monte Carlo (MCMC) methods have been determined to find the best estimation method. The Monte Carlo simulation is used to compare the three methods of estimation based on the ATIIP-censored sample, and also, we made a bootstrap confidence interval estimation. We will analyze data related to the distribution about single carbon fiber and electrical data as real data cases to show how the schemes work in practice.

Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space

In this work, we proposed ultra generalized exponential–hyperbolic potential (UGEHP) and derived various well known exponential–hyperbolic type potentials by setting parameters in UGEHP and using approximation suggested by Greene–Aldrich. The bound state solutions of the multi (D)-dimensional Schrodinger equation for UGEHP have been presented using the parametric Nikiforov–Uvarov method. The approximate analytical bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. We also investigated the rotational vibrational (RV) partition function from the eigenvalue of UGEHP. By the setting parameters, we obtained eigenvalue spectrum and RV partition function for the screened cosine Kratzer potential, screened Kratzer potential, attractive radial potential, quadratic exponential-type potential, Manning Rosen with class of Yukawa potential and Yukawa potential, class of Yukawa potential, mixed class of Yukawa potential, quantum interaction potential or Hulthen–Yukawa inversely quadratic potential, Hulthen plus inversely quadratic exponential Mie-type potential and Hulthen plus exponential Coulombic potential with centrifugal potential barrier. We studied the behavior of energy eigenvalues and RV partition function for the UGEHP. We computed and tabulated numerical results for $$CO, NO, I_2$$ , HCl and LiH diatomic molecules and compared with numerical results available in literature for same diatomic molecules.

Approximated optimal design for a bivariate step-stress accelerated life test with generalized exponential distribution under type-I progressive censoring

PurposeIn this paper, the author proposed an optimization design for a step-stress accelerated life test (SSALT) with two stress variables for the generalized exponential (GE) distribution under progressive type-I censoring.Design/methodology/approachIn this paper, two stress variables were considered. Progressive censoring and accelerated life testing were used to reduce the time and cost of testing. It was assumed that the lifetimes of the test units followed a GE distribution. The effects of changing stress were considered as a cumulative exposure model. A log-linear relationship between the scale parameter of the GE distribution and the stress was proposed. The maximum likelihood estimators and approximate and bootstrap confidence intervals (CIs) for the model parameters were obtained. An optimum test plan was developed using minimization of the asymptotic variance (AV) of the percentile life under the usual operating condition.FindingsAccording to the simulation results, the bootstrap CIs of the model parameters gave more accurate results than approximate CIs through the length of CIs. The sensitivity analysis was performed to illustrate the effect of initial estimates on optimal values that has been studied. Simulation results also indicated that the optimal times were not too sensitive to the initial values of parameters; thus, the proposed design was robust.Originality/valueIn most studies, only one accelerating stress variable is used. Sometimes accelerating one stress variable does not yield enough failure data. Thus, two stress variables may be needed for additional acceleration. In this paper, two stress variables are considered. The inclusion of two stress variables in a test design will lead to a better understanding of the effect of two simultaneously operating stress variables. Also, the author assumes that the failure time of the test units follows a GE distribution. It is observed that the GE distribution can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. Also, most studies in this field have focused on the derivation of optimum test plans. In this paper, the author examined the estimation of model parameters and the optimization of the test design. In this paper, the asymptotic and bootstrap CIs for the model parameters are calculated. In addition, a sensitivity analysis is performed to examine the effect of the changes in the pre-estimated parameters on the optimal hold times. For determining the optimal test plan, due to nonlinearity and complexity of the objective function, the particle swarm optimization (PSO) algorithm is developed to calculate the optimal hold times. In this method, the research speed is very fast and optimization ability is more.

Monitoring Regularly Maintained Systems Based on the Renewal Process with Generalized Exponential Distribution of Time between Events

Abstract In the modern industrial age, regular system maintenance is an integral process because systems, both engineering and nonengineering, deteriorate over time. Statistical process monitoring techniques, especially control charts, are very helpful in monitoring the performance of such systems and consequently help in decision making on whether maintenance is advantageous or the system has reached a phase where no further maintenance is useful. The aim of this article is to introduce an approach that would integrate maintenance decisions with statistical quality monitoring techniques by assuming an exponentiated class of distributions of interarrival times of the renewal process. This article includes generalized exponential, generalized Rayleigh, and exponentiated Pareto distributions as special cases. An extensive simulation study to compute average run length and average length of inspection (or average time to signal) is conducted, and a comparison is given with approximated exponential control charts. The effect of cost for the regularly maintained generalized exponential systems is also discussed in this study. We focus on the generalized exponential distribution because of its practical relevance in reliability and closeness to Weibull distribution. Finally, an illustrative example is also a part of this article.