Generalized Exponential Distribution Research Articles

Stabilization of Semi-Markovian Jumping Uncertain Complex-Valued Networks with Time-Varying Delay: A Sliding-Mode Control Approach

This paper pays close attention to the stabilization issue for delayed uncertain semi-Markovian jumping complex-valued networks via sliding mode control. The concerned corresponding transition rates depend on a positive constant, i.e., sojourn-time, which is not required to obey the general exponential distribution. Combine the generalized Dynkin’s formula with Lyapunov stability theory as well as the characteristics of cumulative distribution functions, a few sufficient criteria are proposed to ascertain the stochastic stability of the obtained sliding mode dynamical system. In addition, design a novel sliding mode controller to ensure all state trajectories of the potential closed-loop system can reach the synthesized sliding mode switching surface in a finite time and maintain there in the subsequent time. In the end of paper, one simple example is presented to verify superiority and feasibility of the provided controller design scheme.

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.

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.

Estimation of step-stress life testing model using time-censoring: A Bayesian approach

Accelerated life tests (ALTs) of highly reliable products or materials are effective testing techniques to gather failure data rapidly in a limited time period. Also, partially accelerated life tests (PALTs) can enable us to achieve this goal without putting all test units under severe conditions. This article considers both frequent and Bayesian estimations of the step-stress PALTs model using time-censored data from generalized exponential distribution (GED). The maximum likelihood and Bayesian estimates of the model parameters are obtained. The posterior means and posterior variances are computed under the squared error (SE) loss function using Lindley’s procedure. The performance of the estimators is evaluated numerically for different parameter values and different sample sizes via their mean squared error (MSE). In addition, the average confidence intervals lengths (ACIL) of the model parameters are also obtained. For illustrative purposes, a simulation study is given.

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.

Conditionally Exponential Prior in Focal Near- and Far-Field EEG Source Localization via Randomized Multiresolution Scanning (RAMUS)

In this paper, we focus on the inverse problem of reconstructing distributional brain activity with cortical and weakly detectable deep components in non-invasive Electroencephalography. We consider a recently introduced hybrid reconstruction strategy combining a hierarchical Bayesian model to incorporate a priori information and the advanced randomized multiresolution scanning (RAMUS) source space decomposition approach to reduce modelling errors, respectively. In particular, we aim to generalize the previously extensively used conditionally Gaussian prior (CGP) formalism to achieve distributional reconstructions with higher focality. For this purpose, we introduce as a hierarchical prior, a general exponential distribution, which we refer to as conditionally exponential prior (CEP). The first-degree CEP corresponds to focality enforcing Laplace prior, but it also suffers from strong depth bias, when applied in numerical modelling, making the deep activity unrecoverable. We sample over multiple resolution levels via RAMUS to reduce this bias as it is known to depend on the resolution of the source space. Moreover, we introduce a procedure based on the physiological a priori knowledge of the brain activity to obtain the shape and scale parameters of the gamma hyperprior that steer the CEP. The posterior estimates are calculated using iterative statistical methods, expectation maximization and iterative alternating sequential algorithm, which we show to be algorithmically similar and to have a close resemblance to the iterative ell _1 and ell _2 reweighting methods. The performance of CEP is compared with the recent sampling-based dipole localization method Sequential semi-analytic Monte Carlo estimation (SESAME) in numerical experiments of simulated somatosensory evoked potentials related to the human median nerve stimulation. Our results obtained using synthetic sources suggest that a hybrid of the first-degree CEP and RAMUS can achieve an accuracy comparable to the second-degree case (CGP) while being more focal. Further, the proposed hybrid is shown to be robust to noise effects and compares well with the dipole reconstructions obtained with SESAME.

Generalized exponential distribution with interval-censored data and time dependent covariate

This study will improve the performance of the Generalized Exponential Distribution (GED) by incorporating time-dependent covariates (TD) in the presence of interval-censored data. Interval-censored data usually arises in clinical and epidemiological studies where lifetime is only known to fall within an interval. Moreover, this study concentrates on the parameters estimation for this distribution. This study will compare the maximum likelihood estimation (MLE) for two distinct models, namely time-dependent covariates model and time-independent covariates model in terms of their bias, standard error (SE) and root mean square error (RMSE) at various attendance probability (AP) and sample sizes. The results indicate that bias, SE and RMSE values of the parameter estimates increase with the increase in attendance probability and decrease with the increase in sample size.

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.

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.

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.

Observed Microphysical Characteristics of Stratiform and Convective Precipitation over an Inland Arid Region of the Qinghai–Tibet Plateau

This study analyzed the microphysical characteristics of stratiform and convective precipitation over an inland arid region of Qinghai–Tibet Plateau in summer for the first time. The observed precipitation data were from the OTT Parsivel2 laser raindrop spectrometer and the raindrop size distribution can be described by a gamma distribution and a general exponential distribution. The results indicate that: (1) compared to the exponential distribution, the gamma distribution is the better function with which to describe the raindrop size distribution in this region; (2) the raindrop sizes are mainly below 1 mm, and the raindrop sizes which contribute most to the rainfall intensity are below 2 mm for stratiform precipitation and convective precipitation; (3) the mean values of microphysical parameters, e.g., rainfall intensity, radar reflectivity factor, and liquid water content, are higher for convective precipitation than stratiform precipitation; and (4) the standard Z–R relationship underestimates the radar reflectivity factor in this region. Overall, the obtained results will enhance our understanding and facilitate future studies regarding the microphysical characteristics of precipitation in such regions. For example, the obtained Z–R relationship can be a reference for estimating the radar reflectivity factor in this region with higher accuracy.

Applying Shrinkage Estimation Technique of P(Y<Max X1, X2,…, Xk) in Case of Generalized Exponential Distribution

This paper concerned with estimation reliability (­ for K components parallel system of the stress-strength model with non-identical components which is subjected to a common stress, when the stress and strength follow the Generalized Exponential Distribution (GED) with unknown shape parameter α and the known scale parameter θ (θ=1) to be common. Different shrinkage estimation methods will be considered to estimate ­ depending on maximum likelihood estimator and prior estimates based on simulation using mean squared error (MSE) criteria. The study approved that the shrinkage estimation using shrinkage weight function was the best.

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.

Type II General Exponential Class of Distributions

In this paper, a new class of continuous distributions with two extra positive parameters is introduced and is called the Type II General Exponential (TIIGE) distribution. Some special models are presented. Asymptotics, explicit expressions for the ordinary and incomplete moments, moment residual life, reversed residual life, quantile and generating functions and stress-strengh reliability function are derived. Characterizations of this family are obtained based on truncated moments, hazard function, conditional expectation of certain functions of the random variable are obtained. The performance of the maximum likelihood estimators in terms of biases, mean squared errors and confidence interval length is examined by means of a simulation study. Two real data sets are used to illustrate the application of the proposed class.

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.

Modified chain sampling plan for assuring percentile life under Weibull distribution and generalized exponential distribution

PurposeThe purpose of this paper is to design a modified chain sampling plan for assuring the product percentile life where the lifetime follows Weibull or generalized exponential distributions (GEDs). In order to reduce the cost of inspection when implementing the proposed modified chain sampling plan, it is also considered the economic aspect of designing of proposed plan in this paper.Design/methodology/approachThe authors have designed the proposed plan on the basis of two points on the operating characteristic (OC) curve approach. The optimization problem is used to determine the plan parameters of the proposed plan so that the specified values of producer’s risk and consumer’s risk are satisfied simultaneously.FindingsThe results we have obtained, confirm that the proposed plan will be very effective in reducing the sample size rather than other existing sampling plans. The OC curves of proposed plan, chain sampling plan and zero acceptance number single sampling plan show that the performance of proposed plan in discriminating the good and poor quality lots is better than other two plans. In this paper, it is proved that the value of number of preceding lots required for current lot disposition plays an important role.Originality/valueThe proposed modified chain sampling plan for assuring the percentile lifetime of the products under Weibull or GEDs is not available in the literature. The proposed plan can be used in all the manufacturing industries to assure the product percentile lifetime with minimum sample size as well as minimum cost.

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.