General Entropy Loss Function Research Articles

Classical and Bayesian estimation of multicomponent stress–strength reliability with power Lindley distribution under progressive first-failure censored samples

This research article presents a methodology for estimating the multicomponent reliability based on progressively first-failure censored samples, where the underlying distribution of stress and strength is modelled using the power Lindley distribution. The classical and Bayesian estimation methods are utilized for estimating the multicomponent reliability. In the Bayesian estimation, the Markov Chain Monte Carlo approximation method is employed to obtain the posterior mean under a generalized entropy loss function. Various intervals including asymptotic confidence, bootstrap-p confidence, bootstrap-t confidence, Bayesian credible, and highest posterior density credible intervals are computed. A simulation study is conducted to evaluate the performance of the proposed methodology. Also, two different real data applications are presented to illustrate the practicality of the approach.

Estimation of multi-stress strength reliability under progressive first failure censoring using generalized inverted exponential distribution

We investigate the estimation of multi-stress strength reliability under progressively first failure censoring for the generalized inverted exponential distribution using classical and Bayesian approaches. Classical estimation employs maximum likelihood estimators with the Expectation-Maximization algorithm to handle incomplete data. Bayesian estimation utilizes Markov Chain Monte Carlo (MCMC) techniques and Tierney-Kadane's approximation to approximate the posterior distribution. Furthermore, A generalized entropy loss function is employed to measure estimation accuracy. Finally, Validation is provided through Monte Carlo simulations and real-world applications. Findings highlight the robustness and accuracy of Bayesian methods, particularly with informative priors, and underscore the strengths of MCMC techniques and Bayesian Credible Intervals in specific aspects of estimation.

Estimating the Parameters of the Generalized Inverted Kumaraswamy Distribution through the Utilization of a First Failure-Censored Sampling Plan

ThispaperfocusesonthedevelopmentofapproximateBayesestimatorsfortheshapeparametersofthegeneralizedinverted Kumaraswamy (GIKum) distribution. The estimators are based on a progressive first-failure censored plan. The study considers both maximum likelihood and Bayesian estimations using a gamma-informative prior distribution for the parameters, as well as the reliability function, hazard rate, and reversed hazard rate functions. To obtain the estimators, the paper employs Lindley’s approximation and utilizes Markov Chain Monte Carlo (MCMC) methods. The Bayes estimators are derived with respect to both symmetric (squared error) and asymmetric (linex and general entropy) loss functions. In order to assess the performance of the proposed estimators, the paper presents numerical results obtained through a simulation study involving different sample sizes.

Maximal entropy prior for the simple step‐stress accelerated test

AbstractThe step‐stress procedure is a popular accelerated test used to analyze the lifetime of highly reliable components. This paper considers a simple step‐stress accelerated test assuming a cumulative exposure model with uncensored lifetime data following a Weibull distribution. The maximum likelihood approach is often used to analyze accelerated stress test data. Another approach is to use the Bayesian inference, which is useful when there is limited data available. In this paper, the parameters of the model are estimated based on the objective Bayesian viewpoint using non‐informative priors. Our main aim is to propose the maximal data information prior (MDIP) presented by Zellner (1984) as an alternative prior to the conventional independent gamma priors for the unknown parameters, in situations where there is little or no a priori knowledge about the parameters. We also obtain the Bayes estimators based on both classes of priors, assuming three different loss functions: square error loss function (SELF), linear‐exponential loss function (LINEX), and generalized entropy loss function (GELF). The proposed MDIP prior is compared with the gamma priors via Monte Carlo simulations by examining their biases and mean square errors under the three loss functions, and coverage probability. Additionally, we employ the Markov Chain Monte Carlo (MCMC) algorithm to extract characteristics of marginal posterior distributions, such as the Bayes estimator and credible intervals. Finally, a real lifetime data is presented to illustrate the proposed methodology.

BAYES ESTIMATION OF EXPONENTIALLY DISTRIBUTED SURVIVAL DATA UNDER SYMMETRIC AND ASYMMETRIC LOSS FUNCTIONS

In a research study, population data are often not available, so the population parameter is unknown. Meanwhile, knowledge about the population parameter is needed to know the characteristics of the studied population. Therefore, it is needed to estimate the parameter of the population which can be estimated by sample data. There are several methods of parameter estimation which are generally classified into classical and Bayesian method. This research studied the Bayesian parameter estimation method to determine the parameters of the exponentially distributed survival data associated with the reliability measure of the estimates under symmetric and asymmetric loss functions for complete sample data in a closed form. The symmetric loss functions used in this research are Squared Error Loss Function (SELF) and Minimum Expected Loss Function (MELF). The asymmetric loss functions used are the General Entropy Loss Function (GELF) and Linex Loss Function (LLF). Performance of some loss functions used in this research are then compared through numerical simulation to select the best loss function in determining the parameter estimation of the exponentially distributed survival data. We also studied which loss function is best for underestimation and overestimation modeling. Based on simulation results, the Bayes estimates using MELF is the best method to estimate population parameters of the exponentially distributed survival data for the overestimation modeling, while LLF is the best for the underestimation modeling. We provided direct application in a case study of fluorescence lamp survival data. The results show that the best method to estimate the parameter of the standard fluorescence life data is using LLF for underestimation with and MELF for overestimation with .

Bayesian inference for the inverse Weibull distribution based on symmetric and asymmetric balanced loss functions with application

In this study, the unified hybrid censored approach is employed to estimate the parameters of the inverse Weibull distribution, as well as the survival and hazard rate functions. Parameter estimates are obtained using both Bayesian and Maximum Likelihood approaches, with Bayesian estimates acquired through Lindley's approximation method using three distinct balanced loss functions. These encompass both symmetric and asymmetric balanced loss functions, specifically the balanced squared error (BSE) loss function, the balanced linear exponential (BLINEX) loss function, and the balanced general entropy (BGE) loss function. We conduct a simulation study to compare the effectiveness of various estimators, and a real-world data analysis is presented to illustrate practical implementation. Ultimately, our findings indicate that Bayesian parameter estimates consistently outperform their Maximum Likelihood counterparts across all methods.

The E-Bayesian and hierarchical Bayesian estimations for the reliability analysis of Kumaraswamy generalized distribution based on upper record values

This paper investigates the E-Bayesian and hierarchical Bayesian estimations of shape parameter and reliability function of Kumaraswamy generalized distribution based on upper record values. The classical estimation method is utilized to deduce the maximum likelihood estimation of unknown parameter and reliability function. Bayesian estimates are derived by using conjugate Gamma prior distributions under quadratic and general entropy loss functions. Furthermore, assuming that hyper-hyperparameters obey three prior distributions, the E-Bayesian estimates of unknown parameters and reliability functions are obtained. The hierarchical Bayesian estimates are obtained by using hierarchical prior distributions. We also explore some characteristics and size relationships of E-Bayesian and hierarchical Bayesian estimations. The performance of E-Bayesian, Hierarchical Bayesian, Bayesian, and maximum likelihood estimations is compared based on the minimum mean square error criterion. Finally, the proposed estimation methods are applied to evaluate the reliability of the specimen under ultrasonic fatigue testing, and the results align with their structures and profiles.

Optimal sampling and statistical inferences for Kumaraswamy distribution under progressive Type-II censoring schemes

In this paper, we study non-Bayesian and Bayesian estimation of parameters for the Kumaraswamy distribution based on progressive Type-II censoring. First, the maximum likelihood estimates and maximum product spacings are derived. In addition, we derive the asymptotic distribution of the parameters and the asymptotic confidence intervals. Second, Bayesian estimators under symmetric and asymmetric loss functions (Squared error, linear exponential, and general entropy loss functions) are also obtained. The Lindley approximation and the Markov chain Monte Carlo method are used to derive the Bayesian estimates. Furthermore, we derive the highest posterior density credible intervals of the parameters. We further present an optimal progressive censoring scheme among different competing censoring scheme using three optimality criteria. Simulation studies are conducted to evaluate the performance of the point and interval estimators. Finally, one application of real data sets is provided to illustrate the proposed procedures.

Bayesian Approach for estimating the unknown Scale parameter of Erlang Distribution Based on General Entropy Loss Function

We are used Bayes estimators for unknown scale parameter when shape Parameter is known of Erlang distribution. Assuming different informative priors for unknown scale parameter. We derived The posterior density with posterior mean and posterior variance using different informative priors for unknown scale parameter which are the inverse exponential distribution, the inverse chi-square distribution, the inverse Gamma distribution, and the standard Levy distribution as prior. And we derived Bayes estimators based on the general entropy loss function (GELF) is used the Simulation method to obtain the results. we generated different cases for the parameters of the Erlang model, for different sample sizes. The estimates have been compared in terms of their mean-squared error (MSE). We concluded that the best estimators of the scale parameterof the Erlang distribution, based on GELF for the shape parameter (c=1,2,3) under inverse gamma prior with for all samples sizes(n) where the true cases of the Erlang model are and according to the smallest values of MSE

Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II

Abstract The dynamic cumulative residual (DCR) entropy is a helpful randomness metric that may be used in survival analysis. A challenging issue in statistics and machine learning is the estimation of entropy measures. This article uses progressive censored type II (PCT-II) samples to estimate the DCR Tsallis entropy (DCRTE) for the moment exponential distribution. The non-Bayesian and Bayesian approaches are the recommended estimating strategies. We obtain the DCRTE Bayesian estimator using the gamma and uniform priors via symmetric and asymmetric (linear exponential and general entropy) loss functions (LoFs). The Metropolis–Hastings algorithm is employed to generate Markov chain Monte Carlo samples from the posterior distribution. The accuracy of different estimates for various sample sizes, is implemented via Monte Carlo simulations. Generally, we note based on the simulation study that, in the majority of cases, the DCRTE Bayesian estimates under general entropy followed by linear exponential LoFs are preferable to the others. The accuracy measure of DCRTE Bayesian estimates using a gamma prior has smaller values than the others using a uniform prior. As sample sizes grow, the Bayesian estimates of the DCRTE are closer to the true value. Finally, analysis of the leukemia data confirmed the proposed estimators.

Reliability Inference of Multicomponent Stress–Strength System Based on Chen Distribution Using Progressively Censored Data

In this paper, we study the inference of the multicomponent stress–strength reliability (MSSR) based on the Chen distribution using progressively Type-II censored data. Both the stress and strength variables follow the Chen distribution with a common second shape parameter. The maximum likelihood estimates and the asymptotic confidence intervals of the MSSR are developed. The bootstrap confidence interval of the MSSR is also constructed. The Bayesian estimation of the MSSR is obtained under the generalized entropy loss function using the Markov Chain Monte Carlo method. To check the effectiveness of the proposed approach, simulation studies are performed. Finally, a real data set is analyzed.

Generalized Bayes Estimation Based on a Joint Type-II Censored Sample from K-Exponential Populations

Generalized Bayes is a Bayesian study based on a learning rate parameter. This paper considers a generalized Bayes estimation to study the effect of the learning rate parameter on the estimation results based on a joint censored sample of type-II exponential populations. Squared error, Linex, and general entropy loss functions are used in the Bayesian approach. Monte Carlo simulations were performed to assess how well the different approaches perform. The simulation study compares the Bayesian estimators for different values of the learning rate parameter and different losses.

Statistical Analysis of Inverse Lindley Data Using Adaptive Type-II Progressively Hybrid Censoring with Applications

This paper deals with the statistical inference of the unknown parameter and some life parameters of inverse Lindley distribution under the assumption that the data are adaptive Type-II progressively censored. The maximum likelihood method is considered to acquire the point and interval estimates of the distribution parameter, reliability, and hazard rate functions. The approximate confidence intervals are also addressed. The delta method is taken into consideration to approximate the variances of the estimators of the reliability and hazard rate functions to get the required intervals. Based on the assumption of gamma prior, we further consider Bayesian estimation of the different parameters. The Bayes estimates are obtained by considering squared error and general entropy loss functions. The Bayes estimates and highest posterior density credible intervals are obtained by employing the Markov chain Monte Carlo procedure. An exhaustive numerical study is conducted to compare the offered estimates with regard to their root means squared error, relative absolute biases, confidence lengths, and coverage probabilities. To explain the suggested methods, two applications are investigated. The numerical findings show that the Bayes estimates perform better than those obtained based on the maximum likelihood method. The Bayesian estimations using the asymmetric loss function give more efficient estimates than the symmetric loss. Finally, the inverse Lindley distribution is recommended to be used as a suitable model to fit airborne communication transceiver and wooden toys data sets when compared with some competitive models including inverse Weibull, inverse gamma and alpha power inverted exponential.

Bayesian estimation for the type-I hybrid xgamma distribution using asymmetric loss function

This article proposes the Bayes estimation of the parameter and reliability function for xgamma distribution in the presence of type-I hybrid censored observations. The Bayes estimate of the parameter has been obtained by assuming informative and non-informative priors using general entropy loss function. Obviously, censoring adds difficulties in estimation procedure; hence the Bayes estimators computed with type-I hybrid censored observation under the mentioned prior often do not assume any standard form. Therefore, Bayes estimates are computed using Tierney-Kadane approximation and Markov Chain Monte Carlo numerical technique. Further, different interval estimates namely asymptotic confidence interval, bootstrap confidence interval and highest posterior density interval along with the width of the interval and coverage probability are also discussed. The maximum likelihood estimate for the same has also been computed using non- linear maximization iterative procedure and compared with corresponding Bayes estimates using Monte Carlo simulations. The comparison of the estimators are made in terms of average loss over whole sample space and corresponding length of the interval. lastly, one medical data set has been considered for the real application of the proposed study.

On Statistical Modeling Using a New Version of the Flexible Weibull Model: Bayesian, Maximum Likelihood Estimates, and Distributional Properties with Applications in the Actuarial and Engineering Fields

In this article, we present a new statistical modification of the Weibull model for updating the flexibility, called the generalized Weibull-Weibull distribution. The new modification of the Weibull model is defined and studied in detail. Some mathematical and statistical functions are studied, such as the quantile function, moments, the information generating measure, the Shannon entropy and the information energy. The joint distribution functions of the two marginal univariate models via the Copula model are provided. The unknown parameters are estimated using the maximum likelihood method and Bayesian method via Monte Carlo simulations. The Bayesian approach is discussed using three different loss functions: the quadratic error loss function, the LINEX loss function, and the general entropy loss function. We perform some numerical simulations to show how interesting the theoretical results are. Finally, the practical application of the proposed model is illustrated by analyzing two applications in the actuarial and engineering fields using corporate data to show the elasticity and advantage of the proposed generalized Weibull-Weibull model. The practical applications show that proposed model is very suitable for modeling actuarial and technical data sets and other related fields.

E-Bayesian inference for xgamma distribution under progressive type II censoring with binomial removals and their applications

ABSTRACT In this article, we propose E-Bayes estimators of the parameter of xgamma distribution under squared error loss function, general entropy loss function, and linear exponential loss function for progressive type II censored data with binomial removals. The proposed estimators, maximum likelihood estimator, and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from xgamma distribution. The proposed methodology is illustrated on two real data sets of bile duct cancer data and the endurance of deep-groove ball bearings data.

Statistical inference for the Power Rayleigh distribution based on adaptive progressive Type-II censored data

<abstract><p>The Power Rayleigh distribution (PRD) is a new extension of the standard one-parameter Rayleigh distribution. To employ this distribution as a life model in the analysis of reliability and survival data, we focused on the statistical inference for the parameters of the PRD under the adaptive Type-II censored scheme. Point and interval estimates for the model parameters and the corresponding reliability function at a given time are obtained using likelihood, Bootstrap and Bayesian estimation methods. A simulation study is conducted in different settings of the life testing experiment to compare and evaluate the performance of the estimates obtained. In addition, the estimation procedure is also investigated in real lifetimes data. The results indicated that the obtained estimates gave an accurate and efficient estimation of the model parameters. The Bootstrap estimates are better than the estimates obtained by the likelihood estimation approach, and estimates obtained using the Markov Chain Monte Carlo method by the Bayesian approach under both the squared error and the general entropy loss functions have priority over other point and interval estimates. Under the adaptive Type-II censoring scheme, concluding results confirmed that the PRD can be effectively used to model the lifetimes in survival and reliability analysis.</p></abstract>

Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms

<abstract><p>Examining life-testing experiments on a product or material usually requires a long time of monitoring. To reduce the testing period, units can be tested under more severe than normal conditions, which are called accelerated life tests (ALTs). The objective of this study is to investigate the problem of point and interval estimations of the Lomax distribution under constant stress partially ALTs based on progressive first failure type-II censored samples. The point estimates of unknown parameters and the acceleration factor are obtained by using maximum likelihood and Bayesian approaches. Since reliability data are censored, the maximum likelihood estimates (MLEs) are derived utilizing the general expectation-maximization (EM) algorithm. In the process of Bayesian inference, the Bayes point estimates as well as the highest posterior density credible intervals of the model parameters and acceleration factor, are reported. This is done by using the Markov Chain Monte Carlo (MCMC) technique concerning both symmetric (squared error) and asymmetric (linear-exponential and general entropy) loss functions. Monte Carlo simulation studies are performed under different sizes of samples for comparison purposes. Finally, the proposed methods are applied to oil breakdown times of insulating fluid under two high-test voltage stress level data.</p></abstract>

Analysis of information measures using generalized type-Ⅰ hybrid censored data

<abstract><p>An entropy measure of uncertainty has a complementary dual function called extropy. In the last six years, this measure of randomness has gotten a lot of attention. It cannot, however, be applied to systems that have survived for some time. As a result, the idea of residual extropy was created. To estimate the extropy and residual extropy, Bayesian and non-Bayesian estimators of unknown parameters of the exponentiated gamma distribution are generated. Bayesian estimators are regarded using balanced loss functions like the balanced squared error, balanced linear exponential and balanced general entropy. We use the Lindley method to get the extropy and residual extropy estimates for the exponentiated gamma distribution based on generalized type-Ⅰ hybrid censored data. To test the effectiveness of the proposed methodologies, a simulation experiment was carried out, and the actual data set was studied for illustrative purposes. In summary, the mean squared error values decrease as the number of failures increases, according to the results obtained. The Bayesian estimates of residual extropy under the balanced linear exponential loss function perform well compared to the other estimates. Alternatively, the Bayesian estimates of the extropy perform well under a balanced general entropy loss function in the majority of situations.</p></abstract>

Multi-Stage Temporal Convolutional Network with Moment Loss and Positional Encoding for Surgical Phase Recognition.

In recent times, many studies concerning surgical video analysis are being conducted due to its growing importance in many medical applications. In particular, it is very important to be able to recognize the current surgical phase because the phase information can be utilized in various ways both during and after surgery. This paper proposes an efficient phase recognition network, called MomentNet, for cholecystectomy endoscopic videos. Unlike LSTM-based network, MomentNet is based on a multi-stage temporal convolutional network. Besides, to improve the phase prediction accuracy, the proposed method adopts a new loss function to supplement the general cross entropy loss function. The new loss function significantly improves the performance of the phase recognition network by constraining un-desirable phase transition and preventing over-segmentation. In addition, MomnetNet effectively applies positional encoding techniques, which are commonly applied in transformer architectures, to the multi-stage temporal convolution network. By using the positional encoding techniques, MomentNet can provide important temporal context, resulting in higher phase prediction accuracy. Furthermore, the MomentNet applies label smoothing technique to suppress overfitting and replaces the backbone network for feature extraction to further improve the network performance. As a result, the MomentNet achieves 92.31% accuracy in the phase recognition task with the Cholec80 dataset, which is 4.55% higher than that of the baseline architecture.