In this present work,we propose the expected Bayesian and hierarchical Bayesian approaches to estimate the shape parameter and hazard rate under a generalized progressive hybrid censoring scheme for the Kumaraswamy di...In this present work,we propose the expected Bayesian and hierarchical Bayesian approaches to estimate the shape parameter and hazard rate under a generalized progressive hybrid censoring scheme for the Kumaraswamy distribution.These estimates have been obtained using gamma priors based on various loss functions such as squared error,entropy,weighted balance,and minimum expected loss functions.An investigation is carried out using Monte Carlo simulation to evaluate the effectiveness of the suggested estimators.The simulation provides a quantitative assessment of the estimates accuracy and efficiency under various conditions by comparing them in terms of mean squared error.Additionally,the monthly water capacity of the Shasta reservoir is examined to offer real-world examples of how the suggested estimations may be used and performed.展开更多
Accelerated life tests play a vital role in reliability analysis,especially as advanced technologies lead to the production of highly reliable products to meet market demands and competition.Among these tests,progress...Accelerated life tests play a vital role in reliability analysis,especially as advanced technologies lead to the production of highly reliable products to meet market demands and competition.Among these tests,progressive-stress accelerated life tests(PSALT)allow for continuous changes in applied stress.Additionally,the generalized progressive hybrid censoring(GPHC)scheme has attracted significant attention in reliability and survival analysis,particularly for handling censored data in accelerated testing.It has been applied to various failure models,including competing risks and step-stress models.However,despite its growing relevance,a notable gap remains in the literature regarding the application of GPHC in PSALT models.This paper addresses that gap by studying PSALT under a GPHC scheme with binomial removal.Specifically,it considers lifetimes following the quasi-Xgamma distribution.Model parameters are estimated using both maximum likelihood and Bayesian methods under gamma priors.Interval estimation is provided through approximate confidence intervals,bootstrap methods,and Bayesian credible intervals.Bayesian estimators are derived under squared error and entropy loss functions,using informative priors in simulation and non-informative priors in real data applications.A simulation study is conducted to evaluate various censoring schemes,with coverage probabilities and interval widths assessed via Monte Carlo simulations.Additionally,Bayesian predictive estimates and intervals are presented.The proposed methodology is illustrated through the analysis of two real-world accelerated life test datasets.展开更多
This paper considers the Bayesian and expected Bayesian（E-Bayesian） estimations of the parameter and reliability function for competing risk model from Gompertz distribution under Type-I progressively hybrid censori...This paper considers the Bayesian and expected Bayesian（E-Bayesian） estimations of the parameter and reliability function for competing risk model from Gompertz distribution under Type-I progressively hybrid censoring scheme（PHCS）. The estimations are obtained based on Gamma conjugate prior for the parameter under squared error（SE） and Linex loss functions. The simulation results are provided for the comparison purpose and one data set is analyzed.展开更多
Maximum product spacing for stress–strength model based on progressive Type-II hybrid censored samples with different cases has been obtained.This paper deals with estimation of the stress strength reliability model ...Maximum product spacing for stress–strength model based on progressive Type-II hybrid censored samples with different cases has been obtained.This paper deals with estimation of the stress strength reliability model R=P(Y<X)when the stress and strength are two independent exponentiated Gumbel distribution random variables with different shape parameters but having the same scale parameter.The stress–strength reliability model is estimated under progressive Type-II hybrid censoring samples.Two progressive Type-II hybrid censoring schemes were used,Case I:A sample size of stress is the equal sample size of strength,and same time of hybrid censoring,the product of spacing function under progressive Type-II hybrid censoring schemes.Case II:The sample size of stress is a different sample size of strength,in which the life-testing experiment with a progressive censoring scheme is terminated at a random time T 2 e0;1T.The maximum likelihood estimation and maximum product spacing estimation methods under progressive Type-II hybrid censored samples for the stress strength model have been discussed.A comparison study with classical methods as the maximum likelihood estimation method is discussed.Furthermore,to compare the performance of various cases,Markov chain Monte Carlo simulation is conducted by using iterative procedures as Newton Raphson or conjugate-gradient procedures.Finally,two real datasets are analyzed for illustrative purposes,first data for the breaking strengths of jute fiber,and the second data for the waiting times before the service of the customers of two banks.展开更多
In this paper, we construct a Bayesian framework combining Type-Ⅰ progressively hybrid censoring scheme and competing risks which are independently distributed as exponentiated Weibull distribution with one scale par...In this paper, we construct a Bayesian framework combining Type-Ⅰ progressively hybrid censoring scheme and competing risks which are independently distributed as exponentiated Weibull distribution with one scale parameter and two shape parameters. Since there exist unknown hyper-parameters in prior density functions of shape parameters, we consider the hierarchical priors to obtain the individual marginal posterior density functions,Bayesian estimates and highest posterior density credible intervals. As explicit expressions of estimates cannot be obtained, the componentwise updating algorithm of Metropolis-Hastings method is employed to compute the numerical results. Finally, it is concluded that Bayesian estimates have a good performance.展开更多
This paper proposes a simple constant-stress accel- erated life test （ALT） model from Burr type XII distribution when the data are Type-I progressively hybrid censored. The maximum likelihood estimation （MLE） of t...This paper proposes a simple constant-stress accel- erated life test （ALT） model from Burr type XII distribution when the data are Type-I progressively hybrid censored. The maximum likelihood estimation （MLE） of the parameters is obtained through the numerical method for solving the likelihood equations. Approxi- mate confidence interval （CI）, based on normal approximation to the asymptotic distribution of MLE and percentile bootstrap Cl is derived. Finally, a numerical example is introduced and then a Monte Carlo simulation study is carried out to illustrate the pro- posed method.展开更多
Under Type-Ⅱ progressively hybrid censoring, this paper discusses statistical inference and optimal design on stepstress partially accelerated life test for hybrid system in presence of masked data. It is assumed tha...Under Type-Ⅱ progressively hybrid censoring, this paper discusses statistical inference and optimal design on stepstress partially accelerated life test for hybrid system in presence of masked data. It is assumed that the lifetime of the component in hybrid systems follows independent and identical modified Weibull distributions. The maximum likelihood estimations(MLEs)of the unknown parameters, acceleration factor and reliability indexes are derived by using the Newton-Raphson algorithm. The asymptotic variance-covariance matrix and the approximate confidence intervals are obtained based on normal approximation to the asymptotic distribution of MLEs of model parameters. Moreover,two bootstrap confidence intervals are constructed by using the parametric bootstrap method. The optimal time of changing stress levels is determined under D-optimality and A-optimality criteria.Finally, the Monte Carlo simulation study is carried out to illustrate the proposed procedures.展开更多
The inference for the dependent competing risks model is studied and the dependent structure of failure causes is modeled by a Marshall-Olkin bivariate Rayleigh distribution. Under generalized progressive hybrid censo...The inference for the dependent competing risks model is studied and the dependent structure of failure causes is modeled by a Marshall-Olkin bivariate Rayleigh distribution. Under generalized progressive hybrid censoring(GPHC), maximum likelihood estimates are established and the confidence intervals are constructed based on the asymptotic theory. Bayesian estimates and the highest posterior density credible intervals are obtained by using Gibbs sampling. Simulation and a real life electrical appliances data set are used for practical illustration.展开更多
In general,simple subsystems like series or parallel are integrated to produce a complex hybrid system.The reliability of a system is determined by the reliability of its constituent components.It is often extremely d...In general,simple subsystems like series or parallel are integrated to produce a complex hybrid system.The reliability of a system is determined by the reliability of its constituent components.It is often extremely difficult or impossible to get specific information about the component that caused the system to fail.Unknown failure causes are instances in which the actual cause of systemfailure is unknown.On the other side,thanks to current advanced technology based on computers,automation,and simulation,products have become incredibly dependable and trustworthy,and as a result,obtaining failure data for testing such exceptionally reliable items have become a very costly and time-consuming procedure.Therefore,because of its capacity to produce rapid and adequate failure data in a short period of time,accelerated life testing(ALT)is the most utilized approach in the field of product reliability and life testing.Based on progressively hybrid censored(PrHC)data froma three-component parallel series hybrid system that failed to owe to unknown causes,this paper investigates a challenging problem of parameter estimation and reliability assessment under a step stress partially accelerated life-test(SSPALT).Failures of components are considered to follow a power linear hazard rate(PLHR),which can be used when the failure rate displays linear,decreasing,increasing or bathtub failure patterns.The Tempered random variable(TRV)model is considered to reflect the effect of the high stress level used to induce early failure data.The maximum likelihood estimation(MLE)approach is used to estimate the parameters of the PLHR distribution and the acceleration factor.A variance covariance matrix(VCM)is then obtained to construct the approximate confidence intervals(ACIs).In addition,studentized bootstrap confidence intervals(ST-B CIs)are also constructed and compared with ACIs in terms of their respective interval lengths(ILs).Moreover,a simulation study is conducted to demonstrate the performance of the estimation procedures and the methodology discussed in this paper.Finally,real failure data from the air conditioning systems of an airplane is used to illustrate further the performance of the suggested estimation technique.展开更多
In this article,we consider the statistical inferences of the unknown parameters of a generalized inverted exponential distribution based on the Type II progressively hybrid censored sample.By applying the expectation...In this article,we consider the statistical inferences of the unknown parameters of a generalized inverted exponential distribution based on the Type II progressively hybrid censored sample.By applying the expectation–maximization(EM)algorithm,the maximum likelihood estimators are developed for estimating the unknown parameters.The observed Fisher information matrix is obtained using the missing information principle,and it can be used for constructing asymptotic con-fidence intervals.By applying the bootstrapping technique,the confidence intervals for the parameters are also derived.Bayesian estimates of the unknown parameters are obtained using the Lindley’s approximation.Monte Carlo simulations are imple-mented and observations are given.Finally,a real data set representing the spread factor of micro-drops is analyzed to illustrative purposes.展开更多
基金funded by Researchers Supporting Project number(RSPD2025R969),King Saud University,Riyadh,Saudi Arabia.
文摘In this present work,we propose the expected Bayesian and hierarchical Bayesian approaches to estimate the shape parameter and hazard rate under a generalized progressive hybrid censoring scheme for the Kumaraswamy distribution.These estimates have been obtained using gamma priors based on various loss functions such as squared error,entropy,weighted balance,and minimum expected loss functions.An investigation is carried out using Monte Carlo simulation to evaluate the effectiveness of the suggested estimators.The simulation provides a quantitative assessment of the estimates accuracy and efficiency under various conditions by comparing them in terms of mean squared error.Additionally,the monthly water capacity of the Shasta reservoir is examined to offer real-world examples of how the suggested estimations may be used and performed.
基金supported and funded by the Deanship of Scientifc Research at ImamMohammad Ibn Saud Islamic University(IMSIU)(grant number IMSIU-DDRSP2503).
文摘Accelerated life tests play a vital role in reliability analysis,especially as advanced technologies lead to the production of highly reliable products to meet market demands and competition.Among these tests,progressive-stress accelerated life tests(PSALT)allow for continuous changes in applied stress.Additionally,the generalized progressive hybrid censoring(GPHC)scheme has attracted significant attention in reliability and survival analysis,particularly for handling censored data in accelerated testing.It has been applied to various failure models,including competing risks and step-stress models.However,despite its growing relevance,a notable gap remains in the literature regarding the application of GPHC in PSALT models.This paper addresses that gap by studying PSALT under a GPHC scheme with binomial removal.Specifically,it considers lifetimes following the quasi-Xgamma distribution.Model parameters are estimated using both maximum likelihood and Bayesian methods under gamma priors.Interval estimation is provided through approximate confidence intervals,bootstrap methods,and Bayesian credible intervals.Bayesian estimators are derived under squared error and entropy loss functions,using informative priors in simulation and non-informative priors in real data applications.A simulation study is conducted to evaluate various censoring schemes,with coverage probabilities and interval widths assessed via Monte Carlo simulations.Additionally,Bayesian predictive estimates and intervals are presented.The proposed methodology is illustrated through the analysis of two real-world accelerated life test datasets.
基金supported by the National Natural Science Foundation of China(7117116471401134+1 种基金71571144)the Natural Science Basic Research Program of Shaanxi Province(2015JM1003)
文摘This paper considers the Bayesian and expected Bayesian（E-Bayesian） estimations of the parameter and reliability function for competing risk model from Gompertz distribution under Type-I progressively hybrid censoring scheme（PHCS）. The estimations are obtained based on Gamma conjugate prior for the parameter under squared error（SE） and Linex loss functions. The simulation results are provided for the comparison purpose and one data set is analyzed.
文摘Maximum product spacing for stress–strength model based on progressive Type-II hybrid censored samples with different cases has been obtained.This paper deals with estimation of the stress strength reliability model R=P(Y<X)when the stress and strength are two independent exponentiated Gumbel distribution random variables with different shape parameters but having the same scale parameter.The stress–strength reliability model is estimated under progressive Type-II hybrid censoring samples.Two progressive Type-II hybrid censoring schemes were used,Case I:A sample size of stress is the equal sample size of strength,and same time of hybrid censoring,the product of spacing function under progressive Type-II hybrid censoring schemes.Case II:The sample size of stress is a different sample size of strength,in which the life-testing experiment with a progressive censoring scheme is terminated at a random time T 2 e0;1T.The maximum likelihood estimation and maximum product spacing estimation methods under progressive Type-II hybrid censored samples for the stress strength model have been discussed.A comparison study with classical methods as the maximum likelihood estimation method is discussed.Furthermore,to compare the performance of various cases,Markov chain Monte Carlo simulation is conducted by using iterative procedures as Newton Raphson or conjugate-gradient procedures.Finally,two real datasets are analyzed for illustrative purposes,first data for the breaking strengths of jute fiber,and the second data for the waiting times before the service of the customers of two banks.
基金Supported by the National Natural Science Foundation of China（71571144,71401134,71171164,11701406） Supported by the International Cooperation and Exchanges in Science and Technology Program of Shaanxi Province（2016KW-033）
文摘In this paper, we construct a Bayesian framework combining Type-Ⅰ progressively hybrid censoring scheme and competing risks which are independently distributed as exponentiated Weibull distribution with one scale parameter and two shape parameters. Since there exist unknown hyper-parameters in prior density functions of shape parameters, we consider the hierarchical priors to obtain the individual marginal posterior density functions,Bayesian estimates and highest posterior density credible intervals. As explicit expressions of estimates cannot be obtained, the componentwise updating algorithm of Metropolis-Hastings method is employed to compute the numerical results. Finally, it is concluded that Bayesian estimates have a good performance.
基金supported by the National Natural Science Foundation of China(7117116470471057)
文摘This paper proposes a simple constant-stress accel- erated life test （ALT） model from Burr type XII distribution when the data are Type-I progressively hybrid censored. The maximum likelihood estimation （MLE） of the parameters is obtained through the numerical method for solving the likelihood equations. Approxi- mate confidence interval （CI）, based on normal approximation to the asymptotic distribution of MLE and percentile bootstrap Cl is derived. Finally, a numerical example is introduced and then a Monte Carlo simulation study is carried out to illustrate the pro- posed method.
基金supported by the National Natural Science Foundation of China(71401134 71571144+1 种基金 71171164)the Program of International Cooperation and Exchanges in Science and Technology Funded by Shaanxi Province(2016KW-033)
文摘Under Type-Ⅱ progressively hybrid censoring, this paper discusses statistical inference and optimal design on stepstress partially accelerated life test for hybrid system in presence of masked data. It is assumed that the lifetime of the component in hybrid systems follows independent and identical modified Weibull distributions. The maximum likelihood estimations(MLEs)of the unknown parameters, acceleration factor and reliability indexes are derived by using the Newton-Raphson algorithm. The asymptotic variance-covariance matrix and the approximate confidence intervals are obtained based on normal approximation to the asymptotic distribution of MLEs of model parameters. Moreover,two bootstrap confidence intervals are constructed by using the parametric bootstrap method. The optimal time of changing stress levels is determined under D-optimality and A-optimality criteria.Finally, the Monte Carlo simulation study is carried out to illustrate the proposed procedures.
基金supported by the China Postdoctoral Science Foundation(2019M650260)the National Natural Science Foundation of China(11501433)。
文摘The inference for the dependent competing risks model is studied and the dependent structure of failure causes is modeled by a Marshall-Olkin bivariate Rayleigh distribution. Under generalized progressive hybrid censoring(GPHC), maximum likelihood estimates are established and the confidence intervals are constructed based on the asymptotic theory. Bayesian estimates and the highest posterior density credible intervals are obtained by using Gibbs sampling. Simulation and a real life electrical appliances data set are used for practical illustration.
文摘In general,simple subsystems like series or parallel are integrated to produce a complex hybrid system.The reliability of a system is determined by the reliability of its constituent components.It is often extremely difficult or impossible to get specific information about the component that caused the system to fail.Unknown failure causes are instances in which the actual cause of systemfailure is unknown.On the other side,thanks to current advanced technology based on computers,automation,and simulation,products have become incredibly dependable and trustworthy,and as a result,obtaining failure data for testing such exceptionally reliable items have become a very costly and time-consuming procedure.Therefore,because of its capacity to produce rapid and adequate failure data in a short period of time,accelerated life testing(ALT)is the most utilized approach in the field of product reliability and life testing.Based on progressively hybrid censored(PrHC)data froma three-component parallel series hybrid system that failed to owe to unknown causes,this paper investigates a challenging problem of parameter estimation and reliability assessment under a step stress partially accelerated life-test(SSPALT).Failures of components are considered to follow a power linear hazard rate(PLHR),which can be used when the failure rate displays linear,decreasing,increasing or bathtub failure patterns.The Tempered random variable(TRV)model is considered to reflect the effect of the high stress level used to induce early failure data.The maximum likelihood estimation(MLE)approach is used to estimate the parameters of the PLHR distribution and the acceleration factor.A variance covariance matrix(VCM)is then obtained to construct the approximate confidence intervals(ACIs).In addition,studentized bootstrap confidence intervals(ST-B CIs)are also constructed and compared with ACIs in terms of their respective interval lengths(ILs).Moreover,a simulation study is conducted to demonstrate the performance of the estimation procedures and the methodology discussed in this paper.Finally,real failure data from the air conditioning systems of an airplane is used to illustrate further the performance of the suggested estimation technique.
文摘In this article,we consider the statistical inferences of the unknown parameters of a generalized inverted exponential distribution based on the Type II progressively hybrid censored sample.By applying the expectation–maximization(EM)algorithm,the maximum likelihood estimators are developed for estimating the unknown parameters.The observed Fisher information matrix is obtained using the missing information principle,and it can be used for constructing asymptotic con-fidence intervals.By applying the bootstrapping technique,the confidence intervals for the parameters are also derived.Bayesian estimates of the unknown parameters are obtained using the Lindley’s approximation.Monte Carlo simulations are imple-mented and observations are given.Finally,a real data set representing the spread factor of micro-drops is analyzed to illustrative purposes.
