Progressive Type-II Censored Data Research Articles

Classical and Bayesian inference of inverted modified Lindley distribution based on progressive type-II censoring for modeling engineering data

This paper investigates statistical inferences for product lifetimes following the inverted modified Lindley distribution, utilizing progressive Type-II censored data. The estimation of model parameters employs the maximum likelihood method, complemented by the construction of approximate confidence intervals. Bayesian estimates are also explored, incorporating squared error and linear exponential loss functions with noninformative priors. To approximate Bayes estimates, the proposal presents Gibbs sampling based on the MCMC algorithm. This results in the generation of the greatest posterior density credible intervals for the parameters. A real data analysis is conducted to validate the accuracy of all the models and methods discussed. Finally, computational studies using Monte Carlo simulations are presented to compare the suggested estimators.

A New Test Statistic to Assess the Goodness of Fit of Location-Scale Distribution Based on Progressive Censored Data

The problem of examining how well the data fit a supposed distribution is very important, and it must be confirmed prior to any data analysis, because many data analysis methods assume a specific distribution of data. For this purpose, histograms or Q-Q plots are employed for the assessment of data distribution. Additionally, a GoF TstS utilizes distance measurements between the empirical distribution function and the theoretical cumulative distribution function (cdf) to evaluate data distribution. In life-testing or reliability studies, the observed failure time of test units may not be recorded in some situations. The GoF TstSs for completely observed data can no longer be used in progressive type II censored data (PrCsD). In this paper, we suggest a GoF TstSs and new plot method for the GoF test of symmetric and asymmetric location-scale distribution (LoScD) based on PrCsD. The power of the suggested TstSs is estimated through Monte Carlo (MC) simulations, and it is compared with that of the TstSs using the order statistics (OrSt). Furthermore, we analyzed real data examples (symmetric and asymmetric data).

Investigation the generalized extreme value under liner distribution parameters for progressive type-Ⅱ censoring by using optimization algorithms

<abstract><p>Several random phenomena have been modeled by using extreme value distributions. Based on progressive type-Ⅱ censored data with three different distributions (i.e., fixed, discrete uniform, and binomial random removal), the statistical inference of the generalized extreme value distribution under liner normalization (GEVL distribution) parameters is investigated in this study. Since there is no analytical solution, determining the maximum likelihood parameters for the GEVL distribution is considered to be a problem. Standard numerical methods are frequently insufficient for this dilemma, requiring the use of artificial intelligence algorithms to address this difficulty. Here, nonlinear minimization and a genetic algorithm have been used to tackle that problem. In addition, Lindley approximation and Monte Carlo estimation were implemented via Metropolis-Hastings algorithms to carry out the Bayesian point estimation based on both the squared error loss function and LINEX loss functions. Moreover, the highest posterior density intervals were applied. The proposed theoretical inference techniques have been applied in a numerical simulation and a real-life example.</p></abstract>

Statistical inference for the step-stress model with competing risks from the Kumaraswamy distribution under progressive type-II censoring

In reliability and life analysis, the accelerated life test is frequently used because it can reduce cost and obtain additional reliability and lifetime information. In addition, when a unit fails in a life test, there are often two or more risk factors related to the cause of the failure. In this article, we consider the inference from a step-stress-accelerated life test with competing risks using progressive type-II censored data. Based on the assumption that the parameters affected by stress follow a log-linear model with the stress level, the proportional hazard model with the Kumaraswamy distribution is established. The point estimation of the unknown parameters is derived using maximum likelihood and Bayesian methods. Accordingly, the logarithmic asymptotic confidence and the highest posterior density credible intervals are derived and constructed. Moreover, the algorithm for multi-parameter sampling and simulation technology based on this model is given. Simulation results show that the proposed methods have good performance. In light of the estimates of the parameters, the estimated reliability function under normal conditions can be expressed, and its images under different methods are drawn. Last, a real dataset and a set of simulated data are presented for illustrative purposes.

Reliability Analysis and Its Applications for a Newly Improved Type-II Adaptive Progressive Alpha Power Exponential Censored Sample

Recently, a newly improved Type-II adaptive progressive censoring plan was devised, which can successfully ensure that the test length will not surpass a particular threshold period. In this study, we explore the statistical inference of the alpha power exponential distribution in the context of improved adaptive progressive Type-II censored data. The parameters, reliability, and hazard functions were estimated from both classical and Bayesian viewpoints using this censoring plan. To begin, we applied the maximum likelihood estimation approach to obtain parameter, reliability, and hazard function estimators. Following that, the approximate confidence intervals for the aforementioned metrics were derived, assuming the asymptotic normality traits of the maximum likelihood estimators. Additionally, by employing the Bayesian method via the Markov chain Monte Carlo technique, the point estimators and highest posterior density intervals of various parameters were created based on the symmetric squared error loss. A simulation study that incorporates numerous scenarios was used to assess the effectiveness of various estimation methodologies. The optimal progressive censorship plans are then discussed based on a set of criteria. Finally, three applications from the engineering and medical domains have been offered as examples.

Parameter Estimation of Inverted Exponentiated Half-Logistic Distribution under Progressive Type-II Censored Data with Competing Risks

In this work, the statistical inference of inverted exponentiated half-logistic distribution is considered under the progressive type-II censored data with competing risks. The parameter estimations of the constant scale parameters but different shape parameters that changed due to various risk factors are investigated in this study. The maximum likelihood estimation is deduced to obtain the parameter estimators and asymptotic confidence intervals; meanwhile, the existence and uniqueness of the maximum likelihood estimators are discussed. Subsequently, bootstrap confidence intervals are calculated and presented. The Bayes estimates based on the square error loss function are derived using the Markov Chain Monte Carlo method. Then, the highest posterior density credible intervals are calculated and provided. Simulations are implemented to compare two kinds of estimates. For illustrative purposes, two experimental lifetime datasets from different fields are exhibited and analyzed. Finally, an optimal censoring scheme is suggested under two optimality criteria.

Inference for a constant-stress model under progressive type-II censored data from the truncated normal distribution

Inference for a constant-stress model under progressive type-II censored data from the truncated normal distribution

Bayesian survival analysis of logistic exponential distribution for adaptive progressive Type-II censored data

Bayesian survival analysis of logistic exponential distribution for adaptive progressive Type-II censored 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>

Inference and prediction of progressive Type-II censored data from Unit-Generalized Rayleigh distribution

In this paper, inference and prediction problems are studied under progressively Type-II censored data. When the latent lifetime follows the Unit-Generalized Rayleigh distribution, maximum likelihood estimators of the unknown parameters are established, and corresponding existence and uniqueness are also provided. Besides, the approximate confidence intervals are constructed based on asymptotic approximation theory. For comparison, another alternative generalized point and interval estimates are constructed based on proposed pivotal quantities. Further, point and interval predictions for the censored samples are established by using conventional classical and generalized inferential approaches. Finally, extensive simulation studies are carried out to investigate the performance of different methods, and one real-life example is presented to illustrate the applicability of the obtained results.

Classical and Bayesian Inference of the Inverse Nakagami Distribution Based on Progressive Type-II Censored Samples

This paper explores statistical inferences when the lifetime of product follows the inverse Nakagami distribution using progressive Type-II censored data. Likelihood-based and maximum product of spacing (MPS)-based methods are considered for estimating the parameters of the model. In addition, approximate confidence intervals are constructed via the asymptotic theory using both likelihood and product spacing functions. Based on traditional likelihood and the product of spacing functions, Bayesian estimates are also considered under a squared error loss function using non-informative priors, and Gibbs sampling based on the MCMC algorithm is proposed to approximate the Bayes estimates, where the highest posterior density credible intervals of the parameters are obtained. Numerical studies are presented to compare the proposed estimators using Monte Carlo simulations. To demonstrate the proposed methodology in a real-life scenario, a well-known data set on agricultural machine elevators with high defect rates is also analyzed for illustration.

E-Bayesian Estimations and Its E-MSE For Compound Rayleigh Progressive Type-II Censored Data

Over the past decades, various methods to estimate the unknown parameter, the survival function, and the hazard rate of a statistical distribution have been proposed from the availability of type-II censored data. They are all differing in terms of how the progressive type-II censored data of the underlying distribution are available. In this study, we estimate the parameter, the survival function, and the hazard rate of the compound Rayleigh distribution by using the E-Bayesian estimation when the progressive type-II censored data are available. The resulting estimators are evaluated based on the asymmetric general entropy and the symmetric squared error loss functions. In addition, the E-Bayesian estimators under the different loss functions have been compared through a real data analysis and Monte Carlo simulation studies by calculating the E-MSE of the resulting estimators.

Product of spacing estimation of entropy for inverse Weibull distribution under progressive type-II censored data with applications

This paper seeks to estimate the entropy for the inverse Weibull distribution using progressively Type-II censored data. To reach this objective, the entropy is defined through three entropy measures, namely, Rényi, q-entropy and Shannon entropy, and two estimation methods are used to estimate them. The first approach to estimate these quantities is the method of maximum likelihood. Furthermore, and for the first time, we consider the method of maximum product of spacing to estimate the mentioned entropy measures. Also, a simulation study is carried out and two real data sets are analysed. The numerical outcomes showed that the maximum likelihood provides good point estimates while the interval estimates based on the maximum product of spacing method have the shortest confidence interval lengths.

Reliability inference for stress-strength model based on inverted exponential Rayleigh distribution under progressive Type-II censored data

In this paper, stress-strength model is studied for an inverted exponential Rayleigh distribution (IERD) when the latent failure times are progressively Type-II censored. When both strength and stress random variables follow common IERD scale parameters, the maximum likelihood estimate of stress-strength reliability (SSR) is established and the associated approximate confidence interval is also constructed using the asymptotic distribution theory and delta method. By constructing pivotal quantities, another alternative generalized estimates for SSR are also proposed for comparison. Moreover, when there are arbitrary strength and stress parameters, likelihood and generalized pivotal based estimates are also presented. In addition, testing problem is gave for comparing the equality of different strength and stress parameters. Finally, simulation study and a real data example are provided for illustration.

Non-parametric estimation of the extropy and the entropy measures based on progressive type-II censored data with testing uniformity

ABSTRACT The extropy measure is a complementary dual function of Shannon entropy which was proposed by Lad et al. Extropy: complementary dual of entropy. Stat Sci. 2015;30:40–58. This measure of uncertainty have received a considerable attention in the last five years. In this paper, several methods of estimation for the extropy and entropy measures based a progressively Type-II censored data are derived. Simulation studies have been performed to see the effectiveness of the proposed methods, and two real data sets have been analysed for illustrative purposes. Extropy and entropy-based tests of uniformity are also proposed along with their simulated critical values. The power values of the proposed tests are simulated and studied under various alternatives.

Bayesian Estimation of Entropy for Burr Type XII Distribution under Progressive Type-II Censored Data

With the rapid development of statistics, information entropy is proposed as an important indicator used to quantify information uncertainty. In this paper, maximum likelihood and Bayesian methods are used to obtain the estimators of the entropy for a two-parameter Burr type XII distribution under progressive type-II censored data. In the part of maximum likelihood estimation, the asymptotic confidence intervals of entropy are calculated. In Bayesian estimation, we consider non-informative and informative priors respectively, and asymmetric and symmetric loss functions are both adopted. Meanwhile, the posterior risk is also calculated to evaluate the performances of the entropy estimators against different loss functions. In a numerical simulation, the Lindley approximation and the Markov chain Monte Carlo method were used to obtain the Bayesian estimates. In turn, the highest posterior density credible intervals of the entropy were derived. Finally, average absolute bias and mean square error were used to evaluate the estimators under different methods, and a real dataset was selected to illustrate the feasibility of the above estimation model.

Accelerated life tests for modified Kies exponential lifetime distribution: binomial removal, transformers turn insulation application and numerical results

<abstract> This paper is concerned with statistical inference of multiple constant-stress testing for progressive type-II censored data with binomial removal. The failure times of the test units are assumed to be independent and follow the modified Kies exponential (MKEx) distribution. The maximum likelihood method as well as Bayes method are used to derive both point and interval estimates of the parameters. Furthermore, a real data application for transformers turn insulation is used to illustrate the proposed methods. Moreover, this real data set is used to show that MKEx distribution can be a possible alternative model to the exponential, generalized exponential and Weibull distributions. Finally, simulation studies are carried out to investigate the accuracy of the different estimation methods. </abstract>

On Progressive Type-II Censored Samples from Alpha Power Exponential Distribution

In this paper the two-parameter α -power exponential distribution is studied. We study the two-parameter α -power exponential μ , λ distribution with the location parameter μ &gt; 0 and scale parameter λ &gt; 0 under progressive Type-II censored data with fixed shape parameter α . We estimate the maximum likelihood estimators of these unknown parameters numerically since it cannot be solved analytically. We use the approximate best linear unbiased estimators μ ∗ and λ ∗ , as an initial guesses to obtain the MLEs μ ^ and λ ^ . We estimate the interval estimation of these unknowns’ parameters. Monte Carlo simulations are performed and data examples have been provided for illustration and comparison.

Assessing the effect of E-Bayesian inference for Poisson inverse exponential distribution parameters under different loss functions and its application

This paper present E-Bayesian and Bayesian estimators of parameters of Poisson inverse exponential distribution (PIED) under Squared error loss function (SELF), General entropy loss function (GELF) and Linear Exponential loss function (LINEX) for progressive type-II censored data with binomial removals (PT-II CBRs). The E-Bayesian and corresponding Bayesian estimators are compared in terms of their risks based on simulated samples from PIED. The proposed methodology is applied to survival time of multiple myeloma patients data.

Goodness-of-fit test for exponentiality based on spacings for general progressive Type-II censored data

There have been numerous tests proposed to determine whether or not the exponential model is suitable for a given data set. In this article, we propose a new test statistic based on spacings to test whether the general progressive Type-II censored samples are from exponential distribution. The null distribution of the test statistic is discussed and it could be approximated by the standard normal distribution. Meanwhile, we propose an approximate method for calculating the expectation and variance of samples under null hypothesis and corresponding power function is also given. Then, a simulation study is conducted. We calculate the approximation of the power based on normality and compare the results with those obtained by Monte Carlo simulation under different alternatives with distinct types of hazard function. Results of simulation study disclose that the power properties of this statistic by using Monte Carlo simulation are better for the alternatives with monotone increasing hazard function, and otherwise, normal approximation simulation results are relatively better. Finally, two illustrative examples are presented.