Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Estimation of parameters of Kumaraswamy-Exponential distribution under progressive type-II censoring

In this paper, the problem of estimating unknown parameters of a two-parameter Kumaraswamy-Exponential (Kw-E) distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. Lindley's approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.     

Two-sample prediction for progressively Type-II censored Weibull lifetimes

Prediction on the basis of censored data has an important role in many fields. This article develops a non-Bayesian two-sample prediction based on a progressive Type-II right censoring scheme. We obtain the maximum likelihood (ML) prediction in a general form for lifetime models including the Weibull distribution. The Weibull distribution is considered to obtain the ML predictor (MLP), the ML prediction estimate (MLPE), the asymptotic ML prediction interval (AMLPI), and the asymptotic predictive ML intervals of the sth-order statistic in a future random sample (Y_s) drawn independently from the parent population, for an arbitrary progressive censoring scheme. To reach this aim, we present three ML prediction methods namely the numerical solution, the EM algorithm, and the approximate ML prediction. We compare the performances of the different methods of ML prediction under asymptotic normality and bootstrap methods by Monte Carlo simulation with respect to biases and mean square prediction errors (MSPEs) of the MLPs of Y_s as well as coverage probabilities (CP) and average lengths (AL) of the AMLPIs. Finally, we give a numerical example and a real data sample to assess the computational comparison of these methods of the ML prediction.     

Model selection criteria for dual-inflated data

ABSTRACT

Inflated data are prevalent in many situations and a variety of inflated models with extensions have been derived to fit data with excessive counts of some particular responses. The family of information criteria (IC) has been used to compare the fit of models for selection purposes. Yet despite the common use in statistical applications, there are not too many studies evaluating the performance of IC in inflated models. In this study, we studied the performance of IC for data with dual-inflated data. The new zero- and K-inflated Poisson (ZKIP) regression model and conventional inflated models including Poisson regression and zero-inflated Poisson (ZIP) regression were fitted for dual-inflated data and the performance of IC were compared. The effect of sample sizes and the proportions of inflated observations towards selection performance were also examined. The results suggest that the Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) are more accurate than the Akaike information criterion (AIC) in terms of model selection when the true model is simple (i.e. Poisson regression (POI)). For more complex models, such as ZIP and ZKIP, the AIC was consistently better than the BIC and CAIC, although it did not reach high levels of accuracy when sample size and the proportion of zero observations were small. The AIC tended to over-fit the data for the POI, whereas the BIC and CAIC tended to under-parameterize the data for ZIP and ZKIP. Therefore, it is desirable to study other model selection criteria for dual-inflated data with small sample size.     

Estimation of the Burr type III distribution with application in unified hybrid censored sample of fracture toughness

In this paper, the statistical inference of the unknown parameters of a Burr Type III (BIII) distribution based on the unified hybrid censored sample is studied. The maximum likelihood estimators of the unknown parameters are obtained using the Expectation–Maximization algorithm. It is observed that the Bayes estimators cannot be obtained in explicit forms, hence Lindley's approximation and the Markov Chain Monte Carlo (MCMC) technique are used to compute the Bayes estimators. Further the highest posterior density credible intervals of the unknown parameters based on the MCMC samples are provided. The new model selection test is developed in discriminating between two competing models under unified hybrid censoring scheme. Finally, the potentiality of the BIII distribution to analyze the real data is illustrated by using the fracture toughness data of the three different materials namely silicon nitride (Si₃N₄), Zirconium dioxide (ZrO₂) and sialon (Si_6?xAl_xO_xN_8?x). It is observed that for the present data sets, the BIII distribution has the better fit than the Weibull distribution which is frequently used in the fracture toughness data analysis.     

On progressively censored generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is considered in this paper. This lifetime distribution is capable of modeling various shapes of failure rates, and hence various shapes of aging criteria. The model can be considered as another useful two-parameter generalization of the IED. Maximum likelihood and Bayes estimates for two parameters of the generalized inverted exponential distribution (GIED) are obtained on the basis of a progressively type-II censored sample. We also showed the existence, uniqueness and finiteness of the maximum likelihood estimates of the parameters of GIED based on progressively type-II censored data. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindley's approximation method and via importance sampling technique. The importance sampling technique is used to compute the Bayes estimates and the associated credible intervals. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. Monte Carlo simulations are performed to compare the performances of the proposed methods and a data set has been analyzed for illustrative purposes.     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

Bayesian model selection for life time data under type II censoring

This paper considers the Bayesian model selection problem in life-time models using type-II censored data. In particular, the intrinsic Bayes factors are calculated for log-normal, exponential, and Weibull lifetime models using noninformative priors under type-II censoring. Numerical examples are given to illustrate our results.     

Estimation of reliability in multicomponent stress–strength based on two parameter exponentiated Weibull Distribution

In this research article, we estimate the multicomponent stress–strength reliability of a system when strength and stress variates are drawn from an exponentiated Weibull distribution with different shape parameters α?and?β, and common shape and scale parameters γ and λ, respectively. We estimate the parameters by using maximum likelihood estimation (MLE) and hence the estimate of reliability obtained applying the MLE method of estimation when samples are drawn from stress and strength distributions. The small sample comparison of the reliability estimates is made through Monte Carlo simulation.     

Estimation of the scale parameter of the Rayleigh distribution with multiply type–II censored sample

Based on a multiply type-II censored sample, the maximum likelihood estimator (MLE) and Bayes estimator for the scale parameter and the reliability function of the Rayleigh distribution are derived. However, since the MLE does not exist an explicit form, an approximate MLE which is the maximizer of an approximate likelihood function will be given. The comparisons among estimators are investigated through Monte Carlo simulations. An illustrative example with the real data concerning the 23 ball bearing in the life test is presented.     

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).     

On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

Inference on progressive-stress model for the exponentiated exponential distribution under type-II progressive hybrid censoring

In this paper, progressive-stress accelerated life tests are applied when the lifetime of a product under design stress follows the exponentiated distribution [G(x)]^α. The baseline distribution, G(x), follows a general class of distributions which includes, among others, Weibull, compound Weibull, power function, Pareto, Gompertz, compound Gompertz, normal and logistic distributions. The scale parameter of G(x) satisfies the inverse power law and the cumulative exposure model holds for the effect of changing stress. A special case for an exponentiated exponential distribution has been discussed. Using type-II progressive hybrid censoring and MCMC algorithm, Bayes estimates of the unknown parameters based on symmetric and asymmetric loss functions are obtained and compared with the maximum likelihood estimates. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared via a simulation study.     

Bias corrected MLEs under progressive type-II censoring scheme

ABSTRACT

Censoring frequently occurs in survival analysis but naturally observed lifetimes are not of a large size. Thus, inferences based on the popular maximum likelihood (ML) estimation which often give biased estimates should be corrected in the sense of bias. Here, we investigate the biases of ML estimates under the progressive type-II censoring scheme (pIIcs). We use a method proposed in Efron and Johnstone [Fisher's information in terms of the hazard rate. Technical Report No. 264, January 1987, Stanford University, Stanford, California; 1987] to derive general expressions for bias corrected ML estimates under the pIIcs. This requires derivation of the Fisher information matrix under the pIIcs. As an application, exact expressions are given for bias corrected ML estimates of the Weibull distribution under the pIIcs. The performance of the bias corrected ML estimates and ML estimates are compared by simulations and a real data application.     

Predicting observables from a general class of distributions

A general class of distributions is proposed to be the underlying population model from which observables are to be predicted using the Bayesian approach. This class of distributions includes, among others, the Weibull, compound Weibull (or three-parameter Burr-type XII), Pareto, beta, Gompertz and compound Gompertz distributions. A proper general prior density function is suggested and the predictive density functions are obtained in the one- and two-sample cases. The informative sample is assumed to be a type II censored sample. Illustrative examples of Weibull (α,β), Burr-type XII (α,β), and Pareto (α,β) distributions are given and compared with the results obtained by previous researchers.     

Estimation based on progressive first-failure censoring from exponentiated exponential distribution

In this paper, point and interval estimations for the parameters of the exponentiated exponential (EE) distribution are studied based on progressive first-failure-censored data. The Bayes estimates are computed based on squared error and Linex loss functions and using Markov Chain Monte Carlo (MCMC) algorithm. Also, based on this censoring scheme, approximate confidence intervals for the parameters of EE distribution are developed. Monte Carlo simulation study is carried out to compare the performances of the different methods by computing the estimated risks (ERs), as well as Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates. Finally, a real data set is introduced and analyzed using EE and Weibull distributions. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the EE model fits the data with the same efficiency as the other model. Point and interval estimation of all parameters are studied based on this real data set as illustrative example.     

Estimation and prediction of Marshall–Olkin extended exponential distribution under progressively type-II censored data

In this paper, we consider Marshall–Olkin extended exponential (MOEE) distribution which is capable of modelling various shapes of failure rates and aging criteria. The purpose of this paper is three fold. First, we derive the maximum likelihood estimators of the unknown parameters and the observed the Fisher information matrix from progressively type-II censored data. Next, the Bayes estimates are evaluated by applying Lindley’s approximation method and Markov Chain Monte Carlo method under the squared error loss function. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We also compute 95% asymptotic confidence interval and symmetric credible interval along with the coverage probability. Third, we consider one-sample and two-sample prediction problems based on the observed sample and provide appropriate predictive intervals under classical as well as Bayesian framework. Finally, we analyse a real data set to illustrate the results derived.     

Reliability estimation in a multicomponent stress–strength model under generalized half-normal distribution based on progressive type-II censoring

In this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example.     

Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples

In this paper, the estimation of parameters for a three-parameter Weibull distribution based on progressively Type-II right censored sample is studied. Different estimation procedures for complete sample are generalized to the case with progressively censored data. These methods include the maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators. We also proposed the use of a censored estimation method with one-step bias-correction to obtain reliable initial estimates for iterative procedures. These methods are compared via a Monte Carlo simulation study in terms of their biases, root mean squared errors and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of inference developed here.