E-Bayesian estimation for the proportional hazard rate model based on record values

In this article, a new parameter estimation method, named E-Bayesian method, is considered to obtain the estimates of the unknown parameter and reliability function based on record values. The maximum likelihood, Bayesian, E-Bayesian, and hierarchical Bayesian estimates of the unknown parameter and reliability function are obtained when the underlying distribution belongs to the proportional hazard rate model. The Bayesian estimates are obtained based on squared error and linear-exponential loss functions. The previously obtained some relations for the E-Bayesian estimates are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations are obtained under the same loss functions. The comparison of the derived estimates are carried out by using Monte Carlo simulations. Real data are analyzed for an illustration of the findings.     

Bayesian Statistical Inference for Weighted Exponential Distribution

Exponential distribution has an extensive application in reliability. Introducing shape parameter to this distribution have produced various distribution functions. In their study in 2009, Gupta and Kundu brought another distribution function using Azzalini's method, which is applicable in reliability and named as weighted exponential (WE) distribution. The parameters of this distribution function have been recently estimated by the above two authors in classical statistics. In this paper, Bayesian estimates of the parameters are derived. To achieve this purpose we use Lindley's approximation method for the integrals that cannot be solved in closed form. Furthermore, a Gibbs sampling procedure is used to draw Markov chain Monte Carlo samples from the posterior distribution indirectly and then the Bayes estimates of parameters are derived. The estimation of reliability and hazard functions are also discussed. At the end of the paper, some comparisons between classical and Bayesian estimation methods are studied by using Monte Carlo simulation study. The simulation study incorporates complete and Type-II censored samples.     

The E-Bayesian and hierarchical Bayesian estimations for the proportional reversed hazard rate model based on record values

In this paper, E-Bayesian and hierarchical Bayesian estimations of the shape parameter, when the underlying distribution belongs to the proportional reversed hazard rate model, are considered. Maximum likelihood, Bayesian and E-Bayesian estimates of the unknown parameter and reliability function are obtained based on record values. The Bayesian estimates are derived based on squared error and linear–exponential loss functions. It is pointed out that some previously obtained order relations of E-Bayesian estimates are inadequate and these results are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations is obtained under the same loss functions. The comparison of the derived estimates is carried out by using Monte Carlo simulations. A real data set is analysed for an illustration of the findings.     

Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach

In this paper, maximum likelihood and Bayes estimators of the parameters, reliability and hazard functions have been obtained for two-parameter bathtub-shaped lifetime distribution when sample is available from progressive Type-II censoring scheme. The Markov chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the model parameters. It has been assumed that the parameters have gamma priors and they are independently distributed. Gibbs within the Metropolis–Hasting algorithm has been applied to generate MCMC samples from the posterior density function. Based on the generated samples, the Bayes estimates and highest posterior density credible intervals of the unknown parameters as well as reliability and hazard functions have been computed. The results of Bayes estimators are obtained under both the balanced-squared error loss and balanced linear-exponential (BLINEX) loss. Moreover, based on the asymptotic normality of the maximum likelihood estimators the approximate confidence intervals (CIs) are obtained. In order to construct the asymptotic CI of the reliability and hazard functions, we need to find the variance of them, which are approximated by delta and Bootstrap methods. Two real data sets have been analyzed to demonstrate how the proposed methods can be used in practice.     

Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application

In this paper, the estimation of parameters for a generalized inverted exponential distribution based on the progressively first-failure type-II right-censored sample is studied. An expectation–maximization (EM) algorithm is developed to obtain maximum likelihood estimates of unknown parameters as well as reliability and hazard functions. Using the missing value principle, the Fisher information matrix has been obtained for constructing asymptotic confidence intervals. An exact interval and an exact confidence region for the parameters are also constructed. Bayesian procedures based on Markov Chain Monte Carlo methods have been developed to approximate the posterior distribution of the parameters of interest and in addition to deduce the corresponding credible intervals. The performances of the maximum likelihood and Bayes estimators are compared in terms of their mean-squared errors through the simulation study. Furthermore, Bayes two-sample point and interval predictors are obtained when the future sample is ordinary order statistics. The squared error, linear-exponential and general entropy loss functions have been considered for obtaining the Bayes estimators and predictors. To illustrate the discussed procedures, a set of real data is analyzed.     

Bayesian and Non Bayesian Estimations on the Exponentiated Modified Weibull Distribution for Progressive Censored Sample

The four-parameter Exponentiated Modified Weibull (EMW) is considered as an important lifetime distribution. Based on progressive Type-II censored sample, maximum likelihood and Bayesian estimators of the parameters, reliability function, and hazard rate function are derived. Two cases are considered: first, the case of one unknown exponent parameter of EMW and second, the case when two parameters of the EMW are both unknown. The Bayes estimators are studied under squared error and LINEX loss functions. The standard Bayes and importance sampling are considered for the estimation. Monte Carlo simulations are performed under different samples sizes and different censoring schemes for investigating and comparing the methods of estimation.     

Inferential statistics on the dynamic system model with time-dependent failure-rate

This study focuses on the classical and Bayesian analysis of a k-components load-sharing parallel system in which components have time-dependent failure rates. In the classical set up, the maximum likelihood estimates of the load-share parameters with their standard errors (SEs) are obtained. (1?γ) 100% simultaneous and two bootstrap confidence intervals for the parameters and system reliability and hazard functions have been constructed. Further, on recognizing the fact that life-testing experiments are very time consuming, the parameters involved in the failure time distribution of the system are expected to follow some random variations. Therefore, Bayes estimates along with their posterior SEs of the parameters and system reliability and hazard functions are obtained by assuming gamma and Jeffrey's priors of the unknown parameters. Markov chain Monte Carlo technique such as Gibbs sampler has been used to obtain Bayes estimates and highest posterior density credible intervals.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Bayesian Estimation for the Exponentiated Weibull Model via Markov Chain Monte Carlo Simulation

Bayesian estimation for the two unknown parameters and the reliability function of the exponentiated Weibull model are obtained based on generalized order statistics. Markov chain Monte Carlo (MCMC) methods are considered to compute the Bayes estimates of the target parameters. Our computations are based on the balanced loss function which contains the symmetric and asymmetric loss functions as special cases. The results have been specialized to the progressively Type-II censored data and upper record values. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation.     

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.     

A new flexible hazard rate distribution: Application and estimation of its parameters

We introduce a new class of flexible hazard rate distributions which have constant, increasing, decreasing, and bathtub-shaped hazard function. This class of distributions obtained by compounding the power and exponential hazard rate functions, which is called the power-exponential hazard rate distribution and contains several important lifetime distributions. We obtain some distributional properties of the new family of distributions. The estimation of parameters is obtained by using the maximum likelihood and the Bayesian methods under squared error, linear-exponential, and Stein’s loss functions. Also, approximate confidence intervals and HPD credible intervals of parameters are presented. An application to real dataset is provided to show that the new hazard rate distribution has a better fit than the other existing hazard rate distributions and some four-parameter distributions. Finally , to compare the performance of proposed estimators and confidence intervals, an extensive Monte Carlo simulation study is conducted.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Estimating the parameters of a Burr distribution under progressive type II censoring

The problem of estimating unknown parameters and reliability function of a two parameter Burr type XII distribution is considered on the basis of a progressively type II censored sample. Several Bayesian estimates are obtained against different symmetric and asymmetric loss functions such as squared error, linex and general entropy. These Bayesian estimates are evaluated by applying the Lindley approximation method. Using simulations, all Bayesian estimates are compared with the corresponding maximum likelihood estimates numerically in terms of their mean square error values and some specific comments are made. Finally, two data sets are analyzed for the purpose of illustration.     

An extended Maxwell distribution: Properties and applications

In this article, we propose an extension of the Maxwell distribution, so-called the extended Maxwell distribution. This extension is evolved by using the Maxwell-X family of distributions and Weibull distribution. We study its fundamental properties such as hazard rate, moments, generating functions, skewness, kurtosis, stochastic ordering, conditional moments and moment generating function, hazard rate, mean and variance of the (reversed) residual life, reliability curves, entropy, etc. In estimation viewpoint, the maximum likelihood estimation of the unknown parameters of the distribution and asymptotic confidence intervals are discussed. We also obtain expected Fisher’s information matrix as well as discuss the existence and uniqueness of the maximum likelihood estimators. The EMa distribution and other competing distributions are fitted to two real datasets and it is shown that the distribution is a good competitor to the compared distributions.     

Posterior analysis of the compound Rayleigh distribution under balanced loss functions for censored data

This paper develops Bayesian analysis in the context of progressively Type II censored data from the compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with associated posterior risks are derived for reliability performances under balanced loss functions by assuming continuous priors for parameters of the distribution. A practical example is used to illustrate the estimation methods. A simulation study has been carried out to compare the performance of estimates. The study indicates that Bayesian estimation should be preferred over maximum likelihood estimation. In Bayesian estimation, the balance general entropy loss function can be effectively employed for optimal decision-making.     

Estimation for the generalized half-normal distribution based on progressive type-II censoring

Based on progressively type-II censored data, the problem of estimating unknown parameters and reliability function of a two-parameter generalized half-normal distribution is considered. Maximum likelihood estimates are obtained by applying expectation-maximization algorithm. Since they do not have closed forms, approximate maximum likelihood estimators are proposed. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX and general entropy are calculated. The Lindley approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, one real data set is analysed.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

The Kumaraswamy generalized gamma distribution with application in survival analysis

We introduce and study the so-called Kumaraswamy generalized gamma distribution that is capable of modeling bathtub-shaped hazard rate functions. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a large number of well-known lifetime special sub-models such as the exponentiated generalized gamma, exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma, generalized Rayleigh, among others. Some structural properties of the new distribution are studied. We obtain two infinite sum representations for the moments and an expansion for the generating function. We calculate the density function of the order statistics and an expansion for their moments. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The usefulness of the new distribution is illustrated in two real data sets.     

Robust Bayesian analysis for parallel system with masked data under inverse Weibull lifetime distribution

Abstract

This paper considers the statistical analysis of masked data in a parallel system with inverse Weibull distributed components under type II censoring. Based on Gamma conjugate prior, the Bayesian estimation as well as the hierarchical Bayesian estimation for the parameters and the reliability function of system are obtained by using the Bayesian theory and the hierarchical Bayesian method. Finally, Monte Carlo simulations are provided to compare the performances of the estimates under different masking probabilities and effective sample sizes.     

Robust Bayesian estimation of a two-parameter exponential distribution under generalized Type-I progressive hybrid censoring

A generalized Type-I progressive hybrid censoring scheme was proposed recently to overcome the limitations of the progressive hybrid censoring scheme. In this article, we provide a robust Bayesian method to estimate the unknown parameters of the two-parameter exponential distribution of a generalized Type-I progressive hybrid censored sample. For each parameter, we derive the marginal posterior density functions and the corresponding Bayesian estimators under the squared error loss function. To assess the proposed method, Monte Carlo simulations are performed using a real dataset.