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.     

Estimation under modified Weibull distribution based on right censored generalized order statistics

In this paper, the maximum likelihood (ML) and Bayes, by using Markov chain Monte Carlo (MCMC), methods are considered to estimate the parameters of three-parameter modified Weibull distribution (MWD(β, τ, λ)) based on a right censored sample of generalized order statistics (gos). Simulation experiments are conducted to demonstrate the efficiency of the proposed methods. Some comparisons are carried out between the ML and Bayes methods by computing the mean squared errors (MSEs), Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates to illustrate the paper. Three real data sets from Weibull(α, β) distribution are introduced and analyzed using the MWD(β, τ, λ) and also using the Weibull(α, β) distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic, {AIC and BIC} to emphasize that the MWD(β, τ, λ) fits the data better than the other distribution. All parameters are estimated based on type-II censored sample, censored upper record values and progressively type-II censored sample which are generated from the real data sets.     

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.     

A fast Monte Carlo expectation–maximization algorithm for estimation in latent class model analysis with an application to assess diagnostic accuracy for cervical neoplasia in women with atypical glandular cells

In this article, we use a latent class model (LCM) with prevalence modeled as a function of covariates to assess diagnostic test accuracy in situations where the true disease status is not observed, but observations on three or more conditionally independent diagnostic tests are available. A fast Monte Carlo expectation–maximization (MCEM) algorithm with binary (disease) diagnostic data is implemented to estimate parameters of interest; namely, sensitivity, specificity, and prevalence of the disease as a function of covariates. To obtain standard errors for confidence interval construction of estimated parameters, the missing information principle is applied to adjust information matrix estimates. We compare the adjusted information matrix-based standard error estimates with the bootstrap standard error estimates both obtained using the fast MCEM algorithm through an extensive Monte Carlo study. Simulation demonstrates that the adjusted information matrix approach estimates the standard error similarly with the bootstrap methods under certain scenarios. The bootstrap percentile intervals have satisfactory coverage probabilities. We then apply the LCM analysis to a real data set of 122 subjects from a Gynecologic Oncology Group study of significant cervical lesion diagnosis in women with atypical glandular cells of undetermined significance to compare the diagnostic accuracy of a histology-based evaluation, a carbonic anhydrase-IX biomarker-based test and a human papillomavirus DNA test.     

Inference on a progressive type I interval-censored truncated normal distribution

In this paper, we consider the problem of making statistical inference for a truncated normal distribution under progressive type I interval censoring. We obtain maximum likelihood estimators of unknown parameters using the expectation-maximization algorithm and in sequel, we also compute corresponding midpoint estimates of parameters. Estimation based on the probability plot method is also considered. Asymptotic confidence intervals of unknown parameters are constructed based on the observed Fisher information matrix. We obtain Bayes estimators of parameters with respect to informative and non-informative prior distributions under squared error and linex loss functions. We compute these estimates using the importance sampling procedure. The highest posterior density intervals of unknown parameters are constructed as well. We present a Monte Carlo simulation study to compare the performance of proposed point and interval estimators. Analysis of a real data set is also performed for illustration purposes. Finally, inspection times and optimal censoring plans based on the expected Fisher information matrix are discussed.     

Estimation for the Parameters of Generalized Half-normal Distribution Based on Progressive Type-I Interval Censoring

Based on progressively Type-I interval censored sample, the problem of estimating unknown parameters of a two parameter generalized half-normal(GHN) distribution is considered. Different methods of estimation are discussed. They include the maximum likelihood estimation, midpoint approximation method, approximate maximum likelihood estimation, method of moments, and estimation based on probability plot. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX, and general entropy is calculated. The Lindley’s approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, analysis is also carried out for a real dataset.     

Estimating the Pareto parameters under progressive censoring data for constant-partially accelerated life tests

Accelerated life testing is widely used in product life testing experiments since it provides significant reduction in time and cost of testing. In this paper, assuming that the lifetime of items under use condition follow the two-parameter Pareto distribution of the second kind, partially accelerated life tests based on progressively Type-II censored samples are considered. The likelihood equations of the model parameters and the acceleration factor are reduced to a single nonlinear equation to be solved numerically to obtain the maximum-likelihood estimates (MLEs). Based on normal approximation to the asymptotic distribution of MLEs, the approximate confidence intervals (ACIs) for the parameters are derived. Two bootstrap CIs are also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose to apply Markov chain Monte Carlo method to tackle this problem, which allows us to construct the credible interval of the involved parameters. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and to compare the performance of different corresponding CIs considered.     

Estimation under inverse Weibull distribution based on Balakrishnan’s unified hybrid censored scheme

In this article, point and interval estimations of the parameters α and β of the inverse Weibull distribution (IWD) have been studied based on Balakrishnan’s unified hybrid censoring scheme (UHCS), see Balakrishnan et al. In point estimation, the maximum likelihood (ML) and Bayes (B) methods have been used. The Bayes estimates have been computed based on squared error loss (SEL) function and Linex loss function and using Markov Chain Monte Carlo (MCMC) algorithm. In interval estimation, a (1 ? τ) × 100% approximate, bootstrap-p, credible and highest posterior density (HPD) confidence intervals (CI′s) for the parameters α and β have been introduced. Based on Monte Carlo simulation, Bayes estimates have been compared with their corresponding maximum likelihood estimates by computing the mean squared errors (MSE′s) of all estimators. Finally, point and interval estimations of all parameters have been studied based on a real data set as an illustrative example.     

Estimation using hybrid censored data from a generalized inverted exponential distribution

ABSTRACT

We consider point and interval estimation of the unknown parameters of a generalized inverted exponential distribution in the presence of hybrid censoring. The maximum likelihood estimates are obtained using EM algorithm. We then compute Fisher information matrix using the missing value principle. Bayes estimates are derived under squared error and general entropy loss functions. Furthermore, approximate Bayes estimates are obtained using Tierney and Kadane method as well as using importance sampling approach. Asymptotic and highest posterior density intervals are also constructed. Proposed estimates are compared numerically using Monte Carlo simulations and a real data set is analyzed for illustrative purposes.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Estimation for Birnbaum–Saunders Distribution in Simple Step Stress–accelerated Life Test with Type-II Censoring

This paper develops the Bayesian estimation for the Birnbaum–Saunders distribution based on Type-II censoring in the simple step stress–accelerated life test with power law accelerated form. Maximum likelihood estimates are obtained and Gibbs sampling procedure is used to get the Bayesian estimates for shape parameter of Birnbaum–Saunders distribution and parameters of power law–accelerated model. Asymptotic normality method and Markov Chain Monte Carlo method are employed to construct the corresponding confidence interval and highest posterior density interval at different confidence level, respectively. At last, the results are compared by using Monte Carlo simulations, and a numerical example is analyzed for illustration.     

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.     

Estimation of the exponential Pareto II distribution parameters

In this article, we estimate the parameters of exponential Pareto II distribution by two new methods. The first one is based on the principle of maximum entropy (POME) and the second is by Kullback–Leibler divergence of survival function (KLS). Monte Carlo simulated data are used to evaluate these methods and compare them with the maximum likelihood method. Finally, we fit this distribution to a set of real data by estimation procedures.     

Estimation from Burr type XII distribution using progressive first-failure censored data

In this paper, a new life test plan called a progressively first-failure-censoring scheme introduced by Wu and Ku? [On estimation based on progressive first-failure-censored sampling, Comput. Statist. Data Anal. 53(10) (2009), pp. 3659–3670] is considered. Based on this type of censoring, the maximum likelihood (ML) and Bayes estimates for some survival time parameters namely reliability and hazard functions, as well as the parameters of the Burr-XII distribution are obtained. The Bayes estimators relative to both the symmetric and asymmetric loss functions are discussed. We use the conjugate prior for the one-shape parameter and discrete prior for the other parameter. Exact and approximate confidence intervals with the exact confidence region for the two-shape parameters are derived. A numerical example using the real data set is provided to illustrate the proposed estimation methods developed here. The ML and the different Bayes estimates are compared via a Monte Carlo simulation study.     

Estimation of P(Y < X) for progressively first-failure-censored generalized inverted exponential distribution

In this article, we consider the problem of estimation of the stress–strength parameter δ?=?P(Y?<?X) based on progressively first-failure-censored samples, when X and Y both follow two-parameter generalized inverted exponential distribution with different and unknown shape and scale parameters. The maximum likelihood estimator of δ and its asymptotic confidence interval based on observed Fisher information are constructed. Two parametric bootstrap boot-p and boot-t confidence intervals are proposed. We also apply Markov Chain Monte Carlo techniques to carry out Bayes estimation procedures. Bayes estimate under squared error loss function and the HPD credible interval of δ are obtained using informative and non-informative priors. A Monte Carlo simulation study is carried out for comparing the proposed methods of estimation. Finally, the methods developed are illustrated with a couple of real data examples.     

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.     

Bayesian estimation and prediction intervals for a Rayleigh distribution under a conjugate prior

This paper is an effort to obtain Bayes estimators of Rayleigh parameter and its associated risk based on a conjugate prior (square root inverted gamma prior) with respect to both symmetric loss function (squared error loss), and asymmetric loss function (precautionary loss function). We also derive the highest posterior density (HPD) interval for the Rayleigh parameter as well as the HPD prediction intervals for a future observation from this distribution. An illustrative example to test how the Rayleigh distribution fits a real data set is presented. Finally, Monte Carlo simulations are performed to compare the performances of the Bayes estimates under different conditions.     

Point and Interval Estimation for Bivariate Normal Distribution Based on Progressively Type-II Censored Data

ABSTRACT

The maximum likelihood estimates (MLEs) of parameters of a bivariate normal distribution are derived based on progressively Type-II censored data. The asymptotic variances and covariances of the MLEs are derived from the Fisher information matrix. Using the asymptotic normality of MLEs and the asymptotic variances and covariances derived from the Fisher information matrix, interval estimation of the parameters is discussed and the probability coverages of the 90% and 95% confidence intervals for all the parameters are then evaluated by means of Monte Carlo simulations. To improve the probability coverages of the confidence intervals, especially for the correlation coefficient, sample-based Monte Carlo percentage points are determined and the probability coverages of the 90% and 95% confidence intervals obtained using these percentage points are evaluated and shown to be quite satisfactory. Finally, an illustrative example is presented.     

Estimation for the three-parameter inverse Gaussian distribution under progressive Type-II censoring

Inverse Gaussian distribution has been used widely as a model in analysing lifetime data. In this regard, estimation of parameters of two-parameter (IG2) and three-parameter inverse Gaussian (IG3) distributions based on complete and censored samples has been discussed in the literature. In this paper, we develop estimation methods based on progressively Type-II censored samples from IG3 distribution. In particular, we use the EM-algorithm, as well as some other numerical methods for determining the maximum-likelihood estimates (MLEs) of the parameters. The asymptotic variances and covariances of the MLEs from the EM-algorithm are derived by using the missing information principle. We also consider some simplified alternative estimators. The inferential methods developed are then illustrated with some numerical examples. We also discuss the interval estimation of the parameters based on the large-sample theory and examine the true coverage probabilities of these confidence intervals in case of small samples by means of Monte Carlo simulations.     

Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.