Optimal Bayesian sampling plans for exponential distributions based on hybrid censored samples

We study variable sampling plans for exponential distributions based on type-I hybrid censored samples. For this problem, two sampling plans based on the non-failure sample proportion and the conditional maximum likelihood estimator are proposed by Chen et al. [J. Chen, W. Chou, H. Wu, and H. Zhou, Designing acceptance sampling schemes for life testing with mixed censoring, Naval Res. Logist. 51 (2004), pp. 597–612] and Lin et al. [C.-T. Lin, Y.-L. Huang, and N. Balakrishnan, Exact Bayesian variable sampling plans for the exponential distribution based on type-I and type-II censored samples, Commun. Statist. Simul. Comput. 37 (2008), pp. 1101–1116], respectively. From the theoretic decision point of view, the preceding two sampling plans are not optimal due to their decision functions not being the Bayes decision functions. In this article, we consider the decision theoretic approach, and the optimal Bayesian sampling plan based on sufficient statistics is derived under a general loss function. Furthermore, for the conjugate prior distribution, the closed-form formula of the Bayes decision rule can be obtained under either the linear or quadratic decision loss. The resulting Bayesian sampling plan has the minimum Bayes risk, and hence it is better than the sampling plans proposed by Chen et al. (2004) and Lin et al. (2008). Numerical comparisons are given and demonstrate that the performance of the proposed Bayesian sampling plan is superior to that of Chen et al. (2004) and Lin et al. (2008).     

Curtailed Bayesian sampling plans for exponential distributions based on Type-II censored samples

This paper studies the problem of designing a curtailed Bayesian sampling plan (CBSP) with Type-II censored data. We first derive the Bayesian sampling plan (BSP) for exponential distributions based on Type-II censored samples in a general loss function. For the conjugate prior with quadratic loss function, an explicit expression for the Bayes decision function is derived. Using the property of monotonicity of the Bayes decision function, a new Bayesian sampling plan modified by the curtailment procedure, called a CBSP, is proposed. It is shown that the risk of CBSP is less than or equal to that of BSP. Comparisons among some existing BSPs and the proposed CBSP are given. Monte Carlo simulations are conducted, and numerical results indicate that the CBSP outperforms those early existing sampling plans if the time loss is considered in the loss function.     

Efficient Bayesian sampling plans for exponential distributions with random censoring

This paper considers Bayesian sampling plans for exponential distribution with random censoring. The efficient Bayesian sampling plan for a general loss function is derived. This sampling plan possesses the property that it may make decisions prior to the end of the life test experiment, and its decision function is the same as the Bayes decision function which makes decisions based on data collected at the end of the life test experiment. Compared with the optimal Bayesian sampling plan of Chen et al. (2004), the efficient Bayesian sampling plan has the smaller Bayes risk due to the less duration time of life test experiment. Computations of the efficient Bayes risks for the conjugate prior are given. Numerical comparisons between the proposed efficient Bayesian sampling plan and the optimal Bayesian sampling plan of Chen et al. (2004) under two special decision losses, including the quadratic decision loss, are provided. Numerical results also demonstrate that the performance of the proposed efficient sampling plan is superior to that of the optimal sampling plan by Chen et al. (2004).     

Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring

From the exact distribution of the maximum likelihood estimator of the average lifetime based on progressive hybrid exponential censored sample, we derive an explicit expression for the Bayes risk of a sampling plan when a quadratic loss function is used. The simulated annealing algorithm is then used to determine the optimal sampling plan. Some optimal Bayes solutions under progressive hybrid and ordinary hybrid censoring schemes are presented to illustrate the effectiveness of the proposed method.     

Bayesian shrinkage estimation based on Rayleigh type-II censored data

In this paper, we construct a Bayes shrinkage estimator for the Rayleigh scale parameter based on censored data under the squared log error loss function. Risk-unbiased estimator is derived and its risk is computed. A Bayes shrinkage estimator is obtained when a prior point guess value is available for the scale parameter. Risk-bias of the Bayes shrinkage estimator is considered. A comparison between the proposed Bayes shrinkage estimator and the risk-unbiased estimator is provided using calculation of the relative efficiency. A numerical example is presented for illustrative and comparative purposes.     

Bayesian inference based on a jointly type-II censored sample from two exponential populations

In this paper, based on a jointly type-II censored sample from two exponential populations, the Bayesian inference for the two unknown parameters are developed with the use of squared-error, linear-exponential and general entropy loss functions. The problem of predicting the future failure times, both point and interval prediction, based on the observed joint type-II censored data, is also addressed from a Bayesian viewpoint. A Monte Carlo simulation study is conducted to compare the Bayesian estimators with the maximum likelihood estimator developed by Balakrishnan and Rasouli [Exact likelihood inference for two exponential populations under joint type-II censoring. Comput Stat Data Anal. 2008;52:2725–2738]. Finally, a numerical example is utilized for the purpose of illustration.     

Progressively censored variables sampling plans for two-parameter exponential distributions

Progressive censoring is quite useful in many practical situations where budget constraints are in place or there is a demand for rapid testing. Balasooriya & Saw (1998) present reliability sampling plans for the two-parameter exponential distribution, based on progressively censored samples. However, the operating characteristic (OC) curve derived in their paper does not depend on the sample size. This seems to be because, in their computations, they forget to consider the proportion of uncensored data, which also has an important influence on the subsequent developments. In consequence, their OC curve is only valid when there is no censoring. In this paper, some modifications are proposed. These are needed to obtain a proper design of the above sampling plan. Whenever at least two uncensored observations are available, the OC curve is derived in closed form and a procedure for determining progressively censored reliability sampling plans is also presented. Finally, the example considered by Balasooriya & Saw is used to illustrate the results developed in this paper for several censoring levels.     

On progressive hybrid censored exponential distribution

In this paper, we introduce a new adaptive Type-I progressive hybrid censoring scheme, which has some advantages over the progressive hybrid censoring schemes already discussed in the literature. Based on an adaptive Type-I progressively hybrid censored sample, we derive the exact distribution of the maximum-likelihood estimator (MLE) of the mean lifetime of an exponential distribution as well as confidence intervals for the failure rate using exact distribution, asymptotic distribution, and three parametric bootstrap resampling methods. Furthermore, we provide computational formula for the expected number of failures and investigate the performance of the point and interval estimation for the failure rate in this case. An alternative simple form for the distribution of the MLE under adaptive Type-II progressive hybrid censoring scheme proposed by Ng et al. [Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme, Naval Res. Logist. 56 (2009), pp. 687–698] is obtained. Finally, from the exact distribution of the MLE, we establish the explicit expression for the Bayes risk of a sampling plan under adaptive Type-II progressive hybrid censoring scheme when a general loss function is used, and present some optimal Bayes solutions under four different progressive hybrid censoring schemes to illustrate the effectiveness of the proposed method.     

Bayesian life test sampling plans for the two parameter exponential distribution

In this paper we consider the determination of Bayesian life test acceptance sampling plans for finite lots when the underlying lifetime distribution is the two parameter exponential. It is assumed that the prior distribution is the natural conjugate prior, that the costs associated with the actions accept and reject are known functions of the lifetimes of the items, and that the cost of testing a sample is proportional to the duration of the test. Type 2 censored sampling is considered where a sample of size n is observed only until the rth failure occurs and the decision of whether to accept or reject the remainder of the lot is made on the basis of the r observed lifetimes. Obtaining the optimal sample size and the optimal censoring number are difficult problems when the location parameter of the distribution is restricted to be non-negative. The case when the positivity restriction on the location parameter is removed has been investigated. An example is provided for illustration.     

On the Bayesian inference of Kumaraswamy distributions based on censored samples

In this article we discuss Bayesian estimation of Kumaraswamy distributions based on three different types of censored samples. We obtain Bayes estimates of the model parameters using two different types of loss functions (LINEX and Quadratic) under each censoring scheme (left censoring, singly type-II censoring, and doubly type-II censoring) using Monte Carlo simulation study with posterior risk plots for each different choices of the model parameters. Also, detailed discussion regarding elicitation of the hyperparameters under the dependent prior setup is discussed. If one of the shape parameters is known then closed form expressions of the Bayes estimates corresponding to posterior risk under both the loss functions are available. To provide the efficacy of the proposed method, a simulation study is conducted and the performance of the estimation is quite interesting. For illustrative purpose, real-life data are considered.     

Acceptance sampling plans based on truncated life tests for weighted exponential distribution

Gupta and Kundu proposed a new class of weighted exponential distributions using the idea of Azzalini. In this article, we develop an acceptance sampling plan for the weighted exponential distribution under a truncated life test. For various acceptance numbers, consumer’s confidence levels and values of the ratio of the experimental time to the specified mean lifetime, the minimum sample size necessary to ensure a certain mean lifetime are obtained. The operating characteristic function values and the associated producer’s risks are also presented. A numerical example is provided to illustrate the acceptance sampling plan.     

Efficient Bayesian sampling plans for exponential distributions with type-I-censored samples

The aim of this paper is to introduce an efficient Bayesian sampling procedure for exponential distribution with type-I censoring. An online inspection method is suggested to reach a Bayes decision prior the termination time of life test. Bayesian sampling plans (BSPs) with quadratic loss function are established to illustrate the use of the proposed method. Some BSPs are tabulated, and the performance of the proposed BSPs is compared with two existing competitive methods. Numerical results indicate that a significant reduction in the experimental time over the conventional BSP can be achieved when the online inspection method is applied.     

Accuracy of approximate progressively censored reliability sampling plans for exponential models

Reliability sampling plans provide an efficient method to determine the acceptability of a product based upon the lifelengths of some test units. Usually, they depend on the producer and consumer’s quality requirements and do not admit closed form solutions. Acceptance sampling plans for one- and two-parameter exponential lifetime models, derived by approximating the operating characteristic curve, are presented in this paper. The accuracy of these approximate plans, which are explicitly expressible and valid for failure and progressive censoring, is assessed. The approximation proposed in the one-parameter case is found to be practically exact. Explicit lower and upper bounds on the smallest sample size are given in the two-parameter case. Some additional advantages are also pointed out.     

Bayesian approach to robust procedures based on censored samples

By adopting; the Bayesian method, we develop in this paper some robust procedures for the one and two-sample location problems based on symmetric Type-II censored samples and by assuming normality for the censored samples. The posterior distributions and the Highest Posterior Density (H.P.D.) intervals are? derived. Finally, we illustrate these procedures by applying the results to Darwin's data and to Brown lee's data     

On selection procedures for positive exponential family distributions based on type-I censored data

We investigate the problem of selecting the best population from positive exponential family distributions based on type-I censored data. A Bayes rule is derived and a monotone property of the Bayes selection rule is obtained. Following that property, we propose an early selection rule. Through this early selection rule, one can terminate the experiment on a few populations early and possibly make the final decision before the censoring time. An example is provided in the final part to illustrate the use of the early selection rule.     

Optimal design of variable acceptance sampling plans for mixture distribution

ABSTRACT

The paper investigates the design of single and sequential variable acceptance sampling plans for a mixture distribution. Mixture distributions are seen in many practical problems such as life testing experiments of electronic components and clinical trials. The sampling plans for this kind of situations are not well addressed in the literature. We first propose a single sampling plan for a distribution which is a mixture of two exponential distributions. An optimization problem which minimizes the total cost of testing at given producer's and consumer's risks is solved to obtain the plan parameters. Two different sequential sampling plans are also defined and plan parameters are obtained by solving corresponding optimization problems. Finally, a case study, a simulation study and a sensitivity analysis are presented to illustrate our sampling plans.     

An objective Bayesian analysis of the two-parameter half-logistic distribution based on progressively type-II censored samples

This paper develops an objective Bayesian analysis method for estimating unknown parameters of the half-logistic distribution when a sample is available from the progressively Type-II censoring scheme. Noninformative priors such as Jeffreys and reference priors are derived. In addition, derived priors are checked to determine whether they satisfy probability-matching criteria. The Metropolis–Hasting algorithm is applied to generate Markov chain Monte Carlo samples from these posterior density functions because marginal posterior density functions of each parameter cannot be expressed in an explicit form. Monte Carlo simulations are conducted to investigate frequentist properties of estimated models under noninformative priors. For illustration purposes, a real data set is presented, and the quality of models under noninformative priors is evaluated through posterior predictive checking.     

Bayesian multiple change-point estimation for exponential distribution with truncated and censored data

This paper considers the multiple change-point estimation for exponential distribution with truncated and censored data by Gibbs sampling. After all the missing data of interest is filled in by some sampling methods such as rejection sampling method, the complete-data likelihood function is obtained. The full conditional distributions of all parameters are discussed. The means of Gibbs samples are taken as Bayesian estimations of the parameters. The implementation steps of Gibbs sampling are introduced in detail. Finally random simulation test is developed, and the results show that Bayesian estimations are fairly accurate.     

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.     

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.