Bayesian inference for the Pareto lifetime model under progressive censoring with binomial removals

This paper considers the estimation and prediction problems when lifetimes are Pareto-distributed and are collected under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a Binomial distribution. The analysis is carried out within the Bayesian context.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

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 inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.     

Optimal designing of skip-lot sampling plan of type SkSP-2 with group acceptance sampling plan as reference plan under Burr-type XII distribution

In this paper, skip-lot sampling plan of type SkSP-2 with group acceptance sampling plans is proposed when the lifetime of the product follows the Burr-type XII distributions. The optimal parameters of the proposed plan are determined when two points on operating characteristics curve namely acceptable quality level and limiting quality level and the number of testers are specified. We also considered the Burr-type XII distribution to find the plan parameters. Several tables are given for practical use. We compare the results of proposed plan with the existing plans. Results are explained using real-world examples.     

Reliability acceptance sampling plans for the Weibull distribution under accelerated Type-I censoring

Type-I censored reliability acceptance sampling plans (RASPs) are developed for the Weibull lifetime distribution with unknown shape and scale parameters such that the producer and consumer risks are satisfied. It is assumed that the life test is conducted at an accelerated condition for which the acceleration factor (AF) is known, and each item is continuously monitored for failure. Sensitivity analyses are also conducted to assess the effect of the uncertainty in the assumed AF on the actual producer and consumer risks, and a method is developed for constructing RASPs that can accommodate the uncertainty in AF.     

A variables repetitive group sampling plan under failure-censored reliability tests for Weibull distribution

We propose a variables repetitive group sampling plan under type-II or failure-censored life testing when the lifetime of a part follows a Weibull distribution with a known shape parameter. The acceptance criteria do not involve unknown scale parameter differently from the existing plans. To determine the design parameters of the proposed plan, the usual approach of using two points on the operating characteristic curve is adopted and an optimization problem is formulated so as to minimize the average number of failures observed. Tables for design parameters are constructed when the quality of parts is represented by the unreliability or the ratio of the mean lifetime to the specified life. It is found that the proposed sampling plan can reduce the sample size significantly than do the single sampling plan.     

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w ₁ is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.     

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).     

A double acceptance sampling plan for generalized log-logistic distributions with known shape parameters

A double acceptance sampling plan for the truncated life test is developed assuming that the lifetime of a product follows a generalized log-logistic distribution with known shape parameters. The zero and one failure scheme is mainly considered, where the lot is accepted if no failures are observed from the first sample and it is rejected if two or more failures occur. When there is one failure from the first sample, the second sample is drawn and tested for the same duration as the first sample. The minimum sample sizes of the first and second samples are determined to ensure that the true median life is longer than the given life at the specified consumer’s confidence level. The operating characteristics are analyzed according to various ratios of the true median life to the specified life. The minimum such ratios are also obtained so as to lower the producer’s risk at the specified level. The results are explained with examples.     

A new policy for designing acceptance sampling plan based on Bayesian inference in the presence of inspection errors

The purpose of this article is to present a new policy for designing an acceptance sampling plan based on the minimum proportion of the lot that should be inspected in the presence of inspection errors. It is assumed that inspection is not perfect and every defective item cannot be detected with complete certainty. The Bayesian method is used for obtaining the probability distribution function of the number of defective items in the lot. To design this model, two constraints of producer risk and consumer risk are considered during the inspection process by using two specified points on operating characteristic curve. In order to illustrate the application of the proposed model, an example is presented. In addition, a sensitivity analysis is performed to analyze the model performance under different scenarios of process parameters and the results are elaborated. Finally, the efficiency of the proposed model is compared with the sampling method of Spencer and Kevan de Lopez (2017) at the same conditions.     

Optimum reliability acceptance sampling plans under progressive type-I interval censoring with random removal using a cost model

A reliability acceptance sampling plan (RASP) is a variable sampling plan, which is used for lot sentencing based on the lifetime of the product under consideration. If a good lot is rejected then there is a loss of sales, whereas if a bad lot is accepted then the post sale cost increases and the brand image of the product is affected. Since cost is an important decision-making factor, adopting an economically optimal RASP is indispensable. This work considers the determination of an asymptotically optimum RASP under progressive type-I interval censoring scheme with random removal (PICR-I). We formulate a decision model for lot sentencing and a cost function is proposed that quantifies the losses. The cost function includes the cost of conducting the life test and warranty cost when the lot is accepted, and the cost of batch disposition when it is rejected. The asymptotically optimal RASP is obtained by minimizing the Bayes risk in a set of decision rules based on the maximum likelihood estimator of the mean lifetime of the items in the lot. For numerical illustration, we consider that lifetimes follow exponential or Weibull distributions.     

Double acceptance sampling plan based on truncated life tests for half exponential power distribution

In this paper, we develop a double acceptance sampling plan for half exponential power distribution when the lifetime experiment is truncated at a prefixed time. The zero and one failure schemes are considered. We obtain the minimum sample sizes of the first and second samples necessary to ensure the specified mean life at the given consumer’s confidence level. The operating characteristic values and the minimum ratios of the mean life to the specified life are also analyzed. Numerical example is provided to illustrate the double acceptance sampling plan.     

A procedure for selection of a gamma-Poisson single sampling plan by attributes

Design and evaluation of sampling plans by attributes and by variables are important aspects in the area of acceptance sampling research. Various procedures for the selection of conventional single sampling by attributes have been developed and are available in the literature. This paper presents a design methodology and tables for the selection of parameters of single sampling plans for specified requirements (strengths) under the conditions of a gamma prior and Poisson sampling distribution. The relative efficiency of gamma-Poisson single sampling plans over conventional plans is discussed through empirical illustrations.     

Optimum constant-stress life test plans for Pareto distribution under type-I censoring

The paper considers the case of constant-stress partially accelerated life testing (CSPALT) when two stress levels are involved under type-I censoring. The lifetimes of test items are assumed to follow a two-parameter Pareto lifetime distribution. Maximum-likelihood method is used to estimate the parameters of CSPALT model. Confidence intervals for the model parameters are constructed. Optimum CSPALT plans that determine the best choice of the proportion of test units allocated to each stress are developed. Such optimum test plans minimize the generalized asymptotic variance of the maximum-likelihood estimators of the model parameters. For illustration, Monte Carlo simulation studies are presented.     

Bayesian analysis of the linear regression model with an edgeworth series prior distribution

The purpose of the present investigation 1s to observe the effect of departure from normahty of the prior distribution of regresslon parameters on the Bayman analysis of a h e a r regresslon model Assuming an Edgeworth serles prior distribution for the regresslon coefficients and gamma prior for the disturbances precision, the expressions for the posterlor distribution, posterlor mean and Bayes risk under a quadratic loss function are obtalned The results of a numerical evaluation are also analyzed     

Bayesian inference for the randomly censored Weibull distribution

In this paper, we consider the Bayesian inference of the unknown parameters of the randomly censored Weibull distribution. A joint conjugate prior on the model parameters does not exist; we assume that the parameters have independent gamma priors. Since closed-form expressions for the Bayes estimators cannot be obtained, we use Lindley's approximation, importance sampling and Gibbs sampling techniques to obtain the approximate Bayes estimates and the corresponding credible intervals. A simulation study is performed to observe the behaviour of the proposed estimators. A real data analysis is presented for illustrative purposes.     

Double and group acceptance sampling plan for truncated life test based on inverse log-logistic distribution

This paper introduces a double and group acceptance sampling plans based on time truncated lifetimes when the lifetime of an item follows the inverse log-logistic (ILL) distribution with known shape parameter. The operating characteristic function and average sample number (ASN) values of the double acceptance sampling inspection plan are provided. The values of the minimum number of groups and operating characteristic function for various quality levels are obtained for a group acceptance sampling inspection plan. A comparative study between single acceptance sampling inspection plan and double acceptance sampling inspection plan is carried out in terms of sample size. One simulated example and four real-life examples are discussed to show the applicability of the proposed double and group acceptance sampling inspection plans for ILL distributed quality parameters.     

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).