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.     

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.     

Quantile estimation for a progressively censored exponential distribution

Abstract

In this paper, we consider the problem of estimating the quantile of a two-parameter exponential distribution with respect to an arbitrary strictly convex loss function under progressive type II censoring. Inadmissibility of the best affine equivariant (BAE) estimator is established through a conditional risk analysis. In particular we provide dominance results for quadratic, linex and absolute value loss functions. Further, a class of dominating estimators is derived using the IERD (integral expression of risk difference) approach of Kubokawa (1994 Kubokawa, T. 1994. A unified approach to improving equivariant estimators. The Annals of Statistics 22 (1):290–9. doi:10.1214/aos/1176325369.[Crossref], [Web of Science ®] , [Google Scholar]). In sequel the generalized Bayes estimator is shown to improve the BAE estimator.     

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).     

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.     

ASN-minimax double sampling plans for variables

X be a continuous quality characteristic, with one-sided lower specification limit, having either normal distribution with known σ or exponential distribution. We report on an algorithm allowing the calculation of the so-called ASN-Minimax plan. This plan has minimal maximal average sample size among all double sampling plans for variables that obey the classical two-points-condition on the operating characteristic. We give examples and tables of the ASN-Minimax plans in the normal as well as in the exponential case. Received: March 20, 2000; revised version: January 22, 2001     

Optimal Bayesian variable sampling plans for exponential distributions based on modified type-II hybrid censored samples

For the conventional type-II hybrid censoring scheme (HCS) in Childs et al., a Bayesian variable sampling plan among the class of the maximum likelihood estimators was derived by Lin et al. under the loss function, which does not include the cost of experimental time. Instead of taking the conventional type-II hybrid censoring scheme, a persuasive argument leads to taking the modified type-II hybrid censoring scheme (MHCS) if the cost of experimental time is included in the loss function. In this article, we apply the decision-theoretic approach for the concerned acceptance sampling. With the type-II MHCS, based on a sufficient statistics, the optimal Bayesian sampling plan 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 the quadratic decision loss. Numerical study is given to demonstrate the performance of the proposed Bayesian sampling plan.     

The role of acceptance sampling in modern quality control

Acceptance sampling procedures can be employedina program of accept ance control to achievebetterqualityat lowercost. An overview of the application of moder nacceptance samplingplans, schemes, andsys tems is described with emphas is on the iruse in facilitating process control. Sampling procedures must continually bemade to matchexist in gconditions, and to self-destruct whennolonger warranted. In this way they can be usedinane voluti on ary manne rina continuing program of acceptance control for quality improvement. Implemen- tation of process control canbefacilitated by on-going application of acceptance control to achieve the full be nefits of statistical quality control.     

Development of life-test sampling plans for exponential distributions based on accelerated life testing

This paper presents a procedure for developing life-test sampling plans for exponential distributions based upon accelerated life testing(ALT). Type II censoring is assumed at each overstress level. The derived test statistic is shown to be a quotient of two independent random variables, each of which is a rational power of a Chi-square random variable. The distribution of the test statistic is characterized by the H-function, which can be numerically evaluated to obtain desired sampling plans.     

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.     

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.     

A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively censored data

This paper proposes an approximation to the distribution of a goodness-of-fit statistic proposed recently by Balakrishnan et al. [Balakrishnan, N., Ng, H.K.T. and Kannan, N., 2002, A test of exponentiality based on spacings for progressively Type-II censored data. In: C. Huber-Carol et al. (Eds.), Goodness-of-Fit Tests and Model Validity (Boston: Birkhäuser), pp. 89–111.] for testing exponentiality based on progressively Type-II right censored data. The moments of this statistic can be easily calculated, but its distribution is not known in an explicit form. We first obtain the exact moments of the statistic using Basu's theorem and then the density approximants based on these exact moments of the statistic, expressed in terms of Laguerre polynomials, are proposed. A comparative study of the proposed approximation to the exact critical values, computed by Balakrishnan and Lin [Balakrishnan, N. and Lin, C.T., 2003, On the distribution of a test for exponentiality based on progressively Type-II right censored spacings. Journal of Statistical Computation and Simulation, 73 (4), 277–283.], is carried out. This reveals that the proposed approximation is very accurate.     

An optimization model for economic design of sampling plans based on conforming run length considering outgoing quality

In this article, a new economical acceptance sampling model is proposed based on Taguchi loss function. The objective function of the model consists of inspection cost, scrap cost, and Taguchi loss function including producer loss and consumer loss. The expected total cost includes the loss for an inspected item plus the loss for an accepted item which has not been inspected. Decision-making is based on conforming run length. It is assumed that the quality characteristics follow normal distribution. A numerical example is solved for illustrating application of this model. Sensitivity analysis is proposed for illustrating the effect of some important parameters on the objective function. Finally, we compared the results of the proposed method with classical Dodge–Romig sampling plans tables based on average outgoing quality limit. The results confirmed the superiority of proposed model.     

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).     

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.     

Optimal exact design of double acceptance sampling plans by attributes

The design of double acceptance sampling (AS) plans for attributes based on the operating characteristic curve paradigm is usually addressed by enumeration algorithms. These AS plans may be non optimal regarding the sample size to inspect as they were obtained without the requirement that the constraints at the OC curve controlled points are not violated for minimum Average Sample Number (ASN) scenarios. An approach based on mathematical programming is proposed to systematically design double AS plans for attributes, where the characteristics controlled are modelled by binomial or Poisson distributions. Specifically, Mixed Integer Nonlinear Programming (MINLP) formulations are developed and combined with an enumeration algorithm that allows finding ASN minimax optimal plans. A theoretical result is developed with the purpose of assuring the global optimum design is reached by iteration where a convenient solver is used to find local optima. To validate the algorithm, we compare our results with those of tables commonly used for practical purposes, consider different rates of risk, and setups commonly used in Lot Quality Assurance Plans (LQAS) for health monitoring programmes. Finally, we compare AS plans determined for processes described by binomial and Poisson distributions.     

Time truncated acceptance sampling plans for generalized exponential distribution

Acceptance sampling plans for generalized exponential distribution when the lifetime experiment is truncated at a pre-determined time are provided in this article. The tables are provided for the minimum sample size required to ensure a certain median life of the experimental unit when the shape parameter is two. The operating characteristic function values of the sampling plans and the associated producer's risks are also presented. It is shown that the tables presented here can be used if instead of median life, other percentile life is chosen as the criterion or if the shape parameter is not two. Examples are provided for illustrative purposes.     

A family of shrinkage estimators for Weibull shape parameter in censored sampling

In this paper a new class of shrinkage estimators has been introduced for the shape parameter in an independently identically distributed two-parameterWeibull model under censored sampling. The main idea is to incorporate the prior guessed value by correcting the standard estimator, which is essentially an unbiased estimator, with optimally weighted ratios of the guessed value and the standard estimator, instead of considering a convex combination of the standard estimator and the difference of the guessed value and the standard estimator. The resulting estimator dominates the standard estimator in a surprisingly large neighborhood of the guessed value. The suggested estimator has also been compared with the minimum mean squared error estimator and a class of estimators suggested by Singh and Shukla in IAPQR Trans 25(2), 107–118, 2000. It is found that the suggested class of estimators has lesser bias as well as lesser mean squared error than its competitors subject to certain conditions.      

A note on the equivariant estimation of an exponential scale using progressively censored data

We consider the problem of estimating the scale parameter θ$\theta$ of the shifted exponential distribution with unknown location based on a type II progressively censored sample. Under a large class of bowl-shaped loss functions, a smooth estimator, that dominates the minimum risk equivariant estimator of θ$\theta$, is proposed. A numerical study is performed and shows that the improved estimator yields significant risk reduction over the MRE.