Sampling Plans For Exponential Distribution Research Articles

Designing efficient Bayesian sampling plans for exponential distributions based on samples under a (n, t) simple step-stress test

This article studies a method about how to design an efficient Bayesian sampling plan based on samples collected through a (n, t)-sampling scheme with a simple step-stress accelerated life test (ALT). For brevity, such a step-stress accelerated life test is called (n, t)-SSALT. A Bayesian sampling plan (BSP) derived through the (n, t)-SSALT is called BSPA. First, we derive the BSPA with data collected through a (n, t)-SSALT for a general loss function. Given gamma and Jeffreys prior distributions, an explicit expression of the Bayes decision function under a certain loss function is derived. By applying a curtailment procedure to the preceding Bayes decision function, an on-line new Bayes decision function and Bayesian sampling plan, called an efficient Bayesian sampling plan (EBSPA), are constructed. It is shown that the Bayes risk of EBSPA is less than or equal to that of BSPA. This indicates that the EBSPA is more efficient than the BSPA. Comparisons among some BSPAs and the proposed EBSPA are given. The numerical results indicate that, in terms of Bayes risks, the EBSPA significantly outperforms the BSPA.

Optimal curtailed Bayesian sampling plans for exponential distributions with Type-I hybrid censored samples

This paper studies a method for constructing a curtailed Bayesian sampling plan with Type-I hybrid censored samples. Based on the Bayesian sampling plan (BSP) of Liang and Yang (2013), a curtailed Bayesian sampling plan (CBSP) is constructed through an online curtailed censoring process. We show that the resulting associated curtailed Bayes decision function is equivalent to the Bayes decision function of Liang and Yang (2013). It is shown that the risk of CBSP is always less than or equal to that of BSP. Monte Carlo simulations are carried out to illustrate the performance between the two sampling plans BSP and CBSP. Numerical results illustrate that the risk of CBSP has better performance than that of BSP, and hence the CBSP is recommended.

Median ranked acceptance sampling plans for exponential distribution

We develop a ranked acceptance sampling plan by attribute for exponential distribution assuming that the life test is truncated at a pre-assigned time. Two main requirements are essential for the proposed ranked sampling plans; namely; the life times of the test units are assumed to follow the exponential distribution; and the data are selected by using a free cost sampling method, the median ranked set sampling scheme from a large lot. The main advantage of using the median rank set sampling is to reduce producer’s risk, and it is one of the ranked scheme that produces a judgment order statistics that are mutually independent and identically distributed random variables to meet the binomial theory assumptions. The distribution function characterization under the median ranked set sampling scheme is derived assuming that the set size is known; then the minimum sample size necessary to ensure the specified average life are obtained and the operating characteristic values of the sampling plans based on the ranked samples and producer’s risk are presented. A comparisons with sampling plan based on simple random sampling and an illustrative example are given.

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

ABSTRACTFor 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.

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

ABSTRACTThis 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.

Design of Optimal Reliability Acceptance Sampling Plans for Exponential Distribution

Abstract In this paper, an attempt is made to derive the most efficient economic reliability sampling plans for accepting a lot containing identical units having exponentially distributed lifetime with parameter θ. We consider two types of sampling plans, namely, (a) sequential sampling plan ( t 1 , t 2 ) ${(t_1,t_2)}$ and (b) repetitive group sampling plan ( n , t 1 , t 2 ) ${(n,t_1,t_2)}$ . Under plan (a), the lot is rejected when the time between successive failures ( Y r ) ${(Y_r)}$ is less than t 1, and accepted when Y r ≥ t 2 ${Y_r\ge t_2}$ . The testing will continue for t 1 ≤ Y r < t 2 ${t_1\le Y_r< t_2}$ . Also we formulate an optimization problem that minimizes the total expected testing cost. Under plan (b), four different criteria are used to derive the sampling plan. The optimization problem formulated under each criterion is solved using a genetic algorithm to obtain the plan parameters ( n , t 1 , t 2 ) ${(n,t_1,t_2)}$ . Several numerical examples are discussed to illustrate our plans. In addition, a real example is also considered to demonstrate our plan. Finally, we compare the cost of our plan with that of an existing plan in the literature. Our plan has significant potential to reduce the testing cost by about 50%.

Bayesian sampling plans for exponential distributions with interval censored samples

ABSTRACTThis article studies the problem of designing Bayesian sampling plans (BSP) with interval censored samples. First, an algorithm for deriving the conventional BSP is proposed. The BSP is shown to possess some monotonicity. Based on the BSP and using the property of monotonicity, a new sampling plan modified by the curtailment procedure is proposed. The resulting curtailed Bayesian sampling plan (CBSP) can reduce the duration time of life test experiment, and it is optimal in the sense that its associated Bayes risk is smaller than the Bayes risk of the BSP if the cost of the duration time of life test experiment is considered. A numerical example to compute the Bayes risks of BSP and CBSP and related quantities is given. Also, a Monte Carlo simulation study is performed to illustrate the performance of the CBSP compared with the BSP. The simulation results demonstrate that our proposed CBSP has better performance because it has smaller risk. The CBSP is recommended. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 604–616, 2015

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).

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.

Comment on “Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring”

Lin et al. [Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring, J. Stat. Comput. Simul. 81 (2011), pp. 873–882] claimed to have derived exact Bayesian sampling plans for exponential distributions with progressive hybrid censoring. We comment on the accuracy of the design parameters of their proposed sampling plans and the associated Bayes risks given in tables of Lin et al. [Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring, J. Stat. Comput. Simul. 81 (2011), pp. 873–882]. Counter-examples to their claim are provided.

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).

Bayesian sampling plans for exponential distribution based on uniform random censored data

The problem of a single sampling plan with polynomial loss for the exponential distribution based on uniformly distributed random censored data has been considered. A Bayes sampling plan is derived under various schemes of censoring time. It is specially focused on a quadratic loss and an unit time cost is included in the loss. Some optimal Bayes solutions are tabulated and some numerical comparisons between the proposed plan and a known plan under special loss are also made. It is shown that the optimal solutions of the known plan are not Bayes in general.

Bayesian Sampling Plans for Exponential Distribution Based on Type I Censoring Data

We study variable sampling plans for the exponential distribution based on type I censoring data. Using a suitable loss function, a Bayesian variable sampling plan (nB, tB, δB) is derived. For certain prior distributions and loss functions, the numerical values of the Bayesian sampling plans and the associated minimum Bayes risks are tabulated. In terms of Bayes risks, comparisons between the proposed Bayesian sampling plans (nB, tB, δB) and the “Bayesian” variable sampling plans (n0, t0, δLT0) of Lam (1994, Ann. Statist., 22, 696–711) have been made. The numerical results indicate that under the same conditions, the proposed Bayesian sampling plan is superior to that of Lam in the sense that the Bayes risk of (nB, tB, δB) is less than that of (n0, t0, δLT0).

Accelerated life test sampling plans for exponential distributions

Simplified formulae are given for computing type I and type II error probabilities in a two stress level accelerated lif test plan developed by Yum and Kim(1990)Using the new formulae, some acceptance sampling plans for the mean value of anexponential distribution, based on type II censored data, are obtained. The criterion for choosing the censoring numbers, r1and rh,corresponding to a pair of higher-than-normal-use stress levels, is to minimize the sum, r1+rh.An approximate procedure is also included

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.