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On designing an acceptance sampling plan for the Pareto lifetime model
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  This article was downloaded by: [Zeinab Amin]On: 16 October 2014, At: 03:43Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Statistical Computation andSimulation Publication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gscs20 On designing an acceptance samplingplan for the Pareto lifetime model Zeinab Amin a   b  & Maram Salem ba  Department of Mathematics and Actuarial Science , TheAmerican University in Cairo , PO Box 74, New Cairo , 11835 ,Egypt b  Faculty of Economics and Political Science, Department of Statistics , Cairo University , Giza , EgyptPublished online: 27 Jun 2011. To cite this article:  Zeinab Amin & Maram Salem (2012) On designing an acceptance sampling planfor the Pareto lifetime model, Journal of Statistical Computation and Simulation, 82:8, 1115-1133,DOI: 10.1080/00949655.2011.570342 To link to this article: http://dx.doi.org/10.1080/00949655.2011.570342 PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &   Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions    D  o  w  n   l  o  a   d  e   d   b  y   [   Z  e   i  n  a   b   A  m   i  n   ]  a   t   0   3  :   4   3   1   6   O  c   t  o   b  e  r   2   0   1   4   Journal of Statistical Computation and Simulation Vol. 82, No. 8,August 2012, 1115–1133 On designing an acceptance sampling plan for the Paretolifetime model ZeinabAmin a , b * and Maram Salem b a  Department of Mathematics and Actuarial Science, The American University in Cairo, PO Box 74, New Cairo 11835, Egypt;  b Faculty of Economics and Political Science, Department of Statistics,Cairo University, Giza, Egypt  (  Received 5 April 2010; final version received 7 March 2011 )In this paper, the design of reliability sampling plans for the Pareto lifetime model under progressiveType-II right censoring is considered. Sampling plans are derived using the decision theoretic approachwith a suitable loss or cost function that consists of sampling cost, rejection cost, and acceptance cost.The decision rule is based on the estimated reliability function. Plans are constructed within the Bayesiancontext using the natural conjugate prior. Simulations for evaluating the Bayes risk are carried out and theoptimal sampling plans are reported for various sample sizes, observed number of failures and removalprobabilities. Keywords:  Bayes risk; cost function; discretization; natural conjugate prior; Pareto distribution; progres-sive censoring; random removals; reliability sampling plans 1. Introduction Acceptance sampling is the procedure adopted by taking samples from batches or lots of someindustrial products, testing or inspecting these samples, and on the basis of these results decidingwhether the whole batch or lot is satisfactory, and hence accepting or rejecting this lot. Designinga sampling plan involves determining the sample size and the acceptance criteria based on whichthe lot is accepted or rejected. Acceptance sampling procedures are most useful in the case of destructive tests or when inspection is quite costly.Variablessamplingplansundertheassumptionofnormalityhavebeenextensivelydiscussedinthe literature, see, for example, [1]. Kochelakota and Balakrishnan [2] developed robust samplingplans using some modification of the maximum-likelihood estimators for the location and scaleparametersofthenormaldistribution.MoskowtizandTang[3]consideredtheproblemofaknown-standard-deviationBayesianvariablesacceptancesamplingplanforthenormaldistributionbasedon the decision-theoretic approach. *Corresponding author. Email: zeinabha@aucegypt.edu ISSN 0094-9655 print / ISSN 1563-5163 online© 2012 Taylor & Francishttp: // dx.doi.org / 10.1080 / 00949655.2011.570342http: // www.tandfonline.com    D  o  w  n   l  o  a   d  e   d   b  y   [   Z  e   i  n  a   b   A  m   i  n   ]  a   t   0   3  :   4   3   1   6   O  c   t  o   b  e  r   2   0   1   4  1116  Z. Amin and M. Salem In reliability studies distributions with positive support are often used. Gupta and Groll [4]considered the problem of acceptance sampling plans for the gamma distribution under Type-Icensoring.FertigandMann[5]developedaType-IIcensoredsamplingplanforthetwo-parameterWeibull distribution. Schneider [6] designed Type-II censored variables sampling plans for thelog-normal and Weibull distributions, assuming large samples and a lower specification limit.Lam [7] considered a Bayesian Type-II censored one-sided variables sampling plan for the one-andtwo-parameterexponentialdistributions.SimilarplanshavebeendevelopedbyLam[8]fortheexponential distribution underType-I censoring. Chen and Lam [9] developed Bayesian variablessampling plans for the Weibull distribution under Type-I censoring with a lower specificationlimit. Similar plans were derived for the Weibull distribution by Chen  et al . [10] under Type-IIcensoring.In addition to the conventional Types-I and II censoring plans, reliability sampling plans werederived, for a number of distributions, under progressive censoring. Balasooriya and Balakrish-nan [11] constructed reliability sampling plans for the log-normal distribution under progressiveType-II censoring, where the number of units removed at each failure is prefixed. Balasooriya et al . [12] discussed the design of reliability sampling plans for the Weibull life time modelunder Type-II progressively censored data. Tse and Yang [13] considered the design of relia-bility sampling plans for the two-parameter exponential distribution under progressive Type-IIcensoring with random removals, where the number of units removed at each failure follows abinomial distribution. Fernandez [14] discussed the design of reliability sampling plans for thetwo-parameter exponential distribution under progressive Type-II censoring with pre-assignednumber of removals. Huang and Wu [15] designed reliability sampling plans for the exponen-tial distribution under progressive Type-I interval censoring. Kundu [16] considered inferenceand life-testing plans for the Weibull distribution under progressive Type-II censoring with pre-assigned number of removals. Approximate Bayes estimators of the Weibull parameters and thecorresponding credible intervals were also obtained.In this paper, we design reliability sampling plans for the Pareto lifetime model under progres-sive Type-II right censoring within the Bayesian context. The rest of this article is organized asfollows:An overview of the Pareto lifetime model and the progressive censoring scheme is givenin Section 2. In Section 3, the proposed decision-theoretic sampling scheme is discussed. TheproposeddatagenerationtechniquesarediscussedinSection4.Discretizationofthedistributionsof the shape and scale parameters is discussed in Section 5. Graphical displays of the resultingdensity estimates are illustrated in Section 6. Evaluation of the Bayes risk and derivation of theoptimal reliability sampling plans are developed in Section 7. 2. The model Consider data from a classical Pareto distribution with shape parameter  α  and scale parameter  σ  .The corresponding density is f(x ; α,σ) = ασ  α x − (α + 1 ) x > σ, α >  0 , σ >  0 .  (1)The reliability or survival function,  S(x) , assumes the form S(x | α,σ) = 1 − F(x | α,σ) =  σ x  α ,  0  < σ   ≤ x, α >  0 .    D  o  w  n   l  o  a   d  e   d   b  y   [   Z  e   i  n  a   b   A  m   i  n   ]  a   t   0   3  :   4   3   1   6   O  c   t  o   b  e  r   2   0   1   4   Journal of Statistical Computation and Simulation  1117 The natural conjugate prior, called the Power Gamma or modified Lwin prior denoted byPG (v,λ,µ,β) ,proposedbyArnoldandPress[17]willbeused.Thispriorisdescribedasfollows: g(α,σ)  = λ(v)( ln µ  −  λ ln β) v σ  λα − 1 α v µ − α ,α >  0 ,  0  < σ < β,  0  < β λ < µ, v,λ,µ, β >  0 .  (2)ThePowerGammapriorisanaturalconjugatepriorwhichspecifiesaGammaprior,Ga (v, ln µ  − λ ln β) , and a power function prior, PF (λα,β) , for  α  and  σ  | α , respectively. These priors aredescribed as follows: g(α)  = ( ln µ  −  λ ln β) v (v)α v − 1 exp {− α( ln µ  −  λ ln β) }  α >  0 (3)and g(σ  | α)  =  λασ  λα − 1 β − λα 0  < σ < β.  (4)In this paper, we consider progressiveType-II censoring, where n units are initially placed on testandthetestterminatesafter m < n unitshavefailed.Immediatelyafterthe i thfailure, r i  survivingunits are removed from the test at random, where i  =  1 , 2 ,...,m  −  1.This process continues tillthe m thfailure,wherealltheremaining r m  =  n  −  m − 1 i = 1  r i  −  m unitsarecensored. Foradetaileddescription of this sampling scheme, importance and applications, see, for example [18]. In thispaper, we assume that the number of units removed at each failure is a binomial random variable.Hence, the number of units removed at the  i th failure,  R i , assume the following distributions: R 1  ∼  Bin (n  −  m,p),  (5)while R i | r 1 ,r 2 ,...,r i − 1  ∼  Bin  n  −  m  − i − 1  j  = 1 r j  ,p  , i  =  2 , 3 ,...,m  −  1 .  (6)Now, suppose that  n  units are placed on a life-test and  X   =  (X ( 1 ) ,X ( 2 ) ,...,X (m) )  denote thefirst observed  m  failure times, where  X 1 ,X 2 ,...,X n  are independent and identically distributedrandom variables having the common probability density function given in Equation (1). Underprogressive Type-II censoring with the censoring scheme  R  =  (R 1 ,R 2 ,...,R m ) , where  R i  isindependentof  X (i) ,for i  =  1 , 2 ,...,m ,andbyapplyingthePowerGammapriorinEquation(2)the resulting posterior density satisfies g(α,σ  |  x ,  r )  =  PG  m  +  v,m  +  λ  + m  i = 1 r i ,µ m  i = 1 x r i + 1 (i)  ,w  ,  (7)where  w  =  min (x ( 1 ) ,β) .The posterior given in Equation (7) specifies g(α |  x ,  r )  =  Ga  m  +  v, ln µ  + m  i = 1 (r i  +  1 ) ln X (i)  −  (n  +  λ) ln w  . and g(σ  | α,  x ,  r )  =  PF [ (n  +  λ)α,w ] .    D  o  w  n   l  o  a   d  e   d   b  y   [   Z  e   i  n  a   b   A  m   i  n   ]  a   t   0   3  :   4   3   1   6   O  c   t  o   b  e  r   2   0   1   4
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