Designing of Acceptance Double Sampling Plan for Life Test Based on Percentiles of Exponentiated Rayleigh Distribution

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1 International Journal of Statistics an Systems ISSN Volume, Number 3 (7), pp Research Inia Publications Designing of Acceptance Double Sampling Plan for Life Test Base on Percentiles of Exponentiate Rayleigh Distribution K.Praeepaveerakumari an P. Ponneeswari Department of Statistics, Bharathiar University, Coimbatore, Tamil Nau, Inia. Abstract In this paper, the attribute characteristic parameter of acceptance ouble sampling plan base on percentiles is obtaine for life testing when the life time of the prouct follows Exponentiate Rayleigh Distribution (ERD). The minimum sample size which is necessary to ensure a specifie life percentile is obtaine at various levels of consumer s risk. The ratio q that ensures the life at specifie percentile is also obtaine which fixes the proucer s risk at.5. Further, the operating characteristic values are prouce an tabulate for various levels of consumer s risk. Keywors: Double sampling plan, percentiles, life test, operating characteristic function, Rayleigh istribution, Exponentiate Rayleigh Distribution. INTRODUCTION An acceptance sampling plan is a sampling proceure with a set of rules for making ecisions about a lot of proucts. The ecision is base on the number of efectives in a sample. Acceptance sampling plans are classifie into two, such as attribute sampling plan an variable sampling plan. Sampling plans are esigne base on the counts of efectives, it is calle attribute sampling plan, on the other han, if the sampling plans are base on sample average an stanar eviation it is calle variable sampling plan. There are ifferent sampling plans such as single sampling plan, ouble sampling plan, multiple sampling plan, continuous sampling plan, skip-

2 476 K.Praeepaveerakumari an P. Ponneeswari lot sampling plan etc., which are available in literature an are evelope for ifferent real life situations. Acceptance ouble sampling plan is use in this paper since the avantage in the sampling plan is that, if a lot is goo or not goo it can be accepte or rejecte respectively with small initial sample. Any lot which cannot be ecie easily can be examine further by means of aitional sample. Therefore in some cases the multiple sampling proceures provie shortcuts to the ecision without jeoparizing the incoming or outgoing quality of material. The sampling inspection plans which are evelope for taking ecision about a lot base on lifetime of the prouct through trials are calle reliability sampling plans. It s important for a proucer to check whether the lifetime of the prouct satisfies the customer s stanar or not. Usually, if the lifetime of the proucts excees the specifie time then the lot is accepte otherwise it is rejecte. However, the limitation of using this metho is the time uration spent on testing. That is when the lifetime of the prouct is assume too long, it might be a time consuming processes to sentence a lot. Hence, it is normal to terminate the experiment by a pre-assigne time t an the numbers of failures are note. If the preetermine mean life reaches a preetermine probability p* then the lot is accepte which will protect the consumer. Thus, the life test is terminate when (c+) th failure is observe at the pre-assigne time t whichever is earlier. Sampling plan base on truncate life tests have been evelope by several authors. Epstein (954) propose truncate life test sampling plan base on exponential istribution. Further, various authors evelope truncate life test sampling plan base on ifferent istributions, such as, Gooe an Kao (96) using Weibull istribution, Gupta an Groll (96) using gamma istribution, Kantam an Rosaiah (998) using half logistic istribution, Kantam et al., () using log-logistic istribution,, Rosaiah an Kantam (5) using inverse Rayleigh istribution, Balakrishnan et al., (7) using generalize Birnbaum-Sauners istribution, Rosaiah et al., (6) using exponentite log-logistic istribution etc. Aitionally Lio et al (9) have consiere acceptance sampling plans from truncate life tests base on the Brinbaum-Sauners istribution for percentiles an they propose that the acceptance sampling plan base on mean may not satisfy the requirement of engineering on the specific percentile of strength or breaking stress. This explains the material strength of proucts is eteriorate significantly an may not meet the consumer s expectations, resulting engineers to pay more attention towars the percentiles of life time than the mean life. Even when the istributions are not symmetric, the percentiles output gives more an clean information regaring the life of the prouct. An when the istribution is symmetric, 5 th percentile or the meian is equivalent to the mean life. Hence, in such cases, the percentile stuy is the generalization of acceptance sampling plans base on mean life of items. In this context, many authors have propose reliability sampling plans base on percentiles

3 Designing of Acceptance Double Sampling Plan for Life Test 477 using various istributions for example Lio et al,. () using percentiles of Burr type XII istribution, Rao an Kantam () using percentiles of log-logistic istribution, Rao (3 a an 3 b) using percentiles of Marshall-Olkin extene Lomax istribution an Marshall-Olkin extene exponential istribution respectively, Rao an Naiu (4) using percentiles of exponentiate half logistic istribution an so on. Further, Praeepa Veerakumari an Ponneeswari (6) esigne single sampling plan base on the percentiles of exponentiate Rayleigh istribution. In this paper, a ouble sampling plan for life testing is evelope when the life of the prouct follows exponentiate Rayleigh istribution. The purpose of proposing this plan through percentiles may satisfy the customer s expectation by rejecting the lot of low percentile, even when the lot is accepte when mean life of lifetime is consiere. Aslam (7), Aslam & Jun () esigne a ouble acceptance sampling plan for generalize log-logistic istributions with known shape parameters. Rao () propose ouble acceptance sampling plans base on truncate life tests for the Marshall-Olkin extene exponential istribution. Aslam et al. () evelope ouble acceptance sampling plans for Burr type XII istribution percentiles uner the truncate life test. ADSP FOR PERCENTILES OF ERD UNDER TRUNCATED LIFE TEST Gupta et al. (998) propose a moel to failure time ata by F*(t) = [F(t)] θ where F(t) is a baseline istribution function an θ is a positive real number which is erive from Lehman alternatives calle exponentiate istribution. Aballah et al (5) states aing a parameter α (a positive real number) to a cumulative istribution function(cf) F by exponentiation prouces a cf of the so calle Exponetiate Distribution(ED). The cf of ED can be written as follows ) [ F( X; )] [ F( )] G( x) G( X; X... () Kunu an Raqab (5) estimate ifferent estimators for generalize RD. The istribution function of RD is given by, F( t, ) e ( t ), t ;/ Hence, the cumulative istribution function of ERD is given by,... () F( t;, ) ( t ) e, t ;/,... (3) where τ an θ are the scale an shape parameters respectively. The first erivative of any cumulative istribution function is its probability ensity function. Hence the probability ensity function of ERD can be written as,

4 478 K.Praeepaveerakumari an P. Ponneeswari f ( t;, ) t t ( t ) F( t,, ) e t ;/, f ( t;, ) ( t ) t ( t ) e e The qth percentile or the q th quantile of any istribution is given by, Pr (T tq) = q... (4)... (5) t q ln( q ) tq an q are irectly proportional. Let, t q / ln( q )... (*) Replacing the scale parameter (τ) by (*), we get the cumulative istribution function of ERD as, Letting t F( t) e t t q ; t,... (6) t q F( t;, ) e ( ) Taking partial erivative with respect to δ, we have ( ) ( ) e e F( t; )... (7)... (8) Assume that a life test is conucte an will be terminate at time t. A probability P* to reject a ba lot is use to protect consumers. A ba lot means that the true qth percentile t q is below the suppose qth percentile t q that is, t q < t q. The lot is confirme as a goo one if the lifetime ata hol the null hypothesis H t q t : q against the alternative H t q t. The consumer's risk P * is use as the : q significance level for this hypothesis testing an P * is the consumer's confience level. The evelopment of DASP (Aslam an Jun, ) with a truncate censoring scheme is propose as follows:

5 Designing of Acceptance Double Sampling Plan for Life Test 479 () Draw the first ranom sample of size n from the lot an put them on test. If c or fewer failures are observe at the pre-etermine time t, the lot is accepte. Otherwise, the life test is truncate to reject the lot before or at t if (c + ) failures are cumulate before or at t, where c < c. () If the observe number of failures () by t is between c + an c (c inclue), then raw a secon sample of size n for life testing till a prescribe termination time t. The lot is accepte if the cumulate number of failures from two samples () is smaller or equal to c. Otherwise, the lot is rejecte. Let us represent the acceptance ouble sampling plan as (n, n, c, c, δ). Here, ni an ci are the sample size an acceptance number associate with the i th sample respectively, i=,. For the propose acceptance ouble sampling plan, the probability of acceptance of lot is given by, L( p) c n p ( p ) n c n p ( p) ( n ) c n c p ( p) n (9) Where, p is the failure probability before the time t, given a specifie qth ( ) / ( ) t, is obtaine from / F( t; ) e e percentile lifetime q p. Where, t / t q an F( t; ) is a non-ecreasing function of δ since F( t; ) from (8). Accoringly, we have F t; ) F( t; ) t q t. The first sample size n is ( q suppose to be the minimum sample size obtaine from single sampling plan hence the secon sample size n is simulate for the evelope sampling plan by satisfying the conition L(p) (-P*). EXAMPLE Assume that the life istribution is an Exponentiate Rayleigh Distribution an the experimenter is intereste in showing that the true unknown th percentile life t. is at least hrs. Let the shape parameter, θ= an the consumer risk is set to 3-(-p*) =.5. It is esire to stop the experiment at time t=hrs. Then for the acceptance numbers c an c as an respectively then from table, the ouble sampling plan (n, n, c, c, t / t. ) = (4, 43,,, ). This explains, the experiment is one up to hrs an the following ecision is mae ) =, the lot is accepte. ) 3, the lot is rejecte an the inspector shoul avice the management to concentrate on the prouction process for better quality proucts.

6 48 K.Praeepaveerakumari an P. Ponneeswari 3) =, the inspector is suggeste to go for secon sample. The operating characteristic curve for the plan obtaine from table is given as t / t. OC It is observe from the above table that if the true th percentile is.75 times the require th percentile ( t / t.75) the proucer s risk is approximately which nears to zero as the true th percentile approaches.5 times the require th percentile. An the OC curve representing the above table is as follows: L(p) Fig : OC curve for c= an c= at p*=.75, δ= base on the th percentile, =. of exponentiate Rayleigh istribution with θ=. The respective. from the table is.6686 which ensures the proucer s risk at.5. This explains that the prouct can have a th percentile life of.6686 times the specifie th percentile. That is we can say that the probability of the prouct to be accepte is at least.95.

7 Designing of Acceptance Double Sampling Plan for Life Test 48 Table : Minimum Sample Sizes an OC values for Acceptance ouble sampling plan (n, n, c, c, δ) when c=, an c= for th percentile of Exponentiate Rayleigh istribution when θ= p* n n t / t. t. / t

8 48 K.Praeepaveerakumari an P. Ponneeswari Table : Gives the ratio. for accepting the lot with the proucer s risk of.5 when θ= t/t p* CONCLUSION In this paper, the acceptance ouble sampling plan base on percentiles of Exponentiate Rayleigh Distribution is esigne for life testing when the life test is truncate for a pre-efine time. This plan will be useful when the life of the prouct follows Exponentiate Rayleigh Distribution. Useful tables are provie an applie for establishment of the evelope plan. REFERENCES [] Abullah. A, Abel Ghaly, Hanan M, Aly an Rana N. S, (5). Different Estimation Methos for constant stress Accelerate life test uner the Family of the Exponentiate Distributions, Quality an Reliability Engineering International. [] Aslam. M an Jun C.H, (). A ouble acceptance sampling plan for generalize log-logistic istributions with known shape parameters, Journal of Applie Statistics, 37: pp [3] Aslam. M, Mahmoo. Y, Lio YL, Tsai TR an Khan MA, (). Double

9 Designing of Acceptance Double Sampling Plan for Life Test 483 acceptance sampling plans for Burr type XII istribution percentiles uner the truncate life test, Journal of the Operational Research Society, 63, pp: - 7. [4] Balakrishnan. N, Leiva. V, an Lopez. J, (7) Acceptance sampling plans from truncate life tests base on the generalize Birnbaum-Sauners istribution, Communications in Statistics: Simulation an Computation, 36, pp: [5] Epstein B, (954). Truncate life tests in the exponential case, Annals of Mathematical Statistics, 5: pp [6] Gooe. H. P an Kao. J. H. K, (96) Sampling plans base on the Weibull istribution, Proceeings of the 7th National Syposium on Reliability an Qualilty Control, pp. 4 4, Philaelphia, Pa, USA. [7] Gupta. S. S an Groll. P. A, (96). Gamma istribution in acceptance sampling base on life tests, Journal of the American Statistical Association, 56, pp [8] Kantam. R. R. L an Rosaiah. K, (998) Half Logistic istribution in acceptance sampling base on life tests, IAPQR Transactions, vol. 3, no., pp. 7 5, 998. [9] Kantam. R. R. L, Rosaiah. K an Rao G. S, (). Acceptance Sampling base on life tests: Log-logistic moel. Journal of Applie Statistics, 8, pp: -8. [] Kunu. D an Raqab. M. Z, (5). Generalize Rayleigh istribution: ifferent methos an estimations, Computational Statistics an Data Analysis. [] Lio Y.L, Tsai T.-R, an Wu S.-J, (9). Acceptance sampling plans from truncate life tests base on the Birnbaum-Sauners istribution for percentiles, Communications in Statistics -Simulation an Computation, 39, pp: [] Lio Y.L, Tsai T.-R, an Wu S.-J, (). Acceptance sampling plans from truncate life tests base on the birnbaum - sauners istribution for percentiles, Communications in Statistics: Simulation an Computation, 39, pp: [3] Praeepa Veerakumari. K an Ponneeswari. P (6) Designing of acceptance sampling plan for life tests base on percentiles of exponentiate Rayleigh istribution, International journal of current engineering an technology, 6, [4] Ramesh, Gupta. C, Pushpa, Gupta. L an Gupta R. D (998). Moeling failure time ata by Lehman alternatives. Communication in statistics Theory an Methos 7, pp: [5] Rao. G. S, () Double acceptance sampling plans base on truncate life tests for the Marshall-Olkin extene exponential istribution, Austrian

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