Limited Failure Censored Life Test Sampling Plan in Burr Type X Distribution

Transcription

1 Journal of Modern Applied Statistical Methods Volue 15 Issue 2 Article Liited Failure Censored Life Test Sapling Plan in Burr Type X Distribution R. R. L. Kanta Acharya Nagarjuna University, Guntur, Andhra Pradesh, India, anta.rrl@gail.co M. S. Raviuar Konaseea Institute of Medical Sciences & Research Foundation, Andhra Pradesh, India., sr.raaa@gail.co Follow this and additional wors at: Part of the Applied Statistics Coons, Social and Behavioral Sciences Coons, and the Statistical Theory Coons Recoended Citation Kanta, R. R. L. and Raviuar, M. S. (2016) "Liited Failure Censored Life Test Sapling Plan in Burr Type X Distribution," Journal of Modern Applied Statistical Methods: Vol. 15 : Iss. 2, Article 27. DOI: /jas/ Available at: This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCoons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCoons@WayneState.

2 Liited Failure Censored Life Test Sapling Plan in Burr Type X Distribution Cover Page Footnote R.R.L.Kanta and M.S.Raviuar This regular article is available in Journal of Modern Applied Statistical Methods: iss2/27

3 Journal of Modern Applied Statistical Methods Noveber 2016, Vol. 15, No. 2, doi: /jas/ Copyright 2016 JMASM, Inc. ISSN Liited Failure Censored Life Test Sapling Plan in Burr Type X Distribution R. R. L. Kanta Acharya Nagarjuna University Andhra Pradesh, India M. S. Raviuar Konaseea Institute of Medical Sciences & Research Foundation Andhra Pradesh, India The Burr type X distribution is considered as a life tie rando variable of a product whose lots are to be decided for acceptance or otherwise on the basis of saple lifeties drawn fro the lot. The saple is divided into various groups in order to develop a group sapling plan in such a way that the life testing experient is terinated as soon as the first failure in each group is observed. The acceptance criterion based on the theory of order statistics is proposed and is shown to be ore econoical than a criterion proposed in the earlier siilar wors. Keywords: Single sapling, lot acceptance, group sapling plan, truncated life tests, reliability test plans, order statistics Introduction Acceptance sapling is concerned with inspection and decision aing regarding products. Life tests are experients carried out on saple products in order to assess the life tie of an ite (tie to its failure or the tie it stops woring satisfactorily). A coon practice in life test is to terinate the test at a prefixed tie and record the nuber of failures that occurred during that tie period or when a prefixed nuber of failures is realised. The forer terination is generally called truncated life tests/tie censored life test and the latter is called a failure censored life test. If the quality of a product is easured through the life tie, sapling plans to deterine acceptability of a product with respect to life tie are called Reliability Sapling Plans. In life test sapling plans a coon constraint is the duration of total tie spent on testing. Sapling plans based on tie truncated life tests would address this constraint to soe extent. When the life tie rando variable is assued to Dr. Kanta is an Professor of Statistics. Eail hi at: anta.rrl@gail.co. M. S. Raviuar is a Lecturer and Biostatistician. Eail at: sr.raaa@gail.co. 428

4 KANTAM & RAVIKUMAR follow a specific continuous probability distribution, sapling plans are developed by various researchers covering a wide spectru of probability odels. Epstein (1954) was one of the foreost wors about acceptance sapling plans based on truncated life tests with the exponential distribution as the probability odel. Other researchers in this direction are as follows: Goode and Kao (1961) wored with the Weibull odel which includes the exponential distribution as a particular case. Gupta and Groll (1961) and Gupta (1962) considered the gaa and log-noral distributions, respectively. More recently, the studies of Kanta, Rosaiah, and Srinivasa Rao (2001), Balizi (2003), Balizi and El-Masri (2004), Rosaiah and Kanta (2005), Balarishanan, Lieva and López (2007), Asla and Kanta (2008), Srinivasa Rao, Ghitany, and Kanta (2009), Rosaiah, Kanta, and Srinivasa Rao (2009), Srinivasa Rao and Kanta (2010), Lio, Tsai, and Wu (2009), Lio, Tsai, and Wu (2010), Lu (2011), Kanta, Srira, and Suhasini (2012), Srinivasa Rao, Kanta, Rosaiah, and Pratapa Reddy (2012), Srinivasa Rao and Kanta (2013), Kanta and Srira (2013), Subba Rao, Prasad, and Kanta (2013), Kanta, Srira, and Suhasini (2013), Rosaiah, Kanta, Raa Krishnan, and Siva Kuar (2014), Subba Rao, Naga Durgaaba, and Kanta (2014) and the references therein, are related to construction of acceptance sapling plans based on truncated life tests with different probability odels. In all these wors, given the terination tie of a life test, the construction of the sapling plan consists of deterining the iniu nuber of saple ites that are to be life-tested and the acceptance nuber beyond which the observed failures out of the life-tested ites of the saple lead to rejection of the subitted lot, conditioned on pre specified producer s and consuer s riss. However, if a failure censored life test is under consideration, one has to wait till a pre specified nuber of failures out of the saple ites that are being tested is realised. Soeties the life of product ight be quite long possibly resulting in even a failure censored life-testing plan to be long tie consuing. Johnson (1964) proposed a sapling plan in which the experienter can decide to group the test units into several groups and then conduct the life-tests on all the groups siultaneously until the first failure in each group is realised. Based on the recorded first failure tie in each group if a decision process about the acceptance/rejection of subitted lot is developed the procedure ay be naed as Liited Failure Censored Life Test Sapling Plan (LFCLTSP). Balasooriya (1995) developed such a sapling plan for the two paraeter exponential distribution though the specific nae is not given as LFCLTSP. Wu and Tsai (2000), Wu, Tsai, and Ouyang (2001), Jun, Balaurali, and Lee (2006) have proposed LFCLTSP when the underlying lifetie rando variable follows 429

5 LFCLTSP IN BURR TYPE X DISTRIBUTION Weibull distribution, with respective distinct approaches in woring out the paraeters of the sapling plan. The schee of life testing and terination process of LFCLTSP is naed by soe researchers as Sudden death testing (for exaple Pascual & Meeer, 1998; Jun et al., 2006). Liited failure censored life tests is the nae proposed by Wu et al. (2001). Our suggested nae is Liited Failure Censored Life Test Sapling Plan (LFCLTSP). Thus, the purpose of this study is to develop LFCLTSPs for one of the odels of Burr (1942) Burr type X distribution on lines of Jun et al. (2006). A new criterion is also suggested that is ore econoical. Construction of LFCLTSP (Jun et al. 2006) The purpose of proposing LFCLTSP is to reduce testing tie. The total nuber of products to be tested, say N is divided into groups of equal size according to the nuber of available experiental testers. Thus there are n ites in each group and a total of groups ay be considered for this grouping so that N = n. The ites in each group are tested identically and siultaneously on different testers. The first group of ites is run until the first failure occurs. At this point the surviving ites are suspended and reoved fro testing. An equal set of new ites nubering n is next tested until the first failure. This process is repeated until one failure is generated fro each of the groups. In the end, failures are observed while (n 1) ites are suspended. Wu et al. (2001) naed this testing process as liited failure censored life test. The saple inforation so obtained can be utilized for deciding upon the acceptance of the lot fro which the original saple of N is put for testing. According to the characteristics of testers a group size n is usually specified but the total nuber of groups should be deterined. For that a variable sapling plan is proposed by Jun et al. (2006) with the following assuptions/specifications The life tie X follows a Weibull distribution with a nown shape paraeter (). There is a lower specification liit (L) regarding the life tie. p 0 is a desirable lot quality level (proportion of non conforities) at the pre specified producer s ris α. p 1 (> p 0 ) is an undesirable lot quality level (proportion of non conforities) at the pre specified consuer s ris β. 430

6 KANTAM & RAVIKUMAR Sapling Plan The cuulative distribution function (cdf) of the base line distribution (Weibull) is given by 1 exp x F x (1) The fraction non-conforing or unreliability is expressed by p PrX L F L (2) If p is given, the corresponding L is obtained fro w L ln 1 p. (3) The proposed sapling plan of Jun et al. (2006) is as follows: (i) (ii) (iii) (iv) Draw a rando saple of size N = n and allocate n ites to each of the groups. Observe Y i the tie to the first failure in the i th group (i = 1, 2,., ). Calculate the quantity V Y. i1 i Accept the lot if V cl and reject the lot otherwise (c ay be called acceptability constant - a concept siilar to the acceptance nuber in tie truncated reliability test plans). The nuber of groups and the acceptability constant c are called the paraeters of the sapling plan and will be deterined by the following procedure: Since Y i is the first order statistic in a saple of size n fro Weibull distribution with shape paraeter its cdf is given by Yi y ny Pr 1 exp, (4) which is the cdf of a Weibull distribution with shape paraeter and scale 1 paraeter n. Therefore the variables Y follow i.i.d exponential with scale paraeter n and as such V Y i1 i follows a gaa distribution with shape i 431

7 LFCLTSP IN BURR TYPE X DISTRIBUTION paraeter and scale paraeter n. Thus the quantity 2nV follows a chi-square distribution with 2 degrees of freedo so that the probability of acceptance of the lot for a lot quality level p is given by P p Pr V cl p Pr 2nV 2ncL p 1 G 2 ncw, (5) a where w is the solution of equation (3) and G l is the cdf of a chisquare variate with l degrees of freedo. As in Fertig and Mann (1980), the probability of acceptance should be at least (1 - α) at the desirable/acceptable lot quality level p 0 where α is producer s ris. Siilarly, the probability of acceptance should not be ore than β at the undesirable/tolerance lot quality level p 1, where β is consuer s ris. These two rears lead to the following two inequalities G 2ncw 1 (6) If 2 ql, 2 1 1G 2 ncw, (7) denotes the percentile point of tail probability q in the chi-square distribution with l degrees of freedo then, fro (6), (7), 2 2ncw0 1,2 (8) 2 2ncw1,2 (9) which jointly lead to w w. (10) 2 0 1,2 2 1,2 Therefore, can be obtained by the sallest integer satisfying (10). The acceptability constant c can be obtained fro the equality case in either of the expressions (8), (9). It can be noticed that the nuber of groups is deterined independently of the group size n and also of the shape paraeter. Jun et al. 432

8 KANTAM & RAVIKUMAR (2006) have evaluated, nc for α = 0.05 and β = 0.1 at selected cobinations of p 0, p 1. The corresponding table is reproduced below: Table 1. Design paraeters of sapling plans (α = 0.05, β = 0.1) p 0 p 1 g r For the sae of convenience in presentation, this procedure of Jun et al. (2006) is called Method-I and adopts the sae for Burr type X distribution to construct LFCLTSP below. LFCLTSP for Burr type X distributed Lifeties: Method-I Let the life tie of a product be given by Burr type X distribution with shape paraeter so that cdf is given by 2 1 F x e x (11) 433

9 LFCLTSP IN BURR TYPE X DISTRIBUTION Let L denote the p th quantile of a Burr type X variate. i.e., F L If p is given, the corresponding L is obtained fro w L p p (12) 1 ln 1 (13) Product with life tie less than L is considered nonconforing. Suppose the producer and the consuer have an agreeent that lots with nonconforing fraction less than or equal to p 0 are presued to be good and have to be accepted with probability of at least 1 - α. Here α is called producer s ris. Furtherore suppose that lots with non conforing fraction greater than p 1 (> p 0 ) are not acceptable to the consuer and should be rejected with a probability of at least 1 - β. Here β is called consuer s ris. If a rando saple of N ites grouped into groups of size n each is put to test, an LFCLTSP on lines of Jun et al. (2006) can be constructed with the following decision process. Observe Y i the tie to the first failure in the i th group (i = 1, 2,, ). Calculate the quantityv Y. i1 i Accept the lot if V cl and reject the lot otherwise (c ay be called acceptability constant - a concept siilar to the acceptance nuber in tie truncated reliability test plans). In order to get the plan paraeters and c, the percentiles of the sapling distribution of V are needed, which is the su of i.i.d observations on the first order statistic in a rando saple of size n odelled by Burr type X distribution with shape paraeter. In view of the atheatical structure of the Burr type X odel the sapling distribution of V cannot be analytically tractable. Hence, consider the epirical sapling distribution of V for various nown values of the shape paraeter and tabulated the percentiles of V for = 1.5(0.5)3; = 2(1)10; n = 5,10 in Tables 2 through

10 KANTAM & RAVIKUMAR Table 2. Percentiles of V Y i1 i at = p n

11 LFCLTSP IN BURR TYPE X DISTRIBUTION Table 3. Percentiles of V Y at = 2 i1 i p n

12 KANTAM & RAVIKUMAR Table 4. Percentiles of V Y i1 i at = p n

13 LFCLTSP IN BURR TYPE X DISTRIBUTION Table 5. Percentiles of V Y at = 3 i1 i p n

14 KANTAM & RAVIKUMAR If G(.) stands for the cdf of the rando variable V, the percentiles in Tables 1 G q stands for the q th percentile of V 2 through 5 are the values of G -1 (p). If with the shape paraeter the following inequalities are parallel to the expressions (6) through (10). which jointly lead to 0 G ncw (14) G ncw (15) 1 1 ncw G (16) ncw G (17) w w G 1 1 G. (18) Therefore, can be obtained by the sallest integer satisfying (18). The acceptability constant c can be obtained fro the equality case in either of the expressions (16), (17). We have tabulated the values of and c deterined for the sae cobinations of p 0, p 1 as chosen by Jun et al. (2006) and are presented in Tables 6 through 9 for = 1.5(0.5)3. 439

15 LFCLTSP IN BURR TYPE X DISTRIBUTION Table 6. Design paraeters of LFCLTSP (α = 0.05, β = 0.1, = 1.5) p 0 p 1 c n = 5 n =

16 KANTAM & RAVIKUMAR Table 7. Design paraeters of LFCLTSP (α = 0.05, β = 0.1, = 2) p 0 p 1 c n = 5 n =

17 LFCLTSP IN BURR TYPE X DISTRIBUTION Table 8. Design paraeters of LFCLTSP (α = 0.05, β = 0.1, = 2.5) p 0 p 1 c n = 5 n =

18 KANTAM & RAVIKUMAR Table 9. Design paraeters of LFCLTSP (α = 0.05, β = 0.1, = 3) p 0 p 1 c n = 5 n = It ay be noted that is solved as integer values only and, c depend on the shape paraeter of the Burr type X distribution. LFCLTSP for Burr type X distributed Lifeties: Method-II The statistic V Y introduced for the decision process of the sapling plan i1 i sees to have been considered as the total test tie to get the liited failure censored saple Y 1, Y 2,, Y which are first order statistics in independent rando saples of size n each. If Z denotes the axiu of Y 1, Y 2,, Y it ay also be viewed as the total test tie/experiental tie as opined by Kanta and Srinivasa Rao (2004). Hence, larger realized value of Z can be considered as an indication that the products in the subitted lot have longer life propting one to 443

19 LFCLTSP IN BURR TYPE X DISTRIBUTION consider the lot as a good lot for acceptability. In other words Z > cl can be taen as a criterion of acceptance of the lot. Thus, for Method-II the following decision rule is proposed: (i) (ii) Draw a rando saple of size N = n and allocate n ites to each of the groups. Observe Y i the tie to the first failure in the i th group (I = 1, 2,., ). (iii) Identify the quantity Z = Max (Y 1, Y 2, Y 3,,Y ). (iv) Accept the lot if Z cl and reject the lot otherwise (c ay be called acceptability constant - a concept siilar to the acceptance nuber in tie truncated reliability test plans). Using the theory of order statistics, the cdf of Z ay be obtained in a closed for as long as the cdf of the base line distribution is in a closed for. Hence, the percentiles of Z can be used to get the design paraeters, c analytically. For the focal distribution, Burr type X distribution with shape paraeter, the following is the analytical procedure of calculating design paraeters of LFCLTSP by Method-II. The cdf of Burr type X with shape paraeter is 2 F x 1 e x. (19) Let X 1, X 2, X 3,,X n be a rando saple of size n fro (19) The cdf of least of X 1, X 2, X 3,,X n is given by That is, n F x 1 1 F x. 1 (20) F x e x. 1 2 n (21) Y 1, Y 2, Y 3,,Y of the liited failure censored test are now a rando saple of size fro F (1) (x). Hence, the cdf of Z the largest of Y 1, Y 2, Y 3,,Y is given by 444

20 KANTAM & RAVIKUMAR 1 G z F z (22) n i.e., G z e. 2 z (23) As a corollary if = 1 then RHS of (23) becoes 2 nz G z 1 e, (24) which correspondents to the cdf of Z when the base line distribution is the well nown Rayleigh distribution which in turn is a special case of Weibull distribution. The design paraeters and c of LFCLTSP are obtained with the help of percentiles of G () (z) given in (23). If α and β are respectively the producer s and consuer s riss for desirable/acceptable lot quality level p 0, undesirable/lot tolerance quality level p 1 then and c are the solutions of the following two inequalities. where w 0 and w 1 are as defined above. 0 G cw (25) G cw (26) 1 1 The inequalities (25), (26) respectively iply which jointly lead to cw G (27) 1 1 cw G (28) 1 1 G 1 1 w G 0 (29) w 445

21 LFCLTSP IN BURR TYPE X DISTRIBUTION Therefore, can be obtained by the sallest integer satisfying (29). The acceptability constant c can be obtained fro the equality case in either of the expressions (27), (28). The values of and c were analytically deterined for the sae cobinations of p 0, p 1 as chosen by Jun et al. (2006) and are presented in Tables 10 through 13 for = 1.5(0.5)3 along with the values of the design paraeters of LFCLTSP of Method-I also for the sae of coparison. The values of obtained for Method-II can be seen to be consistently saller than or equal to those of Method-I, thus indicating less nuber of ites to be put to life test in Method-II and hence giving a preference to Method-II over Method-I. Table 10. Design paraeters of LFCLTSP of Methods I and II at = 1.5, α = 0.05 and β = 0.1 c p 0 p 1 n = 5 n = 10 n = 5 n = 10 I II I II I II I II

22 KANTAM & RAVIKUMAR Table 11. Design paraeters of LFCLTSP of Methods-I and II at = 2, α = 0.05 and β = 0.1 c p 0 p 1 n = 5 n = 10 n = 5 n = 10 I II I II I II I II

23 LFCLTSP IN BURR TYPE X DISTRIBUTION Table 12. Design paraeters of LFCLTSP of Methods-I and II at = 2.5, α = 0.05 and β = 0.1 c p 0 p 1 n = 5 n = 10 n = 5 n = 10 I II I II I II I II

24 KANTAM & RAVIKUMAR Table 13. Design paraeters of LFCLTSP of Methods-I and II at = 3, α = 0.05 and β = 0.1 c p 0 p 1 n = 5 n = 10 n = 5 n = 10 I II I II I II I II When = 1 Burr type X is a Rayleigh distribution which is a Weibull distribution with shape paraeter = 2. Jun et al. (2006) observed that their LFCLTSP for Weibull distribution is invariant of its shape paraeter. As atter of coparison, design paraeters of LFCLTSP of Method-II were coputed for Burr type X at = 1 also, so that these becoe the paraeters of LFCLTSP for Weibull distribution with shape 2. These are given Table

25 LFCLTSP IN BURR TYPE X DISTRIBUTION Table 14. Design paraeters of LFCLTSP of Method-II at = 1, α = 0.05 and β = 0.1 c p 0 p 1 n = 5 n = 5 n = 5 n = Coparison of Tables 1 and 14 also indicate that Method-II is preferable to Method-I in constructing LFCLTSP for Rayleigh distributed life ties. Illustration The quality assurance in a bearing anufacturing process states that p 0 = 0.01, p 1 = 0.04, α = 0.05, β = 0.1 the nuber of test positions (size of each group, n) = 10. For this inforation Table 2.1 of Jun et al. (2006) suggests = 5, c = 196. Accordingly a rando saple of size N = 50 ites are put to test in five groups with 10 ites in each group. The observed first failure ties in the five groups are Y 1 = 120, Y 2 = 200, Y 3 = 185, Y 4 = 55, Y 5 = 265. Assuing that the life ties follow Weibull distribution with shape paraeter 2 and a lower 450

26 KANTAM & RAVIKUMAR 5 2 specification of L = 100 they have calculated V Y i1 i and the acceptability constant cl 2 = since V < cl 2 they decided the subitted lot to be rejected. Adopting the sae inforation to Burr type X distribution we tae the shape paraeter of Burr type X naely = 1. Then it becoes the Rayleigh distribution which is also a Weibull with shape paraeter 2. For the sae of coparison with the sapling plan of Jun et al. (2006), at the above p 0, p 1, α, β, n = 10, we get fro Table 14 as = 2, and acceptability constant c = then cl = Z = the axiu of 55,120 = 120. Since Z < cl. i.e., 120 < , the lot is to be rejected. Fro this exaple, the approach reached the decision of rejecting the lot by conducting liited failure censored life test for only two groups of 10 ites each, whereas that of Jun et al. (2006) required the experient to be conducted for 5 groups of 10 ites each resulting in higher cost of experientation and larger nuber of destructions. In that way, the Method-II is preferable to the Method-I proposed by Jun et al. (2006). Moreover, it ay be recalled that V, Z are defined as V Y i1 i Z Max Y1 Y2 Y,,,. If c is the acceptability constant and L is the lower specification, Z > cl V > cl. That is acceptance by Method-II iplies acceptance by Method-I, so that as far as acceptance decision is considered Method-II gives a stronger conclusion iplying the sae decision by Method-I. References Asla, M., & Kanta, R. R. L. (2008). Econoic reliability acceptance sapling based on truncated life tests in the Birnbau-Saunders distribution. Paistan Journal of Statistics, 24(4), Balizi, A. (2003). Acceptance sapling based on truncated Life tests in the Pareto distribution of the second ind. Advances and Applications in Statistics, 3(1),

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