Double Acceptance Sampling Plan for Time Truncated Life Tests Based on Transmuted Generalized Inverse Weibull Distribution

Transcription

1 J. Stat. Appl. Pro. 6, No. 1, Journal of Statstcs Applcatons & Probablty An Internatonal Journal Double Acceptance Samplng Plan for Tme Truncated Lfe Tests Based on Transmuted Generalzed Inverse Webull Dstrbuton Amer Ibrahm Al-Omar 1, and Ehsan Zamanzade 2 1 Department of Mathematcs, Faculty of Scence, Al al-bayt Unversty, Mafraq 25113, Jordan 2 Department of Statstcs, Unversty of Isfahan, Isfahan , Iran Receved: 27 Sep. 2016, Revsed: 20 Oct. 2016, Accepted: 25 Oct Publshed onlne: 1 Mar Abstract: In ths paper, a double acceptance samplng plan DASP s developed n terms of truncated lfe tests assumng that the lfetme of a product follows a transmuted generalzed nverse Webull TGIWD dstrbuton. Wth fxng the consumer s confdence level, the mnmum requred sample szes of the frst and second samples to ensure the specfed mean lfe are obtaned. The operatng characterstc functon values and the mnmum ratos of the mean lfe to the specfed lfe are presented. Some tables are provded and ther use s llustrated by a numercal example. Keywords: Double Acceptance Samplng Plan, Tme Truncated Lfe Tests, Transmuted Generalzed Inverse Webull Dstrbuton, Operatng Characterstc Functon, Consumer s Rsk. 1 Introducton In ths paper, we suggest acceptance samplng plans based on transmuted generalzed nverse Webull dstrbuton wth probablty densty functon pdf gven by f T GIW x=αβ γα x β 1 e γα x β 1+λ 2λ e γα x β, x 0, β > 0, γ > 0, α > 0, 1 and λ 1 wth cumulatve dstrbuton functon cdf defned as F TGIW x=e γα x β 1+λ λ e γα x β. 2 The mean and the moment generatng functon of the TGIWD are obtaned; respectvely, as: µ T GIW = γ1/β 1+λ λ 2 1/β Γ 1 1, 3 α β and M X t= r=0 The qth quantle, x q, of the TGIW dstrbuton s t r γ r β r!α r Γ { x q = λ α γ log λ e q Correspondng author e-mal: alomar amer@yahoo.com 1 β r [ ] 1+λ λ 2 r β. 4 γα x β } 1 β. 5

2 2 A. I. Al-Omar, E. Zamanzade: Double acceptance samplng plan based on TGIWD The relablty and hazard rate functons of the transmuted generalzed nverse Webull dstrbuton, respectvely, are gven by R T GIW x = 1 F TGIW x and = 1 e γα x β 1+λ λ e γα x β, 6 H T GIW x = f T GIWx 1 F TGIW x = αβ γα x β 1 e γα x β 1+λ 2λ e γα x β. 7 1 e γα x β 1+λ λ e γα x β The coeffcent of kurtoss, coeffcent of skewness, and the coeffcent of varaton of the TGIW dstrbuton, respectvely can be obtaned as Ψ 1 = µ 4 4µ 3 µ 1 + 6µ 2 µ 1 2 µ2 µ 1 2 2, Ψ 2 = µ 3 3µ 2 µ 1 + 2µ 1 3 µ2 µ 2 µ 1 3/2, Ψ 3 = 1. 8 µ 1 For more nformaton about the transmuted generalzed nverse Webull dstrbuton, we refer the nterested reader to [1]. A double acceptance samplng plan n terms of truncated lfe tests s proposed when the lfetme of a product follows the TGIW dstrbuton. The acceptance samplng plan s an mportant subject n statstcal qualty control. The qualty level of products s a very mportant for both producers and consumers. Snce the screenng of products s mpossble and nfeasble due the cost and tme, a decson about the product can be taken based on a selected sample from the lot. The DASP s used to reduce the sample sze or producer s rsk n the feld of qualty control. DASP should be used f the decson cannot be taken based on the frst sample. Therefore, a second sample should be selected from the lot to make a decson [2]. Several authors consdered the DASP under dfferent lfe tme dstrbutons n ther research; [3] studed the generalzed log-logstc dstrbuton n DASP. [4] proposed DASP based on truncated lfe tests for the Marshall-Olkn extended exponental dstrbuton. [5] suggested DASP based on truncated lfe tests usng generalzed exponental dstrbuton. [6] suggested new ASP for three parameters kappa dstrbuton. [7] proposed ASP for generalzed nverted exponental dstrbuton. [8] suggested DASP when the tme truncated lfe tests follows the Maxwell dstrbuton. [9] suggested an ASP for truncated lfe tests for exponentated Frechet dstrbuton. Ths paper s organzed as follows. The suggested double acceptance samplng plan s presented n Secton 2. The operatng characterstc functon s gven n Secton 3. The mnmum varance ratos to the specfed lfe are provded n Secton 4. The paper s concluded n Secton 5. 2 Double Acceptance Samplng Plan The DASP based on truncated lfe tme can be descrbed as follows: 1- Draw the frst random sample of sze n 1 and put them on test durng tme t 0. If there are c 1 or fewer falures, accept the lot. If c falures are observed, stop the test and reject the lot;.e., c 1 < c If the number of falures by t 1 s between c 1 +1 and c 2, then draw the second sample of sze n 2 and then test the drawn tems durng another tme t 0. If at most c 2 falures are observed from the two samples,.e., n 1 + n 2, accept the lot. Otherwse, reject the lot and termnate the test. Therefore, the DASP conssts of four parameters n 1, n 2, c 1 and c 2. Based on these parameters; the probabltes of acceptance Lp 1 and Lp 2 for the samplng plans n 1,c 1, t and n 2,c 2, t can be calculated under the assumpton that the lot s large enough to use the Bnomal probablty dstrbuton as follows: Lp 1 = c 1 =0 =0 p 1 p, 9

3 J. Stat. Appl. Pro. 6, No. 1, / 3 by Lp 2 = c 2 =2 =0 n2 p 1 p n2, 10 respectvely, where p=ft; µ = F t. µ s gven n 2. Then, the probablty of acceptance n general s gven Lp= c 1 =0 p 1 p n 1 + c 2 =c 1 +1 p 1 p n 1 [ c2 j ] n2j p j 1 p n 2 j. 11 j=0 Note that f c 1 = 0 and c 2 = 2, then the probablty of acceptance s the total of three dfferent probabltes that can be formulated as: PA =PNo falure occurs n the frst sample + P1 falure occurs n fst sample and zero or one falure occurs n the second sample + PTwo falures occur n the frst sample and zero or one falure occurs n second sample. In ths artcle, values of the probablty of acceptance of a lot for a DASP based on TGIW dstrbuton are obtaned at P = 0.75, 0.90, 0.95, 0.99 for t/µ o = 0.628, 0.942, 1.257, 1.571, 2.356, 3.141, 3.927, are presented n Tables 1-2 based on sngle and double acceptance samplng plans, respectvely. These choces of P and t/µ o are computable wth [10], [11], [12], [13], [9], and [14]. 3 Operatng Characterstc Functon The operatng characterstc functon OC s one of the man crtera n the acceptance samplng plan where t represents the probablty of acceptng a lot. The OC measures the effcency of a statstcal hypotheses test n order to accept or reject a lot. The acceptance samplng plan s sutable f ts operatng characterstc functon values approaches to one. The operatng characterstc values of the samplng plan n 1,c 1 = 0,t/µ o for a gven P under the TGIWD wth β = 3, γ = 2 and λ = 0.9 are presented n Table 1. Furthermore, for the acceptance samplng plann 1,n 2,c 1,c 2,t/µ o the OC functon values are provded n Table 2. Based on Tables 1-2, t can be noted that the OC values approach to one, specally for large values of µ / based on sngle and double acceptance samplng plans. However, the values of the probablty based on the suggested DASP are larger than ther counterpart usng the sngle acceptance samplng plan. 4 Mnmum Mean Ratos to the Specfed Lfe and Producer s Rsk The producer s rsk PR s defned as the probablty of rejecton of a good lot..e., µ. For the suggested DASP usng TGIW dstrbuton and for a gven value of the producer s rsk θ, we want to fnd the mnmum qualty level of µ/µ 0 that asserts the PR to be at most θ. Therefore, µ/ s the smallest postve number for whch p=ft; µ=f t. µ satsfes the nequalty c 1 p 1 p + =0 c 2 =c 1 +1 p 1 p n 1 [ c2 ] n2j p j 1 p n 2 j 1 θ 12 j=0 For the proposed acceptance samplng plann 1,n 2,c 1 = 0,c 2 = 2,t/µ o at a specfed consumer s confdence level P, the smallest values of µ/ satsfyng 7 are summarzed n Table 3. Suppose that the expermenter wants to assert that true unknown average lfe s at least 1000 hours wth confdence level Also, suppose that the acceptance numbers for ths case are c 1 = 0 and c 2 = 2 wth sample szes n 1 = 4 and n 2 = 6. Thus, the lot s accepted f wthn 942 hours no falure s detected wth a sample of sze 4. For sngle samplng plan, the probablty of acceptng the lot s 1 wth µ / = 4, 6, 8, 10, 12. The DASP for the same measurements the rato probablty s 1. It s of nterest to note here that as the rato µ / ncreases usng DASP the probablty of acceptance ncreases. The producer s rsk wth respect to tme of experment for DASP usng TGIW dstrbuton for c 1 = 0 and c 2 = 2 are presented n Table 4 for P = For example, when µ / = 2 the unknown average lfe s two tmes of specfed average lfe producer s rsk n the case of tme of experment beng 4712 hours and 628 hours are 0 and , respectvely. It s found that f the qualty level of the product ncreases the producer s rsk decreases.

4 4 A. I. Al-Omar, E. Zamanzade: Double acceptance samplng plan based on TGIWD Table 1: Operatng characterstc values of the samplng plann 1,c 1 = 0,t/µ o for a gven P under the TGIWD wth β = 3, γ = 2 and λ = 0.9. P t/µ o n 1 µ/µ o = Conclusons In ths paper, we assumed that the lfetme of the products follows a transmuted generalzed nverse Webull dstrbuton to suggest a double acceptance samplng plan based on TGIWD. For fxed consumer s confdence level, the mnmum sample szes of the requred frst and second samples to assert the specfed mean lfe are calculated. The operatng characterstc values and the mnmum ratos of the mean lfe to the specfed lfe are tabulated. It s shown that the suggested DASP mght be more useful than the sngle acceptance samplng plan. Acknowledgments The authors would lke to thank the referees and the edtor for valuable and constructve comments.

5 J. Stat. Appl. Pro. 6, No. 1, / 5 Table 2: Operatng characterstc values of the samplng plann 1,n 2,c 1 = 0,c 2 = 2,t/µ o for a gven P under the TGIWD wth β = 3, γ = 2 and λ = 0.9. P t/µ o n 1 n 2 µ/µ o = Table 3: Mnmum rato of σ/σ 0 for the acceptablty of a lot wth producer s rsk of 0.05 under the TGIWD wth β = 3, γ = 2 and λ = 0.9. P t/µ o = References [1] F. Merovc, I. Elbatal and A. Ahmed. The transmuted generalzed nverse Webull dstrbuton. Austran Journal of Statstcs, 432, [2] A. Duncan. Qualty control and ndustral statstcs, 5th ed. Homewood, IL: Rchard D. Irwn [3] M. Aslam and C.-H. Jun. A double acceptance samplng plan for generalzed log-logstc dstrbutons wth known shape parameters. Journal of Appled Statstcs, 373, [4] S.S. Rao. Double acceptance samplng plans based on truncated lfe tests for the Marshall-Olkn extended exponental dstrbuton. Austran Journal of Statstcs, 403, [5] A.S. Ramaswamy and P. Anburajan. Double acceptance samplng based on truncated lfe tests n generalzed exponental dstrbuton. Appled Mathematcal Scences, 664,

6 6 A. I. Al-Omar, E. Zamanzade: Double acceptance samplng plan based on TGIWD Table 4: Producer s rsk wth respect to tme of experment for double acceptance samplng for c 1 = 0 c 2 = 2 and P = 0.95 based on TGIWD wth β = 3, γ = 2 and λ = 0.9. P t/µ o n 1 n 2 µ/µ o = E E-05 0 [6] A.I. Al-Omar. Acceptance samplng plan based on truncated lfe tests for three parameter kappa dstrbuton. Economc Qualty Control, 291, [7], A.I. Al-Omar. Tme truncated acceptance samplng plans for generalzed nverted exponental dstrbuton. Electronc Journal of Appled Statstcal Analyss, 81, [8] W. Gu. Double acceptance samplng plan for tme truncated lfe tests based on Maxwell dstrbuton. Amercan Journal of Mathematcal and Management Scences, 33, [9] A.D. Al-Nasser and A.I. Al-Omar. Acceptance samplng plan based on truncated lfe tests for exponentated Frechet dstrbuton. Journal of Statstcs and Management Systems, 161, [10] S.S. Gupta. Lfe test samplng plans for normal and lognormal dstrbutons, Technometrcs, 42, [11] S.S. Gupta and P.A. Groll. Gamma dstrbuton n acceptance samplng based on lfe tests. Journal of Amercan Statstcal Assocaton 56, [12] N. Balakrshnan, V. Leva and J. Lopez. Acceptance samplng plans from truncated lfe tests based on the generalzed Brnbaum- Saunders dstrbuton. Communcaton n Statstcs-Smulaton and Computaton, 36, [13] R.R.L. Kantam, K. Rosaah and G.S. Rao. Acceptance samplng based on lfe tests: log-logstc model. Journal of Appled Statstcs, 28, [14] A. Baklz, A. El Masr and A. Al-Nasser. Acceptance samplng plans n the Raylegh model. The Korean Communcatons n Statstcs, 121, Amer Ibrahm Al-Omar obtaned hs PhD degree n Statstcs from the Faculty of Scence and Technology, Natonal Unversty of Malaysa, Malaysa, n He was Drector of the Qualty Assurance and Plannng Department at Al al-bayt Unversty, Mafraq, Jordan, from 2010 to Now, he s Vce Dean of Academc Research and Assocate Professor of Statstcs at the Department of Mathematcs, Faculty of Scence at Al al-bayt Unversty, Jordan. Hs current research nterests nclude Ranked Set Samplng, Mssng Data, Statstcal Process Control, Acceptance Samplng Plans and Estmaton. Ehsan Zamanzade receved hs B.Sc of Statstcs n 2006, M.Sc of Mathematcal Statstcs n 2008 and Ph.D of Statstcal Inference n 2012 from Ferdows Unversty of Mashhad, Mashhad, Iran. Hs research nterests nclude ranked set samplng, judgment post stratfcaton and goodness of ft tests