Control Charts for Mean for Non-Normally Correlated Data

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1 Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies i Statistics, Vikram Uiversity, Ujjai, Idia, Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical heory Commos Recommeded Citatio Sigh, J. R. & Dar, A. L. (017). Cotrol charts for mea for o-ormally correlated data. Joural of Moder Applied Statistical Methods, 16(1), doi: 10.37/jmasm/ his Regular Article is brought to you for free ad ope access by the Ope Access Jourals at It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of

3 SINGH & DAR the chart. he purpose of this study is to cosider the power of the cotrol chart ad the effect of correlatio o ype-i error ad the OC fuctio, ad also to cosider relaxig the assumptio of ormality ad cosiderig the productio process to follow a o-ormal distributio represeted by the first four terms of a Edgeworth series. Effect of Correlatio o OC Fuctio for Normal Case Suppose that the observatios x 1, x,, x have a multivariate ormal distributio with E(x i ) = μ, V(x i ) = σ ad ρ is the commo correlatio coefficiet betwee ay x i ad x j, i j. he E x Var x 1 1 (1) where 1 1 () he power of the cotrol chart is judged by its OC fuctio. he cotrol chart for the mea is set up by drawig the cetral lie at the process average θ ad the cotrol limits at k where σ is the process stadard deviatio ad is the sample size. he OC fuctio gives the probability that the cotrol chart idicates the value θ as the process average, whe it is actually ot θ, but 453

4 CONROL CHARS FOR MEAN 454 where is as defied i equatio (). he OC fuctio is derived by itegratig the distributio of the mea with θʹ as the process average betwee the limits of the cotrol chart. For the ormal populatio uder correlated data, 1 f (3) he distributio of the sample mea is give by g (4) where 1 exp ad π r r r t d t t t dt he OC fuctio is obtaied after replacig θ i (4) by θʹ ad itegratig it betwee the limits of the cotrol chart as k k N L d (5) k k N L d (6) Makig the trasformatio y

5 SINGH & DAR ad y γ = t sequetially, the above itegral simplifies to L N k k 1 (7) he error of ype I gives the probability of searchig for assigable causes whe i fact there are o such causes. It is give by k 1 k g d (8) After itegratig above as i the case of the OC fuctio we will get k (9) he Effect of No-Normally Correlated Data o OC Fuctio For o-ormal populatios represeted by the first four terms of a Edgeworth series, f (10) where are the stadardized third ad fourth cumulats, respectively. he distributio of the sample mea for correlated data ca be derived, by followig Gaye (195), as ad 455

6 CONROL CHARS FOR MEAN g (11) he OC fuctio is obtaied after replacig θ i equatio (11) by θʹ ad itegratig it betwee the limits of the cotrol chart, i.e. betwee k Itegratig i the similar way as for the ormal case, we get. L L L L (1) N u b where L N is give by equatio (7). he other two terms of the OC fuctio are give by 13 k 3 k 5 k L u (13) 13 k 3 k 5 k L b (14) he ype-i error for the o-ormal populatio works out to be k 1 k g d c (15) where α as defied by equatio (9) is the ype-i error whe the populatio is ormal ad depedet, ad k c k (16) is the correctio for o-ormality ad depedecies i ype-i error. 456

9 SINGH & DAR able 3, cotiued. K = K = 3 ρ λ3 λ4=0.0 λ4=0.5 λ4=1.0 λ4=.0 λ4=0.0 λ4=0. λ4=0.5 λ4= able 4. Values of OC fuctio for o-ormally correlated data (λ 3, λ 4 ) ρ γ (0.0,0.0) (0.0,0.5) (0.0,1.0) (0.0,.0) (0.5,0.0) (0.5,0.5) (0.5,1.0) (0.5,.0) Refereces Alwa, L. C. (199). Effects of autocorrelatio o cotrol chart performace. Commuicatios i Statistics heory ad Methods, 1(4), doi: /

10 CONROL CHARS FOR MEAN Alwa, L. C., & Roberts, H. V. (1995). he problem of misplaced cotrol limits. Joural of the Royal Statistical Society. Series C (Applied Statistics), 44(3), doi: / Dar, A. L., & Sigh, J. R. (015). he power of -chart i presece of data correlatio. Joural of Reliability ad Statistical Studies, 8(1), Retrieved from Gaye, A. K. (195). O settig up cotrol charts for o-ormal samples. Idia Society for Quality Cotrol Bulleti, 53, Maragah, H. D, & Woodall, W. H. (199). he effect of autocorrelatio o the retrospective -chart. Joural of Statistical Computatio ad Simulatio, 40(1-), 9-4. doi: / Sigh, J. R., Sakle, R., & Ahmad, M. (01). Cotrol charts for mea uder correlated data. Joural of Rajastha Statistical Associatio, 1(1),

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