A note on the interval estimation of c_{pk} with asymmetric tolerances

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1 This article was ownloae by [National Chiao Tung University 國立交通大學 ] On 30 April 014, At 304 Publisher Taylor & Francis Informa Lt Registere in Englan an Wales Registere Number Registere office Mortimer House, Mortimer Street, Lonon W1T 3JH, UK Journal of Nonparametric Statistics Publication etails, incluing instructions for authors an subscription information http// A note on the interval estimation of c_{pk} with asymmetric tolerances G. H. Lin a & W. L. Pearn b a Department of Communication Engineering, National Penghu Institute of Technology, Penghu, Taiwan, ROC b Department of Inustrial Engineering an Management, National Chiao Tung University, Taiwan, ROC Publishe online 7 Oct 010. To cite this article G. H. Lin & W. L. Pearn (00) A note on the interval estimation of c_{pk} with asymmetric tolerances, Journal of Nonparametric Statistics, 146, , DOI / To link to this article http//x.oi.org/ / PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) containe in the publications on our platform. However, Taylor & Francis, our agents, an our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions an views expresse in this publication are the opinions an views of the authors, an are not the views of or enorse by Taylor & Francis. The accuracy of the Content shoul not be relie upon an shoul be inepenently verifie with primary sources of information. Taylor an Francis shall not be liable for any losses, actions, claims, proceeings, emans, costs, expenses, amages, an other liabilities whatsoever or howsoever cause arising irectly or inirectly in connection with, in relation to or arising out of the use of the Content. This article may be use for research, teaching, an private stuy purposes. Any substantial or systematic reprouction, reistribution, reselling, loan, sub-licensing, systematic supply, or istribution in any form to anyone is expressly forbien. Terms & Conitions of access an use can be foun at http//

2 Nonparametric Statistics, 00, Vol. 14(6), pp A NOTE ON THE INTERVAL ESTIMATION OF C pk WITH ASYMMETRIC TOLERANCES G. H. LIN a, * an W. L. PEARN b a Department of Communication Engineering, National Penghu Institute of Technology, Penghu, Taiwan, ROC b Department of Inustrial Engineering an Management, National Chiao Tung University, Taiwan, ROC (Receive May 001 In final form February 00) Pearn an Chen (1998) propose a generalization of the wiely use process capability inex (PCI) C pk to hanle processes with asymmetric tolerances. They investigate the sampling istribution an obtaine the exact formulae for the expecte value an variance of its natural estimator. Recently, Pearn an Lin (000) consiere a ifferent estimator uner ifferent process conition, an investigate the statistical properties of the new estimator. However, their efforts focuse on the small sample properties uner the normality assumption. In this paper, we investigate the large sample properties of its natural estimator uner the general conition. Base on the limiting istribution of the new estimator, we provie an approximate 100ð1 aþ% confience interval of the consiere PCI. The obtaine confience interval provies great benefit to quality engineers on monitoring the process an assessing process performance. Keywors Asymmetric tolerances Process capability inex 1 INTRODUCTION Process capability inex C pk (Kane, 1986) has been wiely use in the manufacturing inustry to provie numerical measures of process potential an performance. As note by many quality control researchers an practitioners, C pk is yiel-base an is inepenent of the target T, which fails to account for process centering with symmetric tolerances, has an even greater problem with asymmetric tolerances. To overcome the problem, Pearn an Chen (1998) consiere a generalization of C pk, referre to as Cpk, which is efine as Cpk ¼ A ð1þ where ¼ minf U L g, A ¼ maxf ðm TÞ= U, ðt mþ= L g, U ¼ USL T is the right han sie tolerance, an L ¼ T LSL is the left han sie tolerance. Clearly, if T ¼ m (symmetric case), then ¼ A ¼jm mj an the generalization Cpk reuces to the original inex C pk. The factors an A ensure that the generalization Cpk obtains its * Corresponing author. ISSN print ISSN online # 00 Taylor & Francis Lt DOI /

3 648 G. H. LIN AND W. L. PEARN maximal value at T (process is on-target) regarless of whether the tolerances are symmetric or asymmetric. THE NATURAL ESTIMATOR OF C pk The natural estimator ^C pk of C pk can be obtaine by replacing m an s by X ¼ P n i¼1 X i=n an S n 1 ¼f P n i¼1 ðx i X Þ =ðn 1Þg 1= respectively in expression (1). ^C pk ¼ ^A 3S n 1 ðþ where ^A ¼ maxf ðx TÞ= U ðt X Þ= L g. Uner the normality assumption, Pearn an Chen (1998) showe that the rth moment (about 0) of ^C pk is Eð ^C r pk Þ¼ ffiffiffiffiffiffiffiffiffi n 1 E ðn r= 1ÞS X r n 1 3 s E max Z Z r j U L j¼0 r j p ffiffiffi j s n Hence, the first two moments of ^C pk are (Pearn an Chen, 1998) ( E ^C pk ¼ 1 Cpk b 1 rffiffiffiffiffi þ exp n 1 6 U L np Fð jjþ ffiffi p ) A 3 U L n s r j Eð ^C pk Þ ¼ n 3 ( ðcpk n 1 Þ þ Fð jjþ " 4A þ 1 # þ 3 s 18n U L 1 rffiffiffiffiffi þ exp þ Fð jjþ 3 U L np 3 3n " # þ Fð jjþ U L 3 n U L " Fð jjþ A þ 1 # 3 3 s 9n U L exp p ffiffiffiffiffi p þ ) ð1 Fð jjþþ jj where ¼ p ffiffiffi n ðm TÞ=s, bn 1 ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =ðn 1ÞfG½ðn 1Þ=Š=G½ðn Þ=Šg.

4 INTERVAL ESTIMATION OF C pk LARGE SAMPLE PROPERTIES OF ^C pk Pearn an Chen (1998) succeee in obtaining the moments of ^C pk. Their investigation focuse on processes with normal istribution. Uner general conitions, the asymptotic behavior of ^C pk is esirable. Let X 1 X... X n be a ranom sample of measurements from a process which has istribution G with mean m an variance s. We note that ^C pk is a continuous function of ðx Sn 1 Þ, an ðx S n 1 Þ converges to ðm s Þ in probability. Therefore, ^C pk is a consistent estimator of C pk. THEOREM 1 Let X 1 X... X n be a ranom sample of measurements from a p process ffiffiffi whose fourth central moment m 4 exists an LSL m USL, then as n!1 n ð ^C pk C pk Þ converges to the following in istribution. (a) Nð0 s pk1þ if m > T, (b) Nð0 s pkþ if m < T, an (c) jvj=ðþ ½ =ð6s 3 ÞŠW if m ¼ T, where s 3 L ðm TÞ L ðm TÞ 3 4s 4 if USL T > T LSL s 3 ðm TÞ ðm TÞ 3 4s 4 if USL T ¼ T LSL s 3 U ðm TÞ U ðm TÞ 3 4s 4 if USL T < T LSL s pk ¼ 1 9 m 3 L þðm TÞ L þðm TÞ 3 4s 4 if USL T > T LSL s pk ¼ 1 9 m 3 þðm TÞ þðm TÞ 3 4s 4 if USL T ¼ T LSL s pk ¼ 1 9 m 3 U þðm TÞ U þðm TÞ 3 4s 4 if USL T < T LSL ðv WÞ Nðð0 0Þ SÞ with variance covariance matrix S ¼ s m 3 Proof See Appenix. ^C pk THEOREM is asymptotically unbiase. Proof From Theorem 1, we know that as n!1, Ef n ð ^C pk C pkþg! 0, if m 6¼ T, an ffi Ef n ð ^C pk p C pkþg! ffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi =p =3, if m ¼ T, since EfjVjg ¼ s ð=pþ. Therefore, as n!1, then Eð ^C pk C pk Þ¼ð1= p ffiffi n ÞEf n ð ^C pk C pk Þg! 0 implies that the estimator ^C pk is asymptotically unbiase. THEOREM 3 If the process characteristic follows a normal istribution Nðm s Þ then ^C pk is asymptotically efficient, if m 6¼ T. Proof From Theorem 1, we know that if m 6¼ T, then as n!1, n ð ^C pk C pk Þ converges in istribution to Nð0 s pk1þ,ifm > T, an Nð0 s pkþ,ifm < T. Uner normality assumption,

5 650 G. H. LIN AND W. L. PEARN m 3 ¼ 0, m 4 ¼ 4 implies that n ð ^C pk C pkþ converges in istribution to Nð0 s pknþ, where s pkn ¼ð1=9ÞþðC pk =Þ. The information matrix is IðyÞ ¼Iðm sþ ¼ 1=s 0 Since the Cramer-Rao lower boun 0 1=ðs 4 Þ qcpk qm qc pk qs I 1 ðyþ 6 n 4 qc pk qm qc pk qs ¼ s pkn n is achieve, therefore ^C pk is asymptotically efficient. Since ^C pk is asymptotically efficient for m 6¼ T, efining M 3 ¼ nm 3 ½ðn 1Þðn ÞŠ 1, P an M 4 ¼½nðn n þ 3Þm 4 3nðn 3Þm Š½ðn 1Þðn Þðn 3ÞŠ 1, where m k ¼ n j¼1 ðx j X Þ k =n k ¼ 3 4, we can show that M 3 M 4 are unbiase estimators of m 3 m 4, respectively. If m > T, an estimator for s pk1 is obtaine as ^s pk1 ¼ 1 9 þ M 3 3Sn 1 3 ^C pk1 þ M 4 S 4 n 1 4S 4 n 1 ^C pk1 where ^C pk1 ¼½ ðx TÞŠ=ð3S n 1 Þ. Using this estimator, an approximate 100ð1 aþ% one-sie confience interval of Cpk can be constructe as ^s ^C pk1 pk1 z a 1 ð4þ n where z a represents the upper ath quantile of the stanar normal istribution. Similarly, if m < T, an estimator for s pk is obtaine as ^s pk ¼ 1 9 M 3 ^C 3Sn 1 3 pk þ M 4 Sn 1 4 ^C 4Sn 1 4 pk where ^C pk ¼½ þðx TÞŠ=ð3S n 1 Þ. Using the estimator an its limiting istribution, an approximate 100ð1 aþ% one-sie confience interval of Cpk can be constructe as ^C pk ð3þ ð5þ ^s pk z a 1 ð6þ n 4 AN EXAMPLE Consier the following example taken from a manufacturer an supplier in Taiwan exporting high-en auio speaker components incluing rubber ege, Pulux ege, Kevlar cone, honeycomb an many others. The prouction specifications for a particular moel of Pulux ege are ðlsl T USLÞ ¼ð Þ. A total of 90 observations were collecte which are isplaye in Table I. If the true Cpk value fell into expressions (4) or (6), we conclue that the process is capable, otherwise, the process is incapable uner the given confience level.

6 INTERVAL ESTIMATION OF C pk 651 TABLE I Collecte Sample Data (90 Observations) We note that the mi-point m ¼ðUSL þ LSLÞ= ¼ 5800 an the half length of the specification interval ¼ðUSL LSLÞ= ¼ 0150, ¼ minðusl T T LSLÞ ¼015. The sample mean x ¼ P 90 i¼1 x i=90 ¼ , sample stanar eviation s n 1 ¼f P 90 i¼1 ðx i xþ = ð90 1Þg 1= ¼00334, A^ ¼ m ¼ , m 3 ¼4910 6, m 4 ¼ , M ¼ , M 3 ¼ an M 4 ¼ Since the sample mean is less then the target value, ^C pk ¼½ðUSL TÞþðx TÞŠ=ð n 1Þ¼ Apply the expression (5), we obtaine ^s pk ¼ From the expression (6), an approximate 95% one-sie confience boun of Cpk is ½138 1Þ. A process with C pk < 100 is calle inaequate, a process with 100 Cpk < 133 is calle capable, a process with 133 C pk < 150 is calle satisfactory, a process with 150 Cpk < 00 is calle excellent. In this example, we have 95% confience to claim that the process is satisfactory. Acknowlegement The research was partially supporte by National Science Council of the Republic of China. (NSC E ) References Kane, V. E. (1986). Process capability inices. Journal of Quality Technology, 18, Pearn, W. L. an Chen, K. S. (1998). A new generalization of process capability inex C pk. Journal of Applie Statistics, 5, Pearn, W. L. an Lin, G. H. (000). Estimating capability inex C pk for processes with asymmetric tolerances. Communications in Statistics-Theory an Methos, 9, Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley an Sons, New York, pp APPENDIX Proof of Theorem 1 (I) If USL T > T LSL, then Cpk ¼ð L jm TjÞ=ðÞ an ^C pk ¼ ð L jx TjÞ=ð3S n 1 Þ. Case (a) We first consier the case with m > T. We efine the function g L1 ðx yþ ¼ p ð L ðx TÞÞ=ð3 ffiffi y Þ, where x > T an y > 0. Note that gl1 is a real-value an ifferentiable function for x > T an y > 0, with qg L1 qx ðms Þ ¼ 1 an qg L1 qy ðms Þ ¼ 1 L ðm TÞ s

7 65 G. H. LIN AND W. L. PEARN Define D L1 ¼ qg L1 qx ðms Þ qg L1. Note that qy DL1 6¼ð0 0Þ. ðms Þ Then, we have ffi n ð ^C pk C pk Þ¼ p ffiffi n fgl1 ðx Sn 1 Þ g L1ðm s Þg converges to Nð0 s pk1þ in istribution (Serfling, 1980), where s pk1 ¼ D L1SD L1 ¼ 1 9 þ m 3 L ðm TÞ L ðm TÞ 3 4s 4 Case (b) For the case with m < T, we efine the function g L ðx yþ ¼ L þðx TÞ p 3 ffiffi y where x < T an y > 0. Note that g L is also a real-value an ifferentiable function for all x < T an y > 0, with qg L qx ðms Þ ¼ 1 an qg L qy ðms Þ ¼ 1 L þðm TÞ s Define D L ¼ qg L qx ðms Þ qg L. Note that qy DL 6¼ð0 0Þ. ðms Þ Then, we have n ð ^C pk C pk Þ¼ p ffiffiffi n fgl ðx Sn 1 Þ g Lðm s Þg converges to Nð0 s pk Þ in istribution (Serfling, 1980), where s pk ¼ D LSD L ¼ 1 9 m 3 L þðm TÞ L þðm TÞ 3 4s 4 Case (c) For the case with m ¼ T, ffi n ð ^C pk C pk Þ¼ n jx mj L n ðs n 1 s Þ ðs þ S n 1 Þ 3S n 1 Since n ðx m S n 1 s Þ converges to ðv W ÞNðð0 0Þ SÞ in istribution (Serfling, 1980), an ð 1=ð3S n 1 Þ L =ððs þ S n 1 ÞS n 1 Þ converges to ð 1=ðÞ L =ð6s 3 ÞÞ in probability (Serfling, 1980), then, we have n ð ^C pk C pkþ converges to WL ¼ jvj=ðþ ð L=ð6s 3 ÞÞW in istribution (Serfling, 1980), where ðv W Þ Nðð0 0Þ SÞ with variance covariance matrix S ¼ s m 3 ffi Therefore, if USL T > T LSL, then n ð ^C pk C pkþ converges to the following in istribution. S n 1 (a) Nð0 s pk1þ,ifm > T, (b) Nð0 s pkþ,ifm < T, (c) jvj=ðþ ð L =ð6s 3 ÞÞW,ifm¼T,

8 INTERVAL ESTIMATION OF C pk 653 where s 3 L ðm TÞ L ðm TÞ 3 4s 4 s pk ¼ 1 9 m 3 L þðm TÞ L þðm TÞ 3 4s 4 ðv WÞ Nðð0 0Þ SÞ with variance covariance matrix S ¼ s m 3 (II) If USL T < T LSL, then Cpk ¼ð U jm TjÞ=ðÞ an ^C pk ¼ð U jx TjÞ=ð3S n 1 Þ. Applying the same techniques use in (I) with an g U1 ðx yþ ¼ U ðx TÞ 3 p ffiffi y for m > T g U ðx yþ ¼ U þðx p TÞ 3 ffiffi for m < T y As n!1 WU ¼ jvj U 6s 3 W for m ¼ T ffi n ð ^C pk Nð0 C pkþ converges to ( s pk1 Nð0 s pk Þ for m > T in istribution Þ for m < T Therefore, if USL T < T LSL, then as n!1, n ð ^C pk C pkþ converges to the following in istribution. (a) Nð0 s pk1þ,ifm > T, (b) Nð0 s pkþ,ifm < T, (c) jvj=ðþ ð U =6s 3 ÞW,ifm¼T, where s 3 U ðm TÞ U ðm TÞ 3 4s 4 s pk ¼ 1 9 m 3 U þðm TÞ U þðm TÞ 3 4s 4 ðv WÞ Nðð0 0Þ SÞ with variance covariance matrix S ¼ s m 3 (III) If USL T ¼ T LSL, then Cpk ¼ð jm mjþ=ðþ, an ^C pk ¼ ð jx mjþ=ð3s n 1 Þ Again, we apply the same technique with ðx mþ g M1 ðx yþ ¼ 3 ffiffi þðx mþ p for m > m g M ðx yþ ¼ p y 3 ffiffi for m < m y

9 654 G. H. LIN AND W. L. PEARN an Then as n!1 WM ¼ jvj ðþ 6s 3 W for m ¼ m n ð ^C pk Nð0 C pkþ converges to ( s pk1 Nð0 s pk Þ for m > m in istribution Þ for m < m Therefore, if USL T ¼ T LSL, then as n!1, n ð ^C pk C pkþ converges to the following in istribution. (a) Nð0 s pk1þ,ifm > T, (b) Nð0 s pkþ,ifm < T, (c) jvj= ð=6s 3 ÞW, ifm ¼ T, where s 3 ðm TÞ ðm TÞ 3 4s 4 s pk ¼ 1 9 m 3 þðm TÞ þðm TÞ 3 4s 4 ðv WÞ Nðð0 0Þ SÞ with variance covariance matrix S ¼ s m 3