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Journal of Nonparametric Statistics
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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 Industrial Engineering and Management , National Chiao Tung University , Taiwan, ROC
Published online: 27 Oct 2010.
To cite this article: G. H. Lin & W. L. Pearn (2002) A note on the interval estimation of c_{pk} with asymmetric tolerances, Journal of Nonparametric Statistics, 14:6, 647-654, DOI: 10.1080/10485250215318
To link to this article: http://dx.doi.org/10.1080/10485250215318
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A NOTE ON THE INTERVAL ESTIMATION OF C
pkWITH ASYMMETRIC TOLERANCES
G. H. LINa,* and W. L. PEARNb
a
Department of Communication Engineering, National Penghu Institute of Technology, Penghu, Taiwan, ROC;bDepartment of Industrial Engineering and Management, National Chiao Tung
University, Taiwan, ROC
(Received May 2001; In final form February 2002)
Pearn and Chen (1998) proposed a generalization of the widely used process capability index (PCI) Cpkto handle
processes with asymmetric tolerances. They investigated the sampling distribution and obtained the exact formulae for the expected value and variance of its natural estimator. Recently, Pearn and Lin (2000) considered a different estimator under different process condition, and investigated the statistical properties of the new estimator. However, their efforts focused on the small sample properties under the normality assumption. In this paper, we investigate the large sample properties of its natural estimator under the general condition. Based on the limiting distribution of the new estimator, we provide an approximate 100ð1 aÞ% confidence interval of the considered PCI. The obtained confidence interval provides great benefit to quality engineers on monitoring the process and assessing process performance.
Keywords: Asymmetric tolerances; Process capability index
1 INTRODUCTION
Process capability index Cpk(Kane, 1986) has been widely used in the manufacturing
indus-try to provide numerical measures of process potential and performance. As noted by many quality control researchers and practitioners, Cpkis yield-based and is independent of the
tar-get T, which fails to account for process centering with symmetric tolerances, has an even greater problem with asymmetric tolerances. To overcome the problem, Pearn and Chen (1998) considered a generalization of Cpk, referred to as Cpk, which is defined as:
Cpk ¼d
A
3s ; ð1Þ
where d¼minfd
U; dLg, A¼maxfdðm T Þ=dU, dðT mÞ=dLg, dU¼USL T is the
right hand side tolerance, and dL¼T LSL is the left hand side tolerance. Clearly, if
T ¼ m (symmetric case), then d¼d; A ¼ jm mj and the generalization C
pk reduces to
the original index Cpk. The factors d and A ensure that the generalization Cpk obtains its
* Corresponding author.
ISSN 1048-5252 print; ISSN 1029-0311 online # 2002 Taylor & Francis Ltd DOI: 10.1080/1048525021000002109
maximal value at T (process is on-target) regardless of whether the tolerances are symmetric or asymmetric.
2 THE NATURAL ESTIMATOR OF Cpk
The natural estimator ^CC
pk of Cpk can be obtained by replacing m and s by X ¼
Pn i¼1Xi=n
and Sn1¼ fPni¼1ðXiX Þ2=ðn 1Þg1=2 respectively in expression (1).
^ C Cpk ¼d ^AA 3Sn1 ; ð2Þ where ^AA ¼maxfdðX T Þ=d
U; dðT X Þ=dLg. Under the normality assumption, Pearn
and Chen (1998) showed that the rth moment (about 0) of ^CC pk is: Eð ^CCrpkÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi n 1 p 3 E ðn 1ÞS2 n1 s2 r=2Xr j¼0 r j d s j d ffiffiffi n p rj E max Z dU ; Z dL rj
Hence, the first two moments of ^CC
pk are (Pearn and Chen, 1998):
E ^CCpk¼ 1 bn1 ( Cpk 1 6 d dU þd dL ffiffiffiffiffiffi 2 np r exp d 2 2 : FðjdjÞ 3 d dU d dL d ffiffiffi n p 2A s ) ; Eð ^CCpkÞ2¼ n 3 n 1 ( ðCpkÞ2þ FðjdjÞ 3 d 3s 4A s þ 1 18n d dU 2 þ d dL 2 " # : 1 3 d 3s d dU þd dL ffiffiffiffiffiffi 2 np r exp d 2 2 þ FðjdjÞ 3 d2 3n d dU 2 þ d dL 2 " # FðjdjÞ 3 d 3s 2d n d dU d dL 2 3 FðjdjÞ 3 A s 2 þ1 9n d dU 2 d dL 2 " # dffiffiffiffiffiffi 2p p exp d 2 2 þ d 2jdjð1 2FðjdjÞÞ ) ; where d ¼pffiffiffinðm T Þ=s, bn1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=ðn 1Þ p fG½ðn 1Þ=2=G½ðn 2Þ=2g.
648 G. H. LIN AND W. L. PEARN
3 LARGE SAMPLE PROPERTIES OF ^CC pk
Pearn and Chen (1998) succeeded in obtaining the moments of ^CC
pk. Their investigation
focused on processes with normal distribution. Under general conditions, the asymptotic behavior of ^CC
pk is desirable. Let X1; X2;. . . ; Xnbe a random sample of measurements from
a process which has distribution G with mean m and variance s2. We note that ^CC
pk is a
con-tinuous function of ðX ; S2
n1Þ, and ðX ; Sn12 Þ converges to ðm; s2Þ in probability. Therefore,
^ C C
pk is a consistent estimator of Cpk.
THEOREM1 Let X1; X2;. . . ; Xnbe a random sample of measurements from a process whose
fourth central moment m4 exists and LSL m USL, then as n ! 1;pffiffiffinð ^CC pkCpk Þ
converges to the following in distribution. (a) N ð0; s2 pk1Þ; if m > T , (b) N ð0; s2 pk2Þ; if m < T , and (c) jV j=ð3sÞ ½d=ð6s3ÞW ; if m ¼ T , where s2pk1¼1 9þ m3 3s3 dL ðm T Þ 3s þm4s 4 4s4 dL ðm T Þ 3s 2 ; if USL T > T LSL; s2pk1¼1 9þ m3 3s3 d ðm T Þ 3s þm4s 4 4s4 d ðm T Þ 3s 2 ; if USL T ¼ T LSL; s2pk1¼1 9þ m3 3s3 dU ðm T Þ 3s þm4s 4 4s4 dU ðm T Þ 3s 2 ; if USL T < T LSL; s2pk2¼1 9 m3 3s3 dLþ ðm T Þ 3s þm4s 4 4s4 dLþ ðm T Þ 3s 2 ; if USL T > T LSL; s2pk2¼1 9 m3 3s3 d þ ðm T Þ 3s þm4s 4 4s4 d þ ðm T Þ 3s 2 ; if USL T ¼ T LSL; s2 pk2¼ 1 9 m3 3s3 dUþ ðm T Þ 3s þm4s 4 4s4 dUþ ðm T Þ 3s 2 ; if USL T < T LSL;
ðV ; W Þ N ðð0; 0Þ; SÞ with variance–covariance matrix S ¼ s
2 m
3
m3 m4s4
. Proof See Appendix.
THEOREM2 CC^
pk is asymptotically unbiased.
Proof From Theorem 1, we know that as n ! 1, Efpffiffiffinð ^CC
pkCpk Þg !0, if m 6¼ T , and Efpffiffiffinð ^CC pkCpkÞg ! ffiffiffiffiffiffiffiffi 2=p p
=3, if m ¼ T , since EfjV jg ¼ spffiffiffiffiffiffiffiffiffiffiffið2=pÞ. Therefore, as n ! 1, then Eð ^CC pkCpkÞ ¼ ð1= ffiffiffi n p ÞEfpffiffiffinð ^CC
pkCpk Þg !0 implies that the estimator ^CCpk is
asymptotically unbiased.
THEOREM3 If the process characteristic follows a normal distribution N ðm; s2Þ; then ^CC pkis
asymptotically efficient, if m 6¼ T .
Proof From Theorem 1, we know that if m 6¼ T , then as n ! 1,pffiffiffinð ^CC
pkCpkÞconverges
in distribution to N ð0; s2pk1Þ, if m > T , and N ð0; s2pk2Þ, if m < T . Under normality assumption,
m3¼0, m4¼3s4 implies that
ffiffiffi n p
ð ^CC
pkCpkÞconverges in distribution to N ð0; s2pknÞ, where
s2
pkn¼ ð1=9Þ þ ðC2pk=2Þ. The information matrix is
IðyÞ ¼ Iðm; sÞ ¼ 1=s
2 0
0 1=ð2s4Þ
: Since the Cramer-Rao lower bound
qC pk qm qC pk qs2 I1ðyÞ n qC pk qm qC pk qs2 2 6 6 4 3 7 7 5¼ s2 pkn n is achieved, therefore ^CC pk is asymptotically efficient. Since ^CC
pk is asymptotically efficient for m 6¼ T, defining M3¼nm3½ðn 1Þðn 2Þ1,
and M4¼ ½nðn22n þ 3Þm43nð2n 3Þm22 ½ðn 1Þðn 2Þðn 3Þ
1, where m k¼
Pn
j¼1 ðXjX Þk=n; k ¼ 2; 3; 4, we can show that M3; M4 are unbiased estimators of
m3;m4, respectively. If m > T , an estimator for s2
pk1 is obtained as: ^s s2pk1¼1 9þ M3 3S3 n1 ^ C Cpk1þ M4S 4 n1 4S4 n1 ^ C C2pk1; ð3Þ where ^CC pk1¼ ½d ðX T Þ=ð3S
n1Þ. Using this estimator, an approximate 100ð1 aÞ%
one-sided confidence interval of C
pk can be constructed as:
^ C Cpk1s^s pk1 ffiffiffi n p za; 1 ; ð4Þ
where za represents the upper ath quantile of the standard normal distribution.
Similarly, if m < T , an estimator for s2
pk2is obtained as: ^s s2 pk2¼ 1 9 M3 3S3 n1 ^ C Cpk2þ M4S 4 n1 4S4 n1 ^ C C2pk2; ð5Þ where ^CC
pk2¼ ½dþ ðX T Þ=ð3Sn1Þ. Using the estimator and its limiting distribution, an
approximate 100ð1 aÞ% one-sided confidence interval of C
pk can be constructed as:
^ C Cpk2 s^s pk2 ffiffiffi n p za; 1 : ð6Þ 4 AN EXAMPLE
Consider the following example taken from a manufacturer and supplier in Taiwan exporting high-end audio speaker components including rubber edge, Pulux edge, Kevlar cone, honey-comb and many others. The production specifications for a particular model of Pulux edge are ðLSL; T ; USLÞ ¼ ð5:650; 5:835; 5:950Þ. A total of 90 observations were collected which are displayed in Table I. If the true C
pk value fell into expressions (4) or (6), we
conclude that the process is capable, otherwise, the process is incapable under the given confidence level.
650 G. H. LIN AND W. L. PEARN
We note that the mid-point m ¼ ðUSL þ LSLÞ=2 ¼ 5:800 and the half length of the speci-fication interval d ¼ ðUSL LSLÞ=2 ¼ 0:150, d ¼minðUSL T ; T LSLÞ ¼ 0:125. The
sample mean x ¼P90i¼1xi=90 ¼ 5:83033, sample standard deviation sn1¼ f
P90
i¼1ðxixÞ2=
ð90 1Þg1=2¼0:02334, ^A¼0:00290. m2¼5:39 104, m3¼2:49 106, m4¼7:64 107,
M2¼5:45 104, M3¼2:87 108 and M4¼7:79 107. Since the sample mean is less
then the target value, ^CC
pk2¼ ½ðUSL T Þþðx T Þ=ð3sn1Þ ¼1:57563. Apply the expression
(5), we obtained ^ss
pk2¼1:11739. From the expression (6), an approximate 95% one-sided
confidence bound of C
pk is ½1:38; 1Þ. A process with C
pk< 1:00 is called ‘‘inadequate’’,
a process with 1:00 C
pk< 1:33 is called ‘‘capable’’, a process with 1:33 Cpk< 1:50 is
called ‘‘satisfactory’’, a process with 1:50 C
pk< 2:00 is called ‘‘excellent’’. In this
exam-ple, we have 95% confidence to claim that the process is satisfactory. Acknowledgement
The research was partially supported by National Science Council of the Republic of China. (NSC-90-2218-E-346-001)
References
Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18, 41–52.
Pearn, W. L. and Chen, K. S. (1998). A new generalization of process capability index Cpk. Journal of Applied
Statistics, 25, 79–88.
Pearn, W. L. and Lin, G. H. (2000). Estimating capability index Cpkfor processes with asymmetric tolerances.
Communications in Statistics-Theory and Methods, 29, 2593–2604.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley and Sons, New York, pp. 1–125.
APPENDIX Proof of Theorem 1
(I) If USL T > T LSL, then C
pk¼ ðdL jm T jÞ=ð3sÞ and CC^pk¼
ðdL jX T jÞ=ð3Sn1Þ.
Case (a) We first consider the case with m > T . We define the function gL1ðx; yÞ ¼
ðdL ðx T ÞÞ=ð3 ffiffiffiy
p
Þ, where x > T and y > 0. Note that gL1 is a real-valued and
differ-entiable function for x > T and y > 0, with qgL1 qx ðm;s2Þ ¼ 1 3s; and qgL1 qy ðm;s2Þ ¼ 1 2s2 dL ðm T Þ 3s :
TABLE I Collected Sample Data (90 Observations).
5.88 5.83 5.84 5.80 5.89 5.81 5.84 5.83 5.82 5.83 5.81 5.82 5.85 5.81 5.81 5.81 5.84 5.82 5.80 5.84 5.86 5.87 5.82 5.87 5.80 5.81 5.85 5.84 5.83 5.86 5.81 5.81 5.82 5.83 5.85 5.80 5.86 5.82 5.86 5.83 5.80 5.77 5.82 5.85 5.84 5.82 5.85 5.81 5.86 5.79 5.84 5.83 5.80 5.83 5.81 5.83 5.81 5.85 5.83 5.88 5.82 5.87 5.80 5.82 5.83 5.81 5.84 5.79 5.85 5.85 5.84 5.84 5.80 5.82 5.84 5.85 5.86 5.81 5.81 5.85 5.86 5.81 5.81 5.83 5.85 5.85 5.82 5.83 5.86 5.81
Define DL1¼ qgqxL1 ðm;s2Þ;qgL1 qy ðm;s2Þ . Note that DL16¼ ð0; 0Þ. Then, we have ffiffiffi n p ð ^CCpkCpkÞ ¼pnffiffiffifgL1ðX ; Sn12 Þ gL1ðm; s2Þg converges to N ð0; s2
pk1Þin distribution (Serfling, 1980), where
s2pk1¼DL1SDL1¼ 1 9þ m3 3s3 dL ðm T Þ 3s þm4s 4 4s4 dL ðm T Þ 3s 2 :
Case (b) For the case with m < T , we define the function gL2ðx; yÞ ¼
dLþ ðx T Þ
3 ffiffiffiyp ;
where x < T and y > 0. Note that gL2is also a real-valued and differentiable function for all
x < T and y > 0, with qgL2 qx ðm;s2Þ ¼ 1 3s; and qgL2 qy ðm;s2Þ ¼ 1 2s2 dLþ ðm T Þ 3s : Define DL2¼ qgqxL2 ðm;s2Þ;qgL2 qy ðm;s2Þ . Note that DL26¼ ð0; 0Þ. Then, we have ffiffiffi n p ð ^CCpk CpkÞ ¼pffiffiffin fgL2ðX ; Sn12 Þ gL2ðm; s2Þgconverges to N ð0; s2pk2Þ
in distribution (Serfling, 1980), where
s2pk2¼DL2SDL2¼ 1 9 m3 3s3 dLþ ðm T Þ 3s þm4s 4 4s4 dLþ ðm T Þ 3s 2 :
Case (c) For the case with m ¼ T , ffiffiffi n p ð ^CCpk CpkÞ ¼ ffiffiffi n p jX mj 3Sn1 dL 3sðs þ Sn1Þ ffiffiffi n p ðS2 n1s2Þ Sn1 :
Since pffiffiffinðX m; S2n1s2Þ converges to ðV ; W Þ N ðð0; 0Þ; SÞ in distribution (Serfling,
1980), and ð1=ð3Sn1Þ; dL=ð3sðs þ Sn1ÞSn1Þ converges to ð1=ð3sÞ; dL=ð6s3ÞÞ in
probability (Serfling, 1980), then, we have pffiffiffinð ^CC
pkCpkÞ converges to
WL¼ jV j=ð3sÞ ðdL=ð6s3ÞÞW in distribution (Serfling, 1980), where ðV ; W Þ
N ðð0; 0Þ; SÞ with variance–covariance matrix S ¼ s2 m3
m3 m4s4
. Therefore, if USL T > T LSL, then pffiffiffinð ^CC
pkCpkÞ converges to the following in
distribution.
(a) N ð0; s2pk1Þ, if m > T , (b) N ð0; s2pk2Þ, if m < T ,
(c) jV j=ð3sÞ ðdL=ð6s3ÞÞW , if m ¼ T ,
652 G. H. LIN AND W. L. PEARN
where s2 pk1¼ 1 9þ m3 3s3 dL ðm T Þ 3s þm4s 4 4s4 dL ðm T Þ 3s 2 ; s2pk2¼1 9 m3 3s3 dLþ ðm T Þ 3s þm4s 4 4s4 dLþ ðm T Þ 3s 2 ;
ðV ; W Þ N ðð0; 0Þ; SÞ with variance–covariance matrix S ¼ s
2 m
3
m3 m4s4
. (II) If USL T < T LSL, then C
pk¼ ðdU jm T jÞ=ð3sÞ and
^ C C
pk¼ ðdU jX T jÞ=ð3Sn1Þ.
Applying the same techniques used in (I) with
gU 1ðx; yÞ ¼
dU ðx T Þ
3 ffiffiffiyp ; for m > T ; gU 2ðx; yÞ ¼
dUþ ðx T Þ 3 ffiffiffiyp ; for m < T ; and WU ¼ jV j 3s dU 6s3 W ; for m ¼ T : As n ! 1; pffiffiffinð ^CCpk CpkÞconverges to N ð0; s 2 pk1Þ for m > T : N ð0; s2 pk2Þ for m < T : ( in distribution:
Therefore, if USL T < T LSL, then as n ! 1,pffiffiffinð ^CC
pkCpkÞconverges to the
follow-ing in distribution. (a) N ð0; s2pk1Þ, if m > T , (b) N ð0; s2pk2Þ, if m < T , (c) jV j=ð3sÞ ðdU=6s3ÞW , if m ¼ T , where s2pk1¼1 9þ m3 3s3 dU ðm T Þ 3s þm4s 4 4s4 dU ðm T Þ 3s 2 ; s2pk2¼1 9 m3 3s3 dUþ ðm T Þ 3s þm4s 4 4s4 dUþ ðm T Þ 3s 2 ;
ðV ; W Þ N ðð0; 0Þ; SÞ with variance–covariance matrix S ¼ s2 m3
m3 m4s4
. (III) If USL T ¼ T LSL, then C
pk ¼ ðd jm mjÞ=ð3sÞ, and CC^ pk¼
ðd jX mjÞ=ð3Sn1Þ:
Again, we apply the same technique with
gM 1ðx; yÞ ¼
d ðx mÞ
3 ffiffiffiyp ; for m > m; gM 2ðx; yÞ ¼
d þ ðx mÞ
3 ffiffiffiyp ; for m < m;
and WM ¼ jV j ð3sÞ d 6s3 W ; for m ¼ m: Then as n ! 1; pffiffiffinð ^CCpkCpkÞconverges to N ð0; s 2 pk1Þ for m > m: N ð0; s2 pk2Þ for m < m: ( in distribution:
Therefore, if USL T ¼ T LSL, then as n ! 1,pffiffiffinð ^CC pkC
pkÞconverges to the
follow-ing in distribution. (a) N ð0; s2 pk1Þ, if m > T , (b) N ð0; s2pk2Þ, if m < T , (c) jV j=3s ðd=6s3ÞW , if m ¼ T , where s2 pk1¼ 1 9þ m3 3s3 d ðm T Þ 3s þm4s 4 4s4 d ðm T Þ 3s 2 ; s2pk2¼1 9 m3 3s3 d þ ðm T Þ 3s þm4s 4 4s4 d þ ðm T Þ 3s 2 ;
ðV ; W Þ N ðð0; 0Þ; SÞ with variance–covariance matrix S ¼ s2 m3
m3 m4s4
.
654 G. H. LIN AND W. L. PEARN
