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Estimation of capability index based on bootstrap method
Microelectron. Reliab., Vol. 36, No. 9, pp. 1141-1153, 1996
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/96 $15.00+.00
Pergamon
0026-2714(95)00211-1
ESTIMATION OF CAPABILITY INDEX BASED ON BOOTSTRAP METHOD KUEY CHUNG CHOP, KYUNG HYUN NAM2 and DONG HO PARK3 ~Department of Computing Science and Statistics, Chosun University, Korea, 2Department of Mathematics and Statistics, University of Nebraska, USA and 3Department of Statistics, Hallym University, Korea (Received f o r Publication 31 October 1995)
Abstract A process capability index C'p,,k is recently proposed to measure the degree of process capability of a system. Cp~.k is a generalized version of the existing indices Cp, Cpk, and , Cpm, which have been studied extensively in the last few years. In this
paper, we present the estimators of Cv,,~k and derive its asymptotic distribution. We also utilize the asymptotic variance of the estimator of Cvmk to derive the two-sided confidence interval for Cpmk based on a percentile-t bootstrap method. Our new confidence intervals are shown to outperform the ones based on the percentile and standard bootstrap method by simulation. Copyright © 1996 Elsevier Science Ltd
Introduction To evaluate the degree of process capability of a system, it is necessary to define a quantitative measure that can explain the performance of the system. The process capability indices Cv, Cpk and Cp,, have been widely used to evaluate the degree of process capability of a system. Such indices are defined as
USL - LSL
Cp -
Cpk
Min(
60"
'
U,5'L - I~ t1 - L,~L '
- - ) , 30
(1)
(2)
and Cp,,, =
USL-
LSL
6{a 2 + (/~- T)2}~" 1141
(3)
K. C. Choi et al.
1142
In the definitions of C,, Cpk, and Cpm, U S L and L S L denote the upper and lower specification limits for the value of characteristic X, to be measured, # and a denote the process mean and standard deviation, respectively, and T is the target value of the process. Boyles [1] gives the detailed numerical and graphical investigation of the relationship between Cp and Cpm and also presents a comprehensive discussion of the relationship between Cpm and Cpk. From the definitions of Cp, Cpk and Cpm, it is noted that Cpk is obtained from Cp by modifying the numerator and C~m is obtained from Cp by modifying the denominator. Combining these two modifications, Pearn, Kotz and Johnson [17] introduce a new index C1,mk as
min(USL
c~m~ =
- I~, # - L S L )
3 { ~ + (~ - T)~}~
(4)
The expression (4) can be rewritten as
Cpmk ~
vh +Cpk (.@)2
(5)
In the denominator of the expression (4), the term (# - T) 2 may be interpreted as an additional penalty to the process capability due to the departure of process mean from its target value. This penalty ensures that Cpmk will be more sensitive to departure of the process mean from the target value than Cpk and, in turn can be more effective to distinguish between off-target and on-target processes. It is easy to derive the following relation of Cpmk and Cp,. :
c,,,,,~ = (1 I~ dM[)c~m.
(6)
The latest index C,mk is to measure the capability of the process by generalizing both Cpk and Cpm in such a way that both target value and tile mid point of specification limits are considered simultaneously. Many authors have proposed and discussed the estimators of Cp, Cpk, Cpm and Cpmk in the literature. These estimators are obtained by replacing/~ with/~ and a by ~, where ~ and & are the sample mean and sample standard deviation, respectively.
Estimation of capability index Chan, Cheng and Spiring [2] [3] [4], Chou and Owen [6], Kotz, Pearn and Johnson [15], Boyles [1] and Pearn, Kotz and Johnson [17], among many others, have studied such estimators and its properties. All estimators suggested by these authors are known to be biased. In most process capability studies, the process measurements are assumed to come from a normal population. However, when the assumption of normality can not be made or is made incorrectly, the statistical properties and procedures formulated under the assumption of normality may be invalid. Kane [13], Gunter [11], Kocherakota, Kocherakota and Kirmani [14], and Franklin and Wasserman [8] discuss the cases when the underlying distribution is nonnormal. Franklin and Wasserman [9] [10] suggest several bootstrap methodologies in the interval estimations of Cp, Cpk, and
Cpm, which allow calculations of their confidence intervals
without assumption of normality. In this paper, we first find the asymptotic variance of the estimator of Cpmk and use it to derive a confidence interval for C'm~k based on a percentile-t bootstrap method. Also we compare our percentile-t bootstrap confidence interval with percentile confidence intervals and standard confidence intervals in terms of its coverage probability and length of interval by simulation study. In Section 2, we give an asymptotic distribution of Cpmk under the condition that the fourth moment about p of the process distribution exists. In Section 3, we derive the percentile-,t bootstrap confidence intervals for Cp,,k, utilizing the asymptotic variance of Cpmk. In Section 4, we conduct a series of simulation studies to compare the confidence intervals based on three types of bootstrap methods. Our study shows that the percentile-t confidence intervals give higher coverage probabilities and shorter confidence intervals than the other bootstrap confidence intervals in general, especially when the sample sizes are relatively small.
Asymptotic Distribution of Cpmk When the process distribution is normal, many authors have obtained not only the exact moments, but also the exact distributions of the estimators C'p, Cpk, C~,~, and Cm~k. Such distributions have been studied by Chou and Owen [6], Chan, Cheng,
1143
1144
K.C. Choi et al. and Spiring [2] and many others. However, except when the process distribution is normal, little work has been done regarding the derivation of the exact distributions of such estimators. That is mainly due to the fact that such derivations are not analytically tractable for the most nonnormal parent distributions. Consequently, the statistical inferences on the indices Cp, Cpk, Cpm, and Cp,~k are somewhat restricted to the situations when the process distribution is normal. Chan, Xiong, and Zhang [5] derive the asymptotic distributions of v~(6'p Cp), x/-n(Cpk - Cpk), and V~.(Cp,~ - Cp,~) under the condition that the fourth moment about # of the process distribution exists. Such asymptotic distributions are used to make some approximate statistical inferences on those capability indices when a large sample is available. To our knowledge, the asymptotic properties of C~,~k have not been discussed in the literature. In this section, we derive the asymptotic distribution of x/'n(Cp,,k - Cp,~k), assuming that the fourth moment about # exists. The proof is straightforward and similar to that in Chan, Xiong, and Zhang [5]. Such asymptotic results are used to derive the bootstrap confidence intervals in the following section.
Theorem 2.1
If #4 exists, then
N(O, O'prnk 2 )
Distribution of
c,,,,,k)
v'-~(d,.~ -
ifM>#
(USL- LSL){Z-2TY+2,uY) N(0, (Tpmk) *:
ifM=tz if M < # ,
(7) where
2 ~pmk = -
l
2
~[a + ( T - # ) 2 ] - 3 ( [ a 2 + ( 7 ' - # ) 2 + ( T - # ) ( d + # - M ) 1 2 a (d+#-M)#3[a 2+(T-~)2
+(T - ~)(d + ~ - M)] +
l ( d + # - M)2(#4_ 4 ) ) z~
2
(s)
Estimation of capability index
1145
and
•
%,~k
[~2 + (T -/*)2]-a([er2 + ( T -
IL)2 -- (T - ~ ) ( d - # + M)J2a 2
+
(d-#+M)#a[~r 2+(T-#)~-(T-#)(d-/t+M)]
+
~(d - / , + M ) ~ ( ~ - ~ ) ) .
Here, M =
USL ,~ LSL
(9)
is the mi(bpoint of the specification limits, (Y, Z)
N((0,0),:~), m = E ( X - ~)', N =
and
denotes "eonw~r-
#a /14 - ~4 gence in distribution" as n --* oe.
Proof. The proof requires a slight modification of ones given in Chan, Xiong, and Zhang [5]. For detailed proofs, see Nam [16].
Percentile-t Bootstrap Confidence Intervals Franklin and W~serman [8] [9] [10] propose three different types of bootstrap confidence intervals for 6'p, Cpk, and Cm,,. The method of bootstrap was introduced by Efron [7] as a nonparametric and computer intensive method. Such method allows calculation of confidence intervals for several parameters of interest without the usual assumption of normality. Danklin and Wasserman [10] recommend to use the standard method because of its robustness, high coverage, and short length of confidence intervals. Hall [12] provides excellent arguments explaining which type of bootstrap confidence interval should be used. He examines seven different bootstrap methods in terms of length and coverage probability of confidence intervals and asserts that percentile-t method is one of the best techniques to establish confidence intervals for the parameters, lie warns that the misuse of standard method instead of percentile4 is analogous to mistakenly looking up normal tables instead of Student's t tables in problems of inferences about a normal mean. Motivated by such observations, we employ Hall's [12] percentile4 method to construct bootstrap confidence intervals for Cp,,k in this section.
1146
K.C. Choiet al. To discuss the estimator of Cp,,k, we assume thai, a random sample X 1 ,
X2,.
• - ,
X n
of the characteristic is available and we denote the bootstrap sample by X~, X~, • - •, X ~ There are three cases of confidence intervals for Cpmk. For the case of M > #, confidence interval for Cp,,,k is
^
1^
^
^
1^
^
( Cpmk -- m - r O'pmk Vpmk,( l +c~)/2, Cpmk -- m - ~ O'pmk Vpmk,(1-cO /2 ) ,
(10)
d - I X - MI G,~k = 3(~(~,~=,(xl - T) 2 + n(32 - T)2) 1/2
(11)
where
and
l
^2 O'pmk
[
s
+ (T - 32)21-3([,s2 + ( T - 32) 2 + ( T - 32)(d + 32 - M ) 1 2 s 2
(a + 32 - M)fi3[s 2 + (T - 2 ) 2 + (T - 32)(d + 32 - M)] +
(12)
l ( d + X - M)2(/~4 - (r4)).
Here ~)p,,~k,~is defined as
P ( v f ~ ( - m k ^.
<_ ~p,,,k,~,) = a,
-
(13)
O'pmk
where C~.~k and &~.~k are the resample version of Cp,,,k and &v,~k, respectively. For the case of M < #, we use the different version of &p.,k value such that
1 2 ~[s + (T - )()2]-a([s2 + ( T - 2 ) 2 - ( T - X ) ( d - X + M ) 1 % 2
^2
O'pmk
( d - X + M)fi3[s 2 + ( T - 2 ) 2 - ( T - X ) ( d - 32 + M)] +
( d - 2 + M)2(fi4 -
(14)
O'4)).
Note that o`p.,k^2of (12) and (14) are the sample versions of (8) and (9). For the last case of M -=- #, Cp.~k is reduced to Cpm and tiros we have the same confidence interval as for Cp,~ which is given as
I ^
^
(G,,, - ,,, ~o`~,~v~.,,(,+~,)., G,,,
I ^
^
•
.-~
(is)
Estimation of capability index
1147
where ~
d
Cpm 3 4 1 / n
Einl(Xi
-- T ) 2
and
^2
USL - LSL
s2 T
%~ = 3 6 ( ~ - 7 ( ~ - - Y ) @
(
1
- 2)~ - (T - X ) ~ + 7 ( / ~ -
¢)).
Here bp~,~ satisfies (
^.
_< bpm,~) = a,
(16)
O'pm
where C ~ and O'~m are obtained from the bootstrap resample. The new percentile-t bootstrap confidence intervals will be compared with the percentile and standard confidence intervals, based on Monte Carlo experiments.
Simulation
Study
To compare the performance of the percentile-t method with the percentile bootstrap method and standard method for constructing confidence interval for Cpmk, we choose the following values : U S L = 6 1 , L S L = 40, T = 49,# = 50, and a = 2. These values yield @ink = 1.491, which represents the "very capable" process. In this section, we compare our percentile-t confidence interval with percentile confidence interval and standard confidence interval by conducting simulation studies for various underlying process distributions. Percentile bootstrap confidence interval is obtained by
(17)
where C~,~k(z ^ * ") is the i-th smallest bootstrap estimate for the capability index Cp,~k e,nd B is the number of bootstrap resamples. Note that here we use a as a confidence coefficient. For example, a 90% confidence interval with/5' = 1000, the interval given in (17) would be approximately
(C~mk(51 ), C'~mk(950))-
K. C. Choiet ah
1148
Here we can use Cpmk(51 ) instead of ~"
¢;.,~(5o) to make this interval cover exactly
90% of 1000 estimates. The a-level standard confidence interval for the capability index Cpmk is given by
(¢,.k +
z~_<, d~,~+ z~<),
(18)
where Z~ is the lOOa percentile point of a standard normal distribution and S~ is the standard deviation of the C~mkli)'s, i = 1, 2 , . . . , B. 6'pink denotes the estimator of Cpmk based on the original sample, which is given in (11). We choose the resample size m = 5, 10, or 15( for the original sample size n = 5),m = 10,20, or 3 0 ( f o r n = 10), a n d r e : 3 0 , 4 0 , or 5 0 ( f o r n = 3 0 ) . B = 1000 bootstrap resamples (each of size m) were drawn from each sample of size n and a 90% bootstrap confidence interval was constructed for each method. It was then determined if the calculated confidence interval of each type contains the true index value and also the length of each confidence interval is evaluated. This single simulation was then replicated N = 1000 times and thus a percentage of times the true value of the index is within the calculated interval could be calculated as well as an average length, and a standard error of the average length of the 90% confidence intervals. We use the following process distributions, each of which is adjusted in order to have the mean # = 50 and standard deviation cr = 2 : Normal, Student's t with 5 and 6 degrees of freedom ( heavy tailed distributions), and Chi-squared with 4 and 5 degrees of freedom ( highly skewed distributions). The reason why these heavy tailed and highly skewed distributions are chosen is that they are troublesome with normal assumption and also they are frequently happen to be in the field as Gunter [11] noticed. In order to make each distribution to have the specified values of # = 50, and a = 2, we transform Student's t and Chi-squared distribution as follows.
X ~ ts
==* v ~ x + 5 0 , ? - - -
X~to
~
18X+50, X
4
Estimation of capability index
1149
and
x ~ xl ~
V~gx + 5 o - v5-~.
In the simulation, especially when n is very small, it happens to have a negative estimate of #4 - a 4 which is undesirable because it produces a negative variance term. To avoid such undesirable situations, we draw B resamples until all those B "2 estimates have positive value of ap,~k.
The simulation results are tabulated in Tables 1-5.
In a normally distributed
process, the percentile-t method provides higher coverage probabilities and shorter confidence intervals. When the original sample size n is very small, it turns out that the percentile-t method has remarkably higher coverage probabilities and shorter length of intervals than other two intervals. As the sample size n increases, differences of coverage probabilities and lengths of intervals among three are somewhat reduced. It is observed that as m increases, it seems that each interval's coverage as well as the mean and the standard error of interval lengths are decreasing.
Table 1: Coverage Probability, Mean Width and Standard Error of the Width for the 90% Bootstrap Confidence Intervals-Normal Process n=5 n=10 n=30 5 10 15 l0 20 30 30 40 50 Coverage .746 .595 .511 .814 . 6 6 8 . 5 7 0 . 8 4 5 . 7 7 2 .737 PB Mean 2.193 1.164 . 8 0 5 1.141 . 7 3 3 . 5 7 2 . 5 7 9 . 4 8 8 .435 S.E. .063 .027 .014 .015 . 0 0 9 . 0 0 6 . 0 0 4 . 0 0 3 .003 Coverage . 8 6 4 . 6 7 4 .554 .883 .711 . 5 9 7 .881 .801 .766 SB Mean 2.690 1.279 . 9 1 0 1.230 . 7 5 7 . 5 9 0 . 5 9 0 . 4 9 5 .441 S.E. .095 .033 .021 .018 . 0 0 9 . 0 0 7 . 0 0 4 . 0 0 4 .003 Coverage .877 . 6 3 7 .501 .890 . 7 0 2 . 5 7 3 . 8 9 7 . 8 3 2 .790 PTB Mean 2.096 1.086 . 7 2 7 1.150 . 7 3 7 .556 . 6 0 4 . 5 0 2 .445 S.E .055 .021 .011 .016 . 0 1 0 . 0 0 6 . 0 0 5 . 0 0 4 .004
Table 2: Coverage Probability, Mean Width and Standard Error of the Width for the 90% Bootstra Confidence Intervals--t5 Process n=5 n=lO n=30 5 10 15 10 20 30 30 40 50 Coverage .678 .558 .456 .725 . 6 0 0 .521 . 8 0 5 . 7 4 5 .680 PB Mean 2.528 1.294 .877 1.382 .861 . 6 3 4 . 7 3 5 . 6 2 8 .550 S.E. .073 .028 .014 .021 .01l . 0 0 7 . 0 0 7 . 0 0 6 .005 Coverage .816 .649 .519 .832 .651 . 5 3 3 . 8 4 4 . 7 8 4 .714 SB Mean 3.005 1.412 1.000 1.507 . 8 8 6 . 6 5 7 . 7 5 2 . 6 3 8 .556 S.E. .095 .034 .021 .025 . 0 1 2 . 0 0 7 . 0 0 7 . 0 0 6 .005 Coverage .805 .616 .431 .824 . 6 4 2 . 5 2 3 . 8 6 6 .811 .740 PTB Mean 2.390 1.185 .796 1.425 . 8 8 8 .621 . 8 4 5 . 7 0 8 .610 S.E. '.088 .023 .012 .023 .013 . 0 0 6 . 0 1 2 . 0 1 0 .009
1150
K.C. Choi et al. Table 3: Covera e Probability, Mean Width and Standard Error of the Width for the 90% Bootstra Confidence Intervals t6 Process n=5 n=10 n=30 5 10 15 10 20 30 30 40 50 .693 .539 .461 .771 .617 .523 .824 .762 .699 Coverage 2.429 1.313 .896 1.304 .842 .629 .698 .599 .525 PB Mean .070 .028 .015 .018 .Oil .007 .006 .005 .005 S.E. .861 .634 .5/9 .867 .655 .546 .850 .787 .72] Coverage 3.409 1.433 1.018 1.411 . 8 6 4 . 6 5 6 .714 .609 .531 SB Mean S.E. .188 .035 .021 .021 .011 .008 .006 .005 .005 Coverage .828 .614 .444 .859 .662 .526 .874 .809 .733 PTB Mean 2.263 1.202 .807 1.363 .869 .616 .788 . 6 6 8 .576 .052 .021 .013 .021 . 0 1 3 ,007 .010 .009 .008 S.E.
Table 4: Coverage Probability, Mean Width and Standard Error of the Width for the 900£ Bootstrap Confidence Intervals )/~ Process n=5 n=10 n=30 5 10 15 10 20 30 30 40 50 Coverage .597 .472 .397 .696 .549 .463 .761 .734 .670 PB Mean 3.389 1.842 1.010 1.844 1.121 .777 .933 .763 :674 S.E. .121 .063 .017 .029 .017 .009 .009 .007 .006 Coverage .805 .595 .476 .816 .615 .524 .829 .781 .704 SB Mean 4.437 1.981 1.218 2.045 1.161 .838 .967 .784 .688 S.E. .242 .068 .026 .034 .017 .010 .010 .007 .006 Coverage .825 .618 .417 .854 .658 .510 .880 .832 .768 PTB Mean 4.044 1.872 1.006 2.060 1.238 .761 1.081 .873 .746 S.E. .118 .047 .015 .033 .020 .008 .015 .011 .010
Table 5: Coverage Probability, Mean Width and Standard Error of the Width for the 90% Bootstrap Confidence Intervals X~ Process n=5 n=10 n=30 5 10 15 10 20 40 50 30 30 Coverage .571 .466 .388 .640 .555 .479 .762 .708 .679 PB Mean 3.245 1.708 .989 1.654 1.035 .751 .870 .733 .637 S.E. .127 .051 .018 .023 .016 .008 .007 .007 .006 Coverage .798 .568 .459 .777 .633 .537 .836 .751 .717 SB Mean 4 . 0 5 1 1.839 1.193 1.824 1.075 .801 .898 .752 .649 S.E. .162 .056 .029 .027 ,016 .010 .OO8 .007 .006 Coverage .813 .603 .412 .808 .677 .531 .874 .820 .771 PTB Mean 3.935 1.769 .981 1.894 1.141 .751 .997 .839 .702 S.E. .155 .040 .015 .028 .019 .009 .012 .012 .009
In a process with a Student's t-distribution, tile percentile bootstrap confidence interval has the lower coverage probabilities and provide the longer lengths than the pereentile-t bootstrap confidence interval when n is small. The magnitude of the standard error for the length of intervals based on the standard method is greater than its counterparts. The lengths of confidence intervals by the standard method are slightly greater than the other confidence intervals and the coverage probabilities are higher for the standard method than those for the percentile-t and percentile
Estimation of capability index method. As for the effect of the degree of freedom, we observe that when the sample size n is equal to 5, the coverage probabilities of confidence intervals are increasing in the all three cases as the degrees of freedom increase. But it is not the case for the mean lengths of confidence intervals. When the sample size changes to 10, then coverage probabilities are increasing and lengths of intervals are decreasing in all three meethods as the degrees of freedom increase. When a proces is highly skewed such as X2 distribution with 4 degrees of freedom, we find similar patterns as in other distributions. Percentile-t method produces a confidence interval with higher coverage probability and shorter length than other methods in general. For a X~ distribution with 4 degrees of freedom the magnitude of standard error of confidence interval length in percentile-t type is the smallest when n is small, whereas the standard error in percentile-t method is a little larger than the others when n is large. As the sample size n increases, the coverage probabilities of all confidence intervals are increasing and the interval lengths are getting shorter as expected. We observe the similar trend for a ~ distribution with 5 degree of freedom. As the resample size m is increasing, coverage probabilities of percentile and percentile-t methods are decreasing. In conclusion, the percentile-t method is prefered to the percentile method or standard method because of its higher coverage probability and shorter length of confidence interval for most cases, especially when the sample size n is small. Therefore the bootstrap confidence interval based on the percentile-t method is recommended when the available sample size is very small due to the high price or its rarity.
References 1. R. A. Boyles, The Taguchi Capability Index, Journal of Quality Technology,
23, 107-126 (1991). 2. L. K. Chan, S. W. Cheng and F. A. Spiring, A New Measure of Process Capability : Cpm, Journal of Quality Tech.nology, 20, 162-175 (1988). 3. L. K. Chain, S. W. Cheng and P. A. Spiring, A Graphical Technique for Process Capability, Transactions ASQC Congress, Dallas, TX. 268-275 (1988).
1151
K. C. Choi et al.
1152
4. L. K. Chan, S. W. Cheng and F. A. Spiring, The Robustness of Process Capability Index Cp to Departures from Normality, In Statistical Theory and Data
Analysis, II (K. Matusita, ed.), North-Holland, Amsterdam, 223 229 (1988). 5. L. K. Chan, Z. Xiong and D. Zhang, On the Asymptotic Distributions of Some Process Capability Indices, Communications in Statistics-Theory and
Methods, 19, 11-18 (1990). 6. Y. Chou and D. B. Owen, On the Distribution of the Estimated Process Capability Indices, Communications in Statistics-Theory and Methods, 18, 45494560 (1989). 7. B. Efron, Bootstrap Methods : Another Look at the Jackknife, Annals of
Statistics, 7, 1 26 (1979). 8. L. A. Franklin and G. Wasserman, Bootstrap Confidence Interval Estimates of Cpk : An Introduction, Communications in Statistics-Simulation and Com-
putation, 20, 231 242 (1991). 9. L. A. Franklin and G. Wasserman, A Note on the Conservative Nature of the Tables of Lower Confidence Limits for Cpk with a Suggested Correction, Com-
munications in Statistics-Simulation and Computation, 21,926-932 (1992). 10. L. A. Franklin and G. Wasserman, Bootstrap Lower Confidence Limits for Capability Indices, Journal of Quality Technology, 24, 196-210 (1992). 11. B. H. Gunter, The Use and Abuse of Cpk, Parts 1-4, Quality Progress, 22(1), 72-73; 22(3), 108-109; 22(5), 79-80; 22(7), 86-87 (1989). 12. P. Hall, Theoretical Comparison of Bootstrap Confidence Intervals, The An-
nals of Statistics, 16, 927-953 (1988). 13. V. E. Kane, Process Capability Indices, Journal of Quality Technology, 18, 41-52. (Corrigenda 265.) (1986). 14. S. K. Kocherlakota, K. Kocherlakota and S. N. U. A. Kirmani, Process Capability Indices Under Non-normality, International Journal of Mathematical
Statistics, 1,175-210 (1992).
Estimation of capability index 15. S. Kotz, W. L. Pearn and N. L. Johnson, Some Process Capability Indicators Are More Reliable Than One Might Think( A Comment on Bissel(1990)),
Applied Statistics, 42, 55-62 11993). 16. K. H. Naro, Trend Changes in Failure Rate and Mean Residual Life : Its Relations and Applications, Ph.D. Dissertation, University of Nebraska, (1995). 17. W. L. Pearn, S. Kotz and N. L. Johnson, Distributional and Inferential Properties of Process Capability Indices, Journal of Quality Technology, 24,216-231 11992).
Acknowledgements
This study was supported in part by the Research Fund and Utilizing Facilities of Advanced Research Center (Environmental Science and Technology) of Chosun University, 1995 and in part by the Basic Science Research Institute Program, Ministry of Education, 1994, Project no. 1403.
1153
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/96 $15.00+.00
Pergamon
0026-2714(95)00211-1
ESTIMATION OF CAPABILITY INDEX BASED ON BOOTSTRAP METHOD KUEY CHUNG CHOP, KYUNG HYUN NAM2 and DONG HO PARK3 ~Department of Computing Science and Statistics, Chosun University, Korea, 2Department of Mathematics and Statistics, University of Nebraska, USA and 3Department of Statistics, Hallym University, Korea (Received f o r Publication 31 October 1995)
Abstract A process capability index C'p,,k is recently proposed to measure the degree of process capability of a system. Cp~.k is a generalized version of the existing indices Cp, Cpk, and , Cpm, which have been studied extensively in the last few years. In this
paper, we present the estimators of Cv,,~k and derive its asymptotic distribution. We also utilize the asymptotic variance of the estimator of Cvmk to derive the two-sided confidence interval for Cpmk based on a percentile-t bootstrap method. Our new confidence intervals are shown to outperform the ones based on the percentile and standard bootstrap method by simulation. Copyright © 1996 Elsevier Science Ltd
Introduction To evaluate the degree of process capability of a system, it is necessary to define a quantitative measure that can explain the performance of the system. The process capability indices Cv, Cpk and Cp,, have been widely used to evaluate the degree of process capability of a system. Such indices are defined as
USL - LSL
Cp -
Cpk
Min(
60"
'
U,5'L - I~ t1 - L,~L '
- - ) , 30
(1)
(2)
and Cp,,, =
USL-
LSL
6{a 2 + (/~- T)2}~" 1141
(3)
K. C. Choi et al.
1142
In the definitions of C,, Cpk, and Cpm, U S L and L S L denote the upper and lower specification limits for the value of characteristic X, to be measured, # and a denote the process mean and standard deviation, respectively, and T is the target value of the process. Boyles [1] gives the detailed numerical and graphical investigation of the relationship between Cp and Cpm and also presents a comprehensive discussion of the relationship between Cpm and Cpk. From the definitions of Cp, Cpk and Cpm, it is noted that Cpk is obtained from Cp by modifying the numerator and C~m is obtained from Cp by modifying the denominator. Combining these two modifications, Pearn, Kotz and Johnson [17] introduce a new index C1,mk as
min(USL
c~m~ =
- I~, # - L S L )
3 { ~ + (~ - T)~}~
(4)
The expression (4) can be rewritten as
Cpmk ~
vh +Cpk (.@)2
(5)
In the denominator of the expression (4), the term (# - T) 2 may be interpreted as an additional penalty to the process capability due to the departure of process mean from its target value. This penalty ensures that Cpmk will be more sensitive to departure of the process mean from the target value than Cpk and, in turn can be more effective to distinguish between off-target and on-target processes. It is easy to derive the following relation of Cpmk and Cp,. :
c,,,,,~ = (1 I~ dM[)c~m.
(6)
The latest index C,mk is to measure the capability of the process by generalizing both Cpk and Cpm in such a way that both target value and tile mid point of specification limits are considered simultaneously. Many authors have proposed and discussed the estimators of Cp, Cpk, Cpm and Cpmk in the literature. These estimators are obtained by replacing/~ with/~ and a by ~, where ~ and & are the sample mean and sample standard deviation, respectively.
Estimation of capability index Chan, Cheng and Spiring [2] [3] [4], Chou and Owen [6], Kotz, Pearn and Johnson [15], Boyles [1] and Pearn, Kotz and Johnson [17], among many others, have studied such estimators and its properties. All estimators suggested by these authors are known to be biased. In most process capability studies, the process measurements are assumed to come from a normal population. However, when the assumption of normality can not be made or is made incorrectly, the statistical properties and procedures formulated under the assumption of normality may be invalid. Kane [13], Gunter [11], Kocherakota, Kocherakota and Kirmani [14], and Franklin and Wasserman [8] discuss the cases when the underlying distribution is nonnormal. Franklin and Wasserman [9] [10] suggest several bootstrap methodologies in the interval estimations of Cp, Cpk, and
Cpm, which allow calculations of their confidence intervals
without assumption of normality. In this paper, we first find the asymptotic variance of the estimator of Cpmk and use it to derive a confidence interval for C'm~k based on a percentile-t bootstrap method. Also we compare our percentile-t bootstrap confidence interval with percentile confidence intervals and standard confidence intervals in terms of its coverage probability and length of interval by simulation study. In Section 2, we give an asymptotic distribution of Cpmk under the condition that the fourth moment about p of the process distribution exists. In Section 3, we derive the percentile-,t bootstrap confidence intervals for Cp,,k, utilizing the asymptotic variance of Cpmk. In Section 4, we conduct a series of simulation studies to compare the confidence intervals based on three types of bootstrap methods. Our study shows that the percentile-t confidence intervals give higher coverage probabilities and shorter confidence intervals than the other bootstrap confidence intervals in general, especially when the sample sizes are relatively small.
Asymptotic Distribution of Cpmk When the process distribution is normal, many authors have obtained not only the exact moments, but also the exact distributions of the estimators C'p, Cpk, C~,~, and Cm~k. Such distributions have been studied by Chou and Owen [6], Chan, Cheng,
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K.C. Choi et al. and Spiring [2] and many others. However, except when the process distribution is normal, little work has been done regarding the derivation of the exact distributions of such estimators. That is mainly due to the fact that such derivations are not analytically tractable for the most nonnormal parent distributions. Consequently, the statistical inferences on the indices Cp, Cpk, Cpm, and Cp,~k are somewhat restricted to the situations when the process distribution is normal. Chan, Xiong, and Zhang [5] derive the asymptotic distributions of v~(6'p Cp), x/-n(Cpk - Cpk), and V~.(Cp,~ - Cp,~) under the condition that the fourth moment about # of the process distribution exists. Such asymptotic distributions are used to make some approximate statistical inferences on those capability indices when a large sample is available. To our knowledge, the asymptotic properties of C~,~k have not been discussed in the literature. In this section, we derive the asymptotic distribution of x/'n(Cp,,k - Cp,~k), assuming that the fourth moment about # exists. The proof is straightforward and similar to that in Chan, Xiong, and Zhang [5]. Such asymptotic results are used to derive the bootstrap confidence intervals in the following section.
Theorem 2.1
If #4 exists, then
N(O, O'prnk 2 )
Distribution of
c,,,,,k)
v'-~(d,.~ -
ifM>#
(USL- LSL){Z-2TY+2,uY) N(0, (Tpmk) *:
ifM=tz if M < # ,
(7) where
2 ~pmk = -
l
2
~[a + ( T - # ) 2 ] - 3 ( [ a 2 + ( 7 ' - # ) 2 + ( T - # ) ( d + # - M ) 1 2 a (d+#-M)#3[a 2+(T-~)2
+(T - ~)(d + ~ - M)] +
l ( d + # - M)2(#4_ 4 ) ) z~
2
(s)
Estimation of capability index
1145
and
•
%,~k
[~2 + (T -/*)2]-a([er2 + ( T -
IL)2 -- (T - ~ ) ( d - # + M)J2a 2
+
(d-#+M)#a[~r 2+(T-#)~-(T-#)(d-/t+M)]
+
~(d - / , + M ) ~ ( ~ - ~ ) ) .
Here, M =
USL ,~ LSL
(9)
is the mi(bpoint of the specification limits, (Y, Z)
N((0,0),:~), m = E ( X - ~)', N =
and
denotes "eonw~r-
#a /14 - ~4 gence in distribution" as n --* oe.
Proof. The proof requires a slight modification of ones given in Chan, Xiong, and Zhang [5]. For detailed proofs, see Nam [16].
Percentile-t Bootstrap Confidence Intervals Franklin and W~serman [8] [9] [10] propose three different types of bootstrap confidence intervals for 6'p, Cpk, and Cm,,. The method of bootstrap was introduced by Efron [7] as a nonparametric and computer intensive method. Such method allows calculation of confidence intervals for several parameters of interest without the usual assumption of normality. Danklin and Wasserman [10] recommend to use the standard method because of its robustness, high coverage, and short length of confidence intervals. Hall [12] provides excellent arguments explaining which type of bootstrap confidence interval should be used. He examines seven different bootstrap methods in terms of length and coverage probability of confidence intervals and asserts that percentile-t method is one of the best techniques to establish confidence intervals for the parameters, lie warns that the misuse of standard method instead of percentile4 is analogous to mistakenly looking up normal tables instead of Student's t tables in problems of inferences about a normal mean. Motivated by such observations, we employ Hall's [12] percentile4 method to construct bootstrap confidence intervals for Cp,,k in this section.
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K.C. Choiet al. To discuss the estimator of Cp,,k, we assume thai, a random sample X 1 ,
X2,.
• - ,
X n
of the characteristic is available and we denote the bootstrap sample by X~, X~, • - •, X ~ There are three cases of confidence intervals for Cpmk. For the case of M > #, confidence interval for Cp,,,k is
^
1^
^
^
1^
^
( Cpmk -- m - r O'pmk Vpmk,( l +c~)/2, Cpmk -- m - ~ O'pmk Vpmk,(1-cO /2 ) ,
(10)
d - I X - MI G,~k = 3(~(~,~=,(xl - T) 2 + n(32 - T)2) 1/2
(11)
where
and
l
^2 O'pmk
[
s
+ (T - 32)21-3([,s2 + ( T - 32) 2 + ( T - 32)(d + 32 - M ) 1 2 s 2
(a + 32 - M)fi3[s 2 + (T - 2 ) 2 + (T - 32)(d + 32 - M)] +
(12)
l ( d + X - M)2(/~4 - (r4)).
Here ~)p,,~k,~is defined as
P ( v f ~ ( - m k ^.
<_ ~p,,,k,~,) = a,
-
(13)
O'pmk
where C~.~k and &~.~k are the resample version of Cp,,,k and &v,~k, respectively. For the case of M < #, we use the different version of &p.,k value such that
1 2 ~[s + (T - )()2]-a([s2 + ( T - 2 ) 2 - ( T - X ) ( d - X + M ) 1 % 2
^2
O'pmk
( d - X + M)fi3[s 2 + ( T - 2 ) 2 - ( T - X ) ( d - 32 + M)] +
( d - 2 + M)2(fi4 -
(14)
O'4)).
Note that o`p.,k^2of (12) and (14) are the sample versions of (8) and (9). For the last case of M -=- #, Cp.~k is reduced to Cpm and tiros we have the same confidence interval as for Cp,~ which is given as
I ^
^
(G,,, - ,,, ~o`~,~v~.,,(,+~,)., G,,,
I ^
^
•
.-~
(is)
Estimation of capability index
1147
where ~
d
Cpm 3 4 1 / n
Einl(Xi
-- T ) 2
and
^2
USL - LSL
s2 T
%~ = 3 6 ( ~ - 7 ( ~ - - Y ) @
(
1
- 2)~ - (T - X ) ~ + 7 ( / ~ -
¢)).
Here bp~,~ satisfies (
^.
_< bpm,~) = a,
(16)
O'pm
where C ~ and O'~m are obtained from the bootstrap resample. The new percentile-t bootstrap confidence intervals will be compared with the percentile and standard confidence intervals, based on Monte Carlo experiments.
Simulation
Study
To compare the performance of the percentile-t method with the percentile bootstrap method and standard method for constructing confidence interval for Cpmk, we choose the following values : U S L = 6 1 , L S L = 40, T = 49,# = 50, and a = 2. These values yield @ink = 1.491, which represents the "very capable" process. In this section, we compare our percentile-t confidence interval with percentile confidence interval and standard confidence interval by conducting simulation studies for various underlying process distributions. Percentile bootstrap confidence interval is obtained by
(17)
where C~,~k(z ^ * ") is the i-th smallest bootstrap estimate for the capability index Cp,~k e,nd B is the number of bootstrap resamples. Note that here we use a as a confidence coefficient. For example, a 90% confidence interval with/5' = 1000, the interval given in (17) would be approximately
(C~mk(51 ), C'~mk(950))-
K. C. Choiet ah
1148
Here we can use Cpmk(51 ) instead of ~"
¢;.,~(5o) to make this interval cover exactly
90% of 1000 estimates. The a-level standard confidence interval for the capability index Cpmk is given by
(¢,.k +
z~_<, d~,~+ z~<),
(18)
where Z~ is the lOOa percentile point of a standard normal distribution and S~ is the standard deviation of the C~mkli)'s, i = 1, 2 , . . . , B. 6'pink denotes the estimator of Cpmk based on the original sample, which is given in (11). We choose the resample size m = 5, 10, or 15( for the original sample size n = 5),m = 10,20, or 3 0 ( f o r n = 10), a n d r e : 3 0 , 4 0 , or 5 0 ( f o r n = 3 0 ) . B = 1000 bootstrap resamples (each of size m) were drawn from each sample of size n and a 90% bootstrap confidence interval was constructed for each method. It was then determined if the calculated confidence interval of each type contains the true index value and also the length of each confidence interval is evaluated. This single simulation was then replicated N = 1000 times and thus a percentage of times the true value of the index is within the calculated interval could be calculated as well as an average length, and a standard error of the average length of the 90% confidence intervals. We use the following process distributions, each of which is adjusted in order to have the mean # = 50 and standard deviation cr = 2 : Normal, Student's t with 5 and 6 degrees of freedom ( heavy tailed distributions), and Chi-squared with 4 and 5 degrees of freedom ( highly skewed distributions). The reason why these heavy tailed and highly skewed distributions are chosen is that they are troublesome with normal assumption and also they are frequently happen to be in the field as Gunter [11] noticed. In order to make each distribution to have the specified values of # = 50, and a = 2, we transform Student's t and Chi-squared distribution as follows.
X ~ ts
==* v ~ x + 5 0 , ? - - -
X~to
~
18X+50, X
4
Estimation of capability index
1149
and
x ~ xl ~
V~gx + 5 o - v5-~.
In the simulation, especially when n is very small, it happens to have a negative estimate of #4 - a 4 which is undesirable because it produces a negative variance term. To avoid such undesirable situations, we draw B resamples until all those B "2 estimates have positive value of ap,~k.
The simulation results are tabulated in Tables 1-5.
In a normally distributed
process, the percentile-t method provides higher coverage probabilities and shorter confidence intervals. When the original sample size n is very small, it turns out that the percentile-t method has remarkably higher coverage probabilities and shorter length of intervals than other two intervals. As the sample size n increases, differences of coverage probabilities and lengths of intervals among three are somewhat reduced. It is observed that as m increases, it seems that each interval's coverage as well as the mean and the standard error of interval lengths are decreasing.
Table 1: Coverage Probability, Mean Width and Standard Error of the Width for the 90% Bootstrap Confidence Intervals-Normal Process n=5 n=10 n=30 5 10 15 l0 20 30 30 40 50 Coverage .746 .595 .511 .814 . 6 6 8 . 5 7 0 . 8 4 5 . 7 7 2 .737 PB Mean 2.193 1.164 . 8 0 5 1.141 . 7 3 3 . 5 7 2 . 5 7 9 . 4 8 8 .435 S.E. .063 .027 .014 .015 . 0 0 9 . 0 0 6 . 0 0 4 . 0 0 3 .003 Coverage . 8 6 4 . 6 7 4 .554 .883 .711 . 5 9 7 .881 .801 .766 SB Mean 2.690 1.279 . 9 1 0 1.230 . 7 5 7 . 5 9 0 . 5 9 0 . 4 9 5 .441 S.E. .095 .033 .021 .018 . 0 0 9 . 0 0 7 . 0 0 4 . 0 0 4 .003 Coverage .877 . 6 3 7 .501 .890 . 7 0 2 . 5 7 3 . 8 9 7 . 8 3 2 .790 PTB Mean 2.096 1.086 . 7 2 7 1.150 . 7 3 7 .556 . 6 0 4 . 5 0 2 .445 S.E .055 .021 .011 .016 . 0 1 0 . 0 0 6 . 0 0 5 . 0 0 4 .004
Table 2: Coverage Probability, Mean Width and Standard Error of the Width for the 90% Bootstra Confidence Intervals--t5 Process n=5 n=lO n=30 5 10 15 10 20 30 30 40 50 Coverage .678 .558 .456 .725 . 6 0 0 .521 . 8 0 5 . 7 4 5 .680 PB Mean 2.528 1.294 .877 1.382 .861 . 6 3 4 . 7 3 5 . 6 2 8 .550 S.E. .073 .028 .014 .021 .01l . 0 0 7 . 0 0 7 . 0 0 6 .005 Coverage .816 .649 .519 .832 .651 . 5 3 3 . 8 4 4 . 7 8 4 .714 SB Mean 3.005 1.412 1.000 1.507 . 8 8 6 . 6 5 7 . 7 5 2 . 6 3 8 .556 S.E. .095 .034 .021 .025 . 0 1 2 . 0 0 7 . 0 0 7 . 0 0 6 .005 Coverage .805 .616 .431 .824 . 6 4 2 . 5 2 3 . 8 6 6 .811 .740 PTB Mean 2.390 1.185 .796 1.425 . 8 8 8 .621 . 8 4 5 . 7 0 8 .610 S.E. '.088 .023 .012 .023 .013 . 0 0 6 . 0 1 2 . 0 1 0 .009
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K.C. Choi et al. Table 3: Covera e Probability, Mean Width and Standard Error of the Width for the 90% Bootstra Confidence Intervals t6 Process n=5 n=10 n=30 5 10 15 10 20 30 30 40 50 .693 .539 .461 .771 .617 .523 .824 .762 .699 Coverage 2.429 1.313 .896 1.304 .842 .629 .698 .599 .525 PB Mean .070 .028 .015 .018 .Oil .007 .006 .005 .005 S.E. .861 .634 .5/9 .867 .655 .546 .850 .787 .72] Coverage 3.409 1.433 1.018 1.411 . 8 6 4 . 6 5 6 .714 .609 .531 SB Mean S.E. .188 .035 .021 .021 .011 .008 .006 .005 .005 Coverage .828 .614 .444 .859 .662 .526 .874 .809 .733 PTB Mean 2.263 1.202 .807 1.363 .869 .616 .788 . 6 6 8 .576 .052 .021 .013 .021 . 0 1 3 ,007 .010 .009 .008 S.E.
Table 4: Coverage Probability, Mean Width and Standard Error of the Width for the 900£ Bootstrap Confidence Intervals )/~ Process n=5 n=10 n=30 5 10 15 10 20 30 30 40 50 Coverage .597 .472 .397 .696 .549 .463 .761 .734 .670 PB Mean 3.389 1.842 1.010 1.844 1.121 .777 .933 .763 :674 S.E. .121 .063 .017 .029 .017 .009 .009 .007 .006 Coverage .805 .595 .476 .816 .615 .524 .829 .781 .704 SB Mean 4.437 1.981 1.218 2.045 1.161 .838 .967 .784 .688 S.E. .242 .068 .026 .034 .017 .010 .010 .007 .006 Coverage .825 .618 .417 .854 .658 .510 .880 .832 .768 PTB Mean 4.044 1.872 1.006 2.060 1.238 .761 1.081 .873 .746 S.E. .118 .047 .015 .033 .020 .008 .015 .011 .010
Table 5: Coverage Probability, Mean Width and Standard Error of the Width for the 90% Bootstrap Confidence Intervals X~ Process n=5 n=10 n=30 5 10 15 10 20 40 50 30 30 Coverage .571 .466 .388 .640 .555 .479 .762 .708 .679 PB Mean 3.245 1.708 .989 1.654 1.035 .751 .870 .733 .637 S.E. .127 .051 .018 .023 .016 .008 .007 .007 .006 Coverage .798 .568 .459 .777 .633 .537 .836 .751 .717 SB Mean 4 . 0 5 1 1.839 1.193 1.824 1.075 .801 .898 .752 .649 S.E. .162 .056 .029 .027 ,016 .010 .OO8 .007 .006 Coverage .813 .603 .412 .808 .677 .531 .874 .820 .771 PTB Mean 3.935 1.769 .981 1.894 1.141 .751 .997 .839 .702 S.E. .155 .040 .015 .028 .019 .009 .012 .012 .009
In a process with a Student's t-distribution, tile percentile bootstrap confidence interval has the lower coverage probabilities and provide the longer lengths than the pereentile-t bootstrap confidence interval when n is small. The magnitude of the standard error for the length of intervals based on the standard method is greater than its counterparts. The lengths of confidence intervals by the standard method are slightly greater than the other confidence intervals and the coverage probabilities are higher for the standard method than those for the percentile-t and percentile
Estimation of capability index method. As for the effect of the degree of freedom, we observe that when the sample size n is equal to 5, the coverage probabilities of confidence intervals are increasing in the all three cases as the degrees of freedom increase. But it is not the case for the mean lengths of confidence intervals. When the sample size changes to 10, then coverage probabilities are increasing and lengths of intervals are decreasing in all three meethods as the degrees of freedom increase. When a proces is highly skewed such as X2 distribution with 4 degrees of freedom, we find similar patterns as in other distributions. Percentile-t method produces a confidence interval with higher coverage probability and shorter length than other methods in general. For a X~ distribution with 4 degrees of freedom the magnitude of standard error of confidence interval length in percentile-t type is the smallest when n is small, whereas the standard error in percentile-t method is a little larger than the others when n is large. As the sample size n increases, the coverage probabilities of all confidence intervals are increasing and the interval lengths are getting shorter as expected. We observe the similar trend for a ~ distribution with 5 degree of freedom. As the resample size m is increasing, coverage probabilities of percentile and percentile-t methods are decreasing. In conclusion, the percentile-t method is prefered to the percentile method or standard method because of its higher coverage probability and shorter length of confidence interval for most cases, especially when the sample size n is small. Therefore the bootstrap confidence interval based on the percentile-t method is recommended when the available sample size is very small due to the high price or its rarity.
References 1. R. A. Boyles, The Taguchi Capability Index, Journal of Quality Technology,
23, 107-126 (1991). 2. L. K. Chan, S. W. Cheng and F. A. Spiring, A New Measure of Process Capability : Cpm, Journal of Quality Tech.nology, 20, 162-175 (1988). 3. L. K. Chain, S. W. Cheng and P. A. Spiring, A Graphical Technique for Process Capability, Transactions ASQC Congress, Dallas, TX. 268-275 (1988).
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K. C. Choi et al.
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4. L. K. Chan, S. W. Cheng and F. A. Spiring, The Robustness of Process Capability Index Cp to Departures from Normality, In Statistical Theory and Data
Analysis, II (K. Matusita, ed.), North-Holland, Amsterdam, 223 229 (1988). 5. L. K. Chan, Z. Xiong and D. Zhang, On the Asymptotic Distributions of Some Process Capability Indices, Communications in Statistics-Theory and
Methods, 19, 11-18 (1990). 6. Y. Chou and D. B. Owen, On the Distribution of the Estimated Process Capability Indices, Communications in Statistics-Theory and Methods, 18, 45494560 (1989). 7. B. Efron, Bootstrap Methods : Another Look at the Jackknife, Annals of
Statistics, 7, 1 26 (1979). 8. L. A. Franklin and G. Wasserman, Bootstrap Confidence Interval Estimates of Cpk : An Introduction, Communications in Statistics-Simulation and Com-
putation, 20, 231 242 (1991). 9. L. A. Franklin and G. Wasserman, A Note on the Conservative Nature of the Tables of Lower Confidence Limits for Cpk with a Suggested Correction, Com-
munications in Statistics-Simulation and Computation, 21,926-932 (1992). 10. L. A. Franklin and G. Wasserman, Bootstrap Lower Confidence Limits for Capability Indices, Journal of Quality Technology, 24, 196-210 (1992). 11. B. H. Gunter, The Use and Abuse of Cpk, Parts 1-4, Quality Progress, 22(1), 72-73; 22(3), 108-109; 22(5), 79-80; 22(7), 86-87 (1989). 12. P. Hall, Theoretical Comparison of Bootstrap Confidence Intervals, The An-
nals of Statistics, 16, 927-953 (1988). 13. V. E. Kane, Process Capability Indices, Journal of Quality Technology, 18, 41-52. (Corrigenda 265.) (1986). 14. S. K. Kocherlakota, K. Kocherlakota and S. N. U. A. Kirmani, Process Capability Indices Under Non-normality, International Journal of Mathematical
Statistics, 1,175-210 (1992).
Estimation of capability index 15. S. Kotz, W. L. Pearn and N. L. Johnson, Some Process Capability Indicators Are More Reliable Than One Might Think( A Comment on Bissel(1990)),
Applied Statistics, 42, 55-62 11993). 16. K. H. Naro, Trend Changes in Failure Rate and Mean Residual Life : Its Relations and Applications, Ph.D. Dissertation, University of Nebraska, (1995). 17. W. L. Pearn, S. Kotz and N. L. Johnson, Distributional and Inferential Properties of Process Capability Indices, Journal of Quality Technology, 24,216-231 11992).
Acknowledgements
This study was supported in part by the Research Fund and Utilizing Facilities of Advanced Research Center (Environmental Science and Technology) of Chosun University, 1995 and in part by the Basic Science Research Institute Program, Ministry of Education, 1994, Project no. 1403.
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