In this paper point estimation of the process capability index C(pk) is considered. In order to improve on the natural estimator, a simple alternative estimator is proposed and its mean square error is evaluated numerically

Nagata, Yasushi

English

Okayama University Scientific Achievement Repository

POINT ESTIMATIONOF. THE PROCESS CAPABILITYINDEX CpkYasushi NAGATAFaculty of Economics, Okayama UniversityAbstractIn this paper point estimation of the process capability index Cpkis considered. In order to improve on the natural estimator, a simplealternative estimator is proposed and its mean square error is evaluatednumerically.1 IntroductionLet XI, X 2 , "', X n be independent random variables from a normaldistribution with unknown mean eand unknown variance u2 • Then, theprocess capability index (PCI) Cpk is defined as. {USL-e e-LSL}Cpk = mm 3u' 3ud -Ie - ml3u (1)where d = (USL - LSL)/2, m = (LSL + USL)/2, and LSL and USLare lower and upper specification limits, respectively. The values of LSLand USL are known. This PCI is frequently used in the activities of thequality control in order to evaluate whether the processes are satisfactoryor not.Althouth other types of definitions for PCPs have been proposed anda lot of studies on these PCPs have been made (see Kotz and John-son(1993)), in this paper we will deal with C pk alone. Because Cpk isone of the most familiar PCPs for the practitioners, especially for theengineers of Japanese industrial compa.nies.Suppose that we are interested in the point estimation of Cpk in termsof the mean square error (MSE) criterion. Since eand u are unknown-325-664parameters, they must be estimated from data. Defining X =Ei=l X;jnand fJ = VEi=l(Xi - X)2/(n -1), a natural estimator of Cpk is• d -IX - mlCpk = . .3a (2)(3)Chou and Owen(1989) derived the MSE ()f Cpk. Kotz and John-son(1993) evaluated the mean and the standard deviation of Cpk numer-ically.In this paper we consider a simple alternative estimator for Cpk inorder to improve on the natural estimator Cpk and evaluate their MSE'snumerically.2 Mean square error of the naturalestimatorIn this section we introduce several notations and summarize known re-sults.Define z = v:n(X - m)/a and let XI be a random variable of chidistribution with degrees of freedom f. Since fJ = aXn-l/..;:n:::T, theestimator (2) is expressed as• d* -lzl/vnCpk = 3XI/Vl 'where d* = d/a and f = n - 1. In this expression, note that z andXI are mutually independent. Define 6 = (e - m)/a. Since z is dis-tributed as N(vn6, 1), the statistic Izi has a folded normal distribution.Elandt(1961) showed thatE(lzl) = 2¢(vn6) - vn6(1 - 2~(vn6)) == e, (4)E(lzI2) = 1 + n62, (5)where ¢ and ~ are the probability density function and the cumulativedistribution function of N(O, 1), respectively. It can be easily shown thatE -r _ f(U - r)/2)(XI ) - 2r / 2 fU /2)-326-(6)POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 665To simplify the notations let us denote r = yinI6I and ( = ylnd*. Then,the MSE of the natural estimator Cpk is• • 2R(Cpk, Cpk) = E{(Cpk - Cpk) }= K{(2 - 2e( + (1 + r2)-2A*«( - r)«( - 0) + (J - 2)«( - r)2 / J}, (7)where K = f/{9n(J - 2)} and A* = (J - 2)E(xjl)/.JJ.The MSE depends on n, 6 and d*. Since e is an even function of 6,the MSE is also an even function of 6 with fixed nand d*. The numericalresults for this MSE will be given in Section 4.Before closing this section we give several values of A* in Table1. (Kotz and Johnson(1993, p.50) reported the values of bf = 1/{V!E(Xjl)} for. f =4(5)59.)Table 1: Values of A*f 4 9A* .627 .85114 19 24 29 34.907 .932 .947 .956 .96339 44.967 .97149.974We can observe from Table 1 that A* increases as f increases and isless than one for at least moderate values of f.3 Components of Cpk and theirestimationThe PCI Cpk is rewritten as(8)where Cp = d* /3. Cp is also one of the traditional and familiar PCPs. Itshould be noted, however, that as Cp does not include the informationabout population mean eit is useful only when ecoincides with m.-327-666Let us call Cp and 161/3 the components of Cpk ' In this section weconsider the estimation of Cp and 161/3.The natural estimators of Cp and 161/3 are.*Cp d d (9)= 3= 3&'181 IX-ml (10)3 3&In order to improve on the natural estimator Cp in terms of the MSEcriterion, let us consider a constant multiple of Cp :<p(A) =ACp = ~Av7d*Xfl.Since the MSE of <p(A) isR(<p(A), Cp) = E{(<p(A) - Cp)2}f d*2 {2 * f - 2 }= 9(/ _ 2) A - 2A A + -f- ,(11)(12)the MSE is minimized at A =A*.Thus, the natural estimator <p(1) = Cp can be improved by ¢(A*)uniformly. Note that the MSE R(¢(A), Cp ) does not depend on 6. Letus define the relative improvement asNote that RI depends on f alone.We give the the values of R(Cp,Cp), R(<p(A*),Cp) and RI for f =9(10)49 in Table 2. We can see from Table 2 that RI is significantly largefor small f and that RI decreases as f increases.Next, let us consider the improvement on the natural estimator 161/3.Consider a constant multiple of 161/3:(14)-328-POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 667Table 2: Values of R(Cp, Cp), R(¢(A*), Cp) and RIf d* R(Cp,Cp) R(¢(A*),Cp) RI(%)9 2 .0432 .0305 29.333 .0972 .0687 29.334 .1729 .1222 29.335 .2701 .1909 29.336 .3889 .2749 29.3319 2 .0152 .0129 15.103 .0341 .0290 15.104 .0607 .0515 15.105 .0948 .0805 15.106 .1365 .1159 15.1029 2 .0091 .0082 10.183 .0204 .0183 10.184 .0363 .0326 10.185 .0567 .0510 10.186 .0817 .0734 10.1839 2 .0065 .0060 7.683 .0145 .0134 7.684 .0258 .0239 7.685 .0404 .0373 7.686 .0582 .0537 7.6849 2 .0050 .0047 6.163 .0113 .0106 6.164 .0201 .0188 6.165 .0313 .0294 6.166 .0451 .0423 6.16It is shown that the MSE of 'Ij;(B) is-329-668Table 3: Values of R(161/3, 161/3), R(7/J(A*) , 161/3) and RIf 161 R(181/3,161/3) R(7/J(A*),161/3) RI (%)9 0 .0143 .0104 27.570.5 .0151 .0107 29.451.0 .0251 .0180 28.341.5 .0386 .0275 28.682.0 .0575 .0409 28.8919 0 .0062 .0054 13.120.5 .0069 .0060 13.601.0 .0100 .0086 13.871.5 .0147 .0126 14.272.0 .0214 .0183 14.5229 0 .0040 .0036 8.600.5 .0045 .0041 8.841.0 .0063 .0057 9.181.5 .0091 .0082 9.492.0 .0131 .0118 9.7039 0 .0029 .0027 6.400.5 .0033 .0031 6.561.0 .0045 .0042 6.861.5 .0066 .0061 7.112.0 .0094 .0087 7.2849 0 .0023 .0022 5.100.5 .0026 .0025 5.231.0 .0036 .0034 5.471.5 .0051 .0048 5.682.0 .0073 .0069 5.83= f {(I + r 2 )B2 _ 2rBA*B + U - 2) r 2 }. (15)9nU - 2) fThis MSE is minimized at B = rBA* /(1 + r 2 )(= B*). As B* depends onthe unknown parameter 6, the estimator 7/J(B*) is not available. Noting,-330-POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 669however, that 8 :::; VI + r 2, it is shown that B* < A* uniformly. SinceA* < 1 from Table 1, it follows thatR (1jJ(B*), I~I) < R (1jJ(A*), I~I) < R (1;1, I~I) , (16)uniformly. As A* does not depend on the unkm.lwn parameter, the esti-mator 1jJ(A*) is available.Thus, the natural estimator 1jJ(I) = 181/3 can be improved by 1jJ(A*)uniformly. Note that the MSE R(1jJ(B), 181/3) does not depend on d*.Let us define the relative improvement RI asRI = R(161/3,181/3} - R(1jJ(A*), 181/3) x 100 (%). (17)R(181/3, 181/3)We give the values of R(161/3, 181/3), R(1jJ(A*), 181/3) and RI for f =9(10)49 in Table 3. We can see from Table 3 that RI is significantly largefor small f and that RI decreases as f increases.In this section we have shown that both of the natural estimators forthe components of Cpk can be improved by some multiples of the naturalestimators, respectively and that the multipliers can be set as the samevalue.4 An alternative estimator for Cpk andits mean square errorIn this section we consider an alternative estimator for Cpk in order toimprove on the natural estimator Cpk.Let us consider the estimatorp(A, B) = ¢(A) -1jJ(B), (18)where ¢(A) and 1jJ(B) were defined in Section 3. Note that p(l, 1) = Cpk'The MSE of p(A, B) is derived asR(p(A, B), Cpk) =E{(p(A, B) - C pk )2}=K {(2A2 - 28(AB + (1 + r 2)B2-2A*«( - r)«(A - BB) +U - 2)«( - r)2/ f}, (19)-331-670Table 4: Values of R(Cpk , Cpk), R(A*Cpk' Cpk )and RI for /=9.The first row: values of R(Cpk , Cpk ),the second row: values of R(A*Cpk , Cpk),the third row: values of RI (%).d* 1610 0.5 1.0 1.5 2.02 .0360 .0354 .0250 .0170 .0143.0409 .0259 .0180 .0123 .0104-13.49 26.77 28.31 27.85 27.573 .0793 .0780 .0575 .0386 .0251.0791 .0565 .0409 .0275 .0180·.31 27.57 28.88 28.68 28.334 .1442 .1421 .1115 .0818 .0575.1325 .1023 .0790 .0581 .04098.11 28.04 29.09 29.02 28.895 .2307 .2279 .1871 .1466 .1115.2012 .1634 .1325 .1039 .079112.77 28.32 29.18 29.16 29.106 .3388 .3353 .2843 .2331 .1871.2852 .2397 .2012 .1650 .132515.82 28.51 29.23 29.22 29.19where K was defined in Section 2.As it was shown in Section 3 that ¢(1) and ¢(1) are improved by¢(A*) and ¢(A*), respectively, let us deal with the estimatorp(A*,A*) = ¢(A*) - ¢(A*) = A*Cpk. (20)This estimator (20) can also be derived in the following way. Differenti-ating R(p(A, B), Cpk) with respect to A and B, we obtain the minimizers-332-POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 671Table 5: Values of R(Cpk,Cpk)' R(A*Cpk,Cpk)and RI for /=19.The first row: values of R(Cpk ,Cpk ),the second row: values of R(A*Cpk,Cpk)'the third row: values of RI (%).d* 1010 0.5 1.0 1.5 2.02 .0154 .0144 .0100 .0072 .0062.0183 .0124 .0086 .0062 .0054-18.97 13.99 13.87 13.38 13.123 .0313 .0296 .0214 .0147 .0100.0344 .0253 .0183 .0126 .0086-9.77 14.46 14.52 14.27 13.874 .0548 .0523 .0403 .0299 .0214.0569 .0446 .0344 .0255 .0183-3.76 14.69 14.80 14.69 14.525 .0859 .0826 .0669 .0527 .0403.0859 .0704 .0569 .0448 .0344.10 14.80 14.92 14.87 14.806 .1246 .1205 .1010 .0830 .0669.1213 .1025 .0859 .0706 .05692.71 14.87 14.98 14.95 14.92A = A*(l - riO = A*(l - lol/d*)(= At) and B = O. Since lol/d* =Ie - mild involves an unknown parameter e, let us estimate it by IX -mild = IzI/(v;:id*). Then, we obtainp(At,O) = ¢(At ) = ~A*/7d*Xfl - ~A*I?IZIXfl, (21)which is equivalent to the estimator (20).-333-672Table 6: Values of R(Cpk' Cpk), R(A*Cpk , Cpk )and RI for 1=29.The first row: values of R(Cpk, Cpk),·the second row: values of R(A*Cpk' Cpl,) ,the third row: values of RI (%).d* 1810 0.5 1.0 1.5 2.02 .0100 .0090 .0063 .0046 .0040.0118 .0082 .0057 .0042 .0036-17.94 9.44 9.18 8.80 8.603 .0198 .0181 .0131 .0091 .0063.0220 .0163 .0118 .0082 .0057-10.94 9.79 9.70 9.49 9.184 .0342 .0317 .0244 .0182 .0131.0363 .0286 .0220 .0164 .0118-6.09 9.95 9.92 9.83 9.705 .0531 .0499 .0403 .0318 .0244.0546 .0449 .0363 .0286 .0220-2.89 10.02 10.02 9.98 9.926 .0765 .0726 .0607 .0499 .0403.0770 .0653 .0546 .0449 .0363-.68 10.07 10.07 10.05 10.02Let us define the relative improvement asRI = R(Cpk' Cpk ).- R(A*Cpk , Cpk) X 100 (%). (22)R(Cpk, Cpk)We give the values of R(Cpk' Cpk), R(A*Cpk , Cpk) and RI for 1 =9(10)49 in Tables 4, 5, 6, 7 and 8, respectively. We selected the samevalues of parameters d* and 8 as in Table 2.4 of Kotz and Johnson(1993).-334-POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 673Table 7: Values of R(Cpk' Cpk), R(A*Cpk' Cpk)and RI for 1=39.The first row: values of R(Cpk' Cpk),the second row: values of R(A*Cpk' Cpk) ,the third row: values of RI (?V).d* 1610 0.5 1.0 1.5 2.02 .0075 .0066 .0045 .0033 .0029.0087 .0061 .0042 .0031 .0027-16.60 7.10 6.86 6.56 6.403 .0146 .0130 .0094 .0066 .0045.0162 .0121 .0087 .0061 .0042-10.84 7.38 7.28 7.11 6.864 .0249 .0227 .0175 .0130 .0094.0266 .0210 .0162 .0121 .0087-6.72 7.51 7.46 7.39 7.285 .0385 .0356 .0288 .0227 .0175.0400 .0329 .0266 .0210 .0162-3.95 7.57 7.55 7.51 7.466 .0553 .0518 .0433 .0356 .0288.0564 .0478 .0400 .0329 .0266-2.02 7.60 7.59 7.57 7.55From Tables 4-8 we observe the foliowingpoints:(a)RI is positiveunless 161 = 0; (b)RI is significantly large when 1 is small and 161 =f:. 0,and RI decreases as 1 increases; (c)the extent of improvement by A*Cpkis comparable to those in Tables 2 and 3 when 161 =f:. O.From above observation, we may conclude that A*Cpk is useful whenwe have some information that 161 is not close to zero.Finally, we give a remark that any member of the class {ACpk; A is-335-674Table 8: Values of R(Cpk' Cpk), R(A*Cpk' Cpk )and RI for 1=49.The first row: values of R(Cpk' Cpk ), .the second row: values of R(A*Cpk' Cpk),the third row: values of RI (%).d* 1810 0.5 1.0 1.5 2.02 .0060 .0051 .0036 .0026 .0023.0069 ,0048 .0034 .0025 .0022-15.44 5.68 5.47 5.22 5.103 .0116 .0102 .0073 .0051 .0036,0128 .0096 .0069 .0048 .0034-10.48 5.92 5.83 5.68 5.474 .0197 .0177 .0136 .0102 .0073.0210 .0166 .0128 .0096 .0069-6.85 6.02 5.98 5.92 5.835 .0303 .0277 .0224 .0177 .0136.0316 .0260 .0210 .0166 .0128-4.38 6.07 6.05 6.02 5.986 .0434 .0402 .0336 .0277 .0224.0445 .0378 .0316 .0260 .0210-2.66 6.10 6.09 6.07 6.05constant} can not improve on the natural estimator Cpk uniformly when1 is more than at least 19 (see Appendix for details), although the natu-ral estimators for the components of Cpk can be improved uniformly. Wehave not been able to prove whether there exist some estimators whichbelong to the class {p(A, B); A and B are constants} and improve on Cpkuniformly. We have evaluated the MSE's of p(A, B)'s for several (A, B)'sin moderately large neighborhood of (A*, A*) numerically. We have, how--336-POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 675ever, found no uniformly improving estimator. The author thinks thatA*bpk is a simple and good choice as an alternative estimator.Since the natural estimator bpk is not a smooth estimator, it is shownfrom the well-known decision theory that it is not admissible. In thispaper we have tried to look for an improving estimator, but we have notbeen able to find such an estimator in the class of simple estimators. Theauthor will continue this study in the future work.AppendixLet us consider the class of estimators {Abpk ; A is constant}. Since itfollows from (19) thatA A 2R(ACpk, Cpk ) = E{(ACpk - Cpk ) }=K{«(2 - 2B( + (1 + r2»A2-2A*«( - r)«( - B)A +U - 2)«( - r)2 / j}, (23)we obtain the minimizerA = A*«( - r)«( - B) (= AI).(2 - 2B( + 1 + r 2 (24)We give the values of AI in Table 9. We observe from Table 9 thatwhen f ~ 19 AI > 1 for some parameter values and that AI < 1 for otherparameter values.If we set the constant A which is smaller than one, it follows that forsome parameter values which implies AI > 1 .Similarly, if we set the constant A which is larger than one, it followsthat for some parameter values which implies Al < 1Thus, no constant multiple of the natural estimator can not improveon the natural estimator uniformly.-337-676Table 9: Values of AIf d* 1810 0.5 1.0 1.5 2.09 2 .9625 .8282 .7740 .6079 .0000 .3 .9248 .8448 .8304 .8149 .77374 .9060 .8489 .8418 .8377 .83035 .8949 .8504 .8458 .8442 .84176 .8875 .8511 .8477 .8469 .845819 2 1.018 .9138 .8877 .7768 .00003 .9888 .9257 .9206 .9118 .88774 .9744 .9290 .9270 .9247 .92065 .9658 .9303 .9292 .9283 .92706 .9602 .9310 .9302 .9298 .929229 2 1.027 .9424 .9252 .8435 .00003 1.003 .9511 .9481 .9421 .92524 .9913 .9535 .9525 .9509 .94815 .9842 .9545 .9540 .9534 .95256 .9795 .9550 .9547 .9544 .954039 2 1.030 .9569 .9439 .8795 .00003 1.009 .9636 .9615 .9568 .94394 .9984 .9655 .9648 .9636 .96155 .9921 .9663 .9660 .9655 .96486 .9880 .9667 .9665 .9663 .966049 2 1.030 .9656 .9551 .9020 .00003 1.011 .9711 .9693 .9656 .95514 1.002 .9726 .9720 .9711 .96935 .9964 .9732 .9730 .9726 .97206 .9926 .9735 .9734 .9732 .9730-338-POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX Cpk 677References[1] Chou,Y.M. and Owen,E.B.(1989). On the qistribution of the esti-mated process capability indices, Communications in Statistics -Theory and Methods, 18, 4549-4560.[2] Elandt,R.C.(1961). The folded normal distribution: two methods ofestimating parameters from moments, Technometrics, 3, 551-562.[3] Kotz,S. and Johnson,N .L.(1993). Process Capability Indices, Chap-man and Hall.-339-

POINT ESTIMATION OF THE PROCESS CAPABILITY INDEX C(pk)

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