Qual. Reliab. Engng. Int. 2001; 17: 279–290 (DOI: 10.1002/qre.378)
ON THE RELIABILITY OF THE ESTIMATED
INCAPABILITY INDEX
W. L. PEARN1∗AND G. H. LIN2
1Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, Republic of China 2Department of Telecommunication Engineering, National Penghu Institute of Technology, Taiwan, Republic of China
SUMMARY
Greenwich and Jahr-Schaffrath (1995) introduced the process incapability index Cpp = Cip+ Cia, which provides an uncontaminated separation between information concerning the process precision (Cip) and process accuracy (Cia). In this paper, we consider the three indices, and investigate the statistical properties of their natural estimators. For the three indices, we obtain their UMVUEs and MLEs, and compare the reliability of the two estimators based on the relative mean square errors. In addition, we construct 90%, 95%, and 99% upper confidence limits, and the maximum values of ˆCppfor which the process is capable 90%, 95%, and 99% of the time. The results obtained in this paper are useful to the practitioners in choosing good estimators and making reliable decisions on judging process capability. Copyright2001 John Wiley & Sons, Ltd.
KEY WORDS: imprecision index; inaccuracy index; UMVUE; MLE; relative mean square error
1. INTRODUCTION
Greenwich and Jahr-Schaffrath [1] introduced the process incapability index Cpp to provide numerical measures on process performance for industrial applications. The index Cppis a simple transformation of C∗pm, a general form of the capability index
Cpm considered by Chan et al [2], which provides an uncontaminated separation between information concerning the process precision and the process accuracy. The index Cppis defined as follows:
Cpp = 1 C∗2 pm = σD2+ µ − T D 2
where µ is the process mean, σ is the process standard deviation, D = min{(USL − T )/3, (T −
LSL)/3}, USL and LSL are the upper and the lower
specification limits, and T is the target value. If we define Cip = (σ/D)2, and Cia = [(µ − T )/D]2, then Cpp can be expressed as Cpp = Cip + Cia. Since Cip measures the process variation relative to the specification tolerance, it has been referred to as the process imprecision index. On the other hand, Cia measures the relative process departure and has been ∗Correspondence to: W. L. Pearn, Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsuch Road, Hsinchu, Taiwan 30050, Republic of China.
referred to as the process inaccuracy index. We note that the mathematical relationships Cip = 1/(Cp)2, and Cia = 9(1−Ca)2can be established, where Cpand
Caare two basic process capability indices considered by Kane [3] and Pearn et al. [4].
In this paper, we consider the three indices Cip,
Cia, and Cpp and investigate the statistical properties of their natural estimators. For Cip, we show that the natural estimator is the UMVUE, which is consistent and asymptotically efficient. We also obtain the MLE (maximum likelihood estimator), which has smaller mean square error than the UMVUE (uniformly minimum variance unbiased estimator), hence it is more reliable, particularly, for short production run applications. For Cia, we show that the natural estimator is the MLE. We also obtain the UMVUE, which is shown to be more reliable than the MLE for applications with n ≥ 4. We show that the UMVUE is consistent and asymptotically efficient. For Cpp, we show that the natural estimator is the MLE and also the UMVUE, which is consistent and asymptotically efficient. In addition, we construct tables of 90%, 95%, and 99% upper confidence limits for Cppbased on the UMVUE. We also construct tables of the maximum values of ˆCpp under µ = T for which the process is capable. The estimators we recommend have all
Received 25 July 2000
the desired statistical properties, and are considered reliable.
2. ESTIMATING PROCESS IMPRECISION To estimate the process imprecision, we consider the natural estimator ˆCip defined as follows, where
Sn−1 = [ni=1(Xi − ¯X)2/(n − 1)]1/2 is the conventional estimator of the process standard deviation σ , ˆCip= 1 n − 1 n i=1 (Xi− ¯X)2 D2 = S2 n−1 D2 The natural estimator ˆCipcan be rewritten as
ˆCip= n − 1Cip (n − 1) ˆCC ip ip = Cip n − 1 n i=1 (Xi− ¯X)2 σ2 If the process follows the normal distribution, then ˆCip is distributed as[Cip/(n − 1)]χn−12 , where χn−12 is a chi-squared distribution with (n− 1) degrees of freedom. The probability density function of ˆCip can be easily derived as
f (y) = {[(n − 1)y/(2Cip)](n−1)/2 × exp[−(n − 1)y/(2Cip)]}
× {y[(n − 1)/2]}−1, for y > 0 The rth moment, the expected value, the variance, and the mean squared error of ˆCipcan be calculated as follows: E( ˆCip)r ={[(n − 1)/2] + r} [(n − 1)/2] 2Cip n − 1 r E( ˆCip) = C ip n − 1 E(χn−12 ) = Cip Var( ˆCip) = C ip n − 1 2 Var(χn−12 ) = C ip n − 1 2 2(n− 1) = 2C 2 ip n − 1 MSE( ˆCip) = E( ˆCip− Cip)2
= Var( ˆCip) + [E( ˆCip) − Cip]2 = 2C
2 ip
n − 1
If the process characteristic is normally distributed, then we can show that the natural estimator ˆCip is
the UMVUE of Cip, which is consistent. We can also show that√n( ˆCip− Cip) converges to N(0, 2Cip2) in distribution, and that ˆCip is asymptotically efficient (see the Appendix for the proofs). Thus, in real-world applications using ˆCip, which has all the desired statistical properties, as an estimate of Cip would be reasonable.
We note that by multiplying the UMVUE ˆCip by the constant cn = (n − 1)/n, we obtain the MLE of Cip. We can show that the MLE ˆCip is consistent, and is asymptotically unbiased. We can also show that √n( ˆC
ip−Cip) converges to N(0, 2Cip2) in distribution, and that ˆCip is asymptotically efficient. Since cn < 1, then ˆCip = cnˆCipunderestimates Cipbut with smaller variance. In fact, we may calculate
MSE( ˆCip) = [(2n − 1)/n2](Cip)2 and obtain
MSE( ˆCip) − MSE( ˆCip ) = [(3n − 1)/n2(n − 1)](C
ip)2> 0, for all n Therefore, the MLE ˆCip has smaller mean squared error than the UMVUE ˆCip, hence it is more reliable, particularly for short production run applications.
Tables1(a)and1(b)display the relative error of the UMVUE ˆCip, defined as[MSER( ˆCip)]1/2= {E[( ˆCip−
Cip)/Cip]2}1/2, for sample sizes n = 2(1)50, and 60(10)550, and some commonly used values of Cip= 1.00, 0.56, 0.44, 0.36, and 0.25, equivalent to Cp = 1.00, 1.33, 1.50, 1.67, and 2.00, covering the range of the precision requirements for most applications.
Precision requirement: Capable: 0.56≤ Cip≤ 1.00 Satisfactory: 0.44≤ Cip≤ 0.56 Good: 0.36≤ Cip≤ 0.44 Excellent: 0.25≤ Cip≤ 0.36 Super: Cip≤ 0.25
The square root of the relative mean squared error is a direct measurement, which presents the expected relative error of the estimation from the true Cip. We note that for UMVUE ˆCip, [MSER( ˆCip)]1/2 = [2/(n−1)]1/2, which is a function of the sample size n only. Therefore,[MSER( ˆCip)]1/2values are the same for all Cip values. For example, with n = 300 we have[MSER( ˆCip)]1/2 = 0.0818. Thus, for n = 300, we expect that the average error of ˆCip would be no greater than 8.18% of the true Cp. Tables2(a)and2(b) display the relative error,[MSER( ˆCip)]1/2, of the MLE
ˆC
Table 1.[MSER( ˆCip)]1/2for various Cip, and sample sizes (a) n= 2(1)50 and (b) n = 60(10)550 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 ****** ****** ****** ****** ****** 60 0.1841 0.1841 0.1841 0.1841 0.1841 2 1.4142 1.4142 1.4142 1.4142 1.4142 70 0.1703 0.1703 0.1703 0.1703 0.1703 3 1.0000 1.0000 1.0000 1.0000 1.0000 80 0.1591 0.1591 0.1591 0.1591 0.1591 4 0.8165 0.8165 0.8165 0.8165 0.8165 90 0.1499 0.1499 0.1499 0.1499 0.1499 5 0.7071 0.7071 0.7071 0.7071 0.7071 100 0.1421 0.1421 0.1421 0.1421 0.1421 6 0.6325 0.6325 0.6325 0.6325 0.6325 110 0.1355 0.1355 0.1355 0.1355 0.1355 7 0.5774 0.5774 0.5774 0.5774 0.5774 120 0.1296 0.1296 0.1296 0.1296 0.1296 8 0.5345 0.5345 0.5345 0.5345 0.5345 130 0.1245 0.1245 0.1245 0.1245 0.1245 9 0.5000 0.5000 0.5000 0.5000 0.5000 140 0.1200 0.1200 0.1200 0.1200 0.1200 10 0.4714 0.4714 0.4714 0.4714 0.4714 150 0.1159 0.1159 0.1159 0.1159 0.1159 11 0.4472 0.4472 0.4472 0.4472 0.4472 160 0.1122 0.1122 0.1122 0.1122 0.1122 12 0.4264 0.4264 0.4264 0.4264 0.4264 170 0.1088 0.1088 0.1088 0.1088 0.1088 13 0.4082 0.4082 0.4082 0.4082 0.4082 180 0.1057 0.1057 0.1057 0.1057 0.1057 14 0.3922 0.3922 0.3922 0.3922 0.3922 190 0.1029 0.1029 0.1029 0.1029 0.1029 15 0.3780 0.3780 0.3780 0.3780 0.3780 200 0.1003 0.1003 0.1003 0.1003 0.1003 16 0.3651 0.3651 0.3651 0.3651 0.3651 210 0.0978 0.0978 0.0978 0.0978 0.0978 17 0.3536 0.3536 0.3536 0.3536 0.3536 220 0.0956 0.0956 0.0956 0.0956 0.0956 18 0.3430 0.3430 0.3430 0.3430 0.3430 230 0.0935 0.0935 0.0935 0.0935 0.0935 19 0.3333 0.3333 0.3333 0.3333 0.3333 240 0.0915 0.0915 0.0915 0.0915 0.0915 20 0.3244 0.3244 0.3244 0.3244 0.3244 250 0.0896 0.0896 0.0896 0.0896 0.0896 21 0.3162 0.3162 0.3162 0.3162 0.3162 260 0.0879 0.0879 0.0879 0.0879 0.0879 22 0.3086 0.3086 0.3086 0.3086 0.3086 270 0.0862 0.0862 0.0862 0.0862 0.0862 23 0.3015 0.3015 0.3015 0.3015 0.3015 280 0.0847 0.0847 0.0847 0.0847 0.0847 24 0.2949 0.2949 0.2949 0.2949 0.2949 290 0.0832 0.0832 0.0832 0.0832 0.0832 25 0.2887 0.2887 0.2887 0.2887 0.2887 300 0.0818 0.0818 0.0818 0.0818 0.0818 26 0.2828 0.2828 0.2828 0.2828 0.2828 310 0.0805 0.0805 0.0805 0.0805 0.0805 27 0.2774 0.2774 0.2774 0.2774 0.2774 320 0.0792 0.0792 0.0792 0.0792 0.0792 28 0.2722 0.2722 0.2722 0.2722 0.2722 330 0.0780 0.0780 0.0780 0.0780 0.0780 29 0.2673 0.2673 0.2673 0.2673 0.2673 340 0.0768 0.0768 0.0768 0.0768 0.0768 30 0.2626 0.2626 0.2626 0.2626 0.2626 350 0.0757 0.0757 0.0757 0.0757 0.0757 31 0.2582 0.2582 0.2582 0.2582 0.2582 360 0.0746 0.0746 0.0746 0.0746 0.0746 32 0.2540 0.2540 0.2540 0.2540 0.2540 370 0.0736 0.0736 0.0736 0.0736 0.0736 33 0.2500 0.2500 0.2500 0.2500 0.2500 380 0.0726 0.0726 0.0726 0.0726 0.0726 34 0.2462 0.2462 0.2462 0.2462 0.2462 390 0.0717 0.0717 0.0717 0.0717 0.0717 35 0.2425 0.2425 0.2425 0.2425 0.2425 400 0.0708 0.0708 0.0708 0.0708 0.0708 36 0.2390 0.2390 0.2390 0.2390 0.2390 410 0.0699 0.0699 0.0699 0.0699 0.0699 37 0.2357 0.2357 0.2357 0.2357 0.2357 420 0.0691 0.0691 0.0691 0.0691 0.0691 38 0.2325 0.2325 0.2325 0.2325 0.2325 430 0.0683 0.0683 0.0683 0.0683 0.0683 39 0.2294 0.2294 0.2294 0.2294 0.2294 440 0.0675 0.0675 0.0675 0.0675 0.0675 40 0.2265 0.2265 0.2265 0.2265 0.2265 450 0.0667 0.0667 0.0667 0.0667 0.0667 41 0.2236 0.2236 0.2236 0.2236 0.2236 460 0.0660 0.0660 0.0660 0.0660 0.0660 42 0.2209 0.2209 0.2209 0.2209 0.2209 470 0.0653 0.0653 0.0653 0.0653 0.0653 43 0.2182 0.2182 0.2182 0.2182 0.2182 480 0.0646 0.0646 0.0646 0.0646 0.0646 44 0.2157 0.2157 0.2157 0.2157 0.2157 490 0.0640 0.0640 0.0640 0.0640 0.0640 45 0.2132 0.2132 0.2132 0.2132 0.2132 500 0.0633 0.0633 0.0633 0.0633 0.0633 46 0.2108 0.2108 0.2108 0.2108 0.2108 510 0.0627 0.0627 0.0627 0.0627 0.0627 47 0.2085 0.2085 0.2085 0.2085 0.2085 520 0.0621 0.0621 0.0621 0.0621 0.0621 48 0.2063 0.2063 0.2063 0.2063 0.2063 530 0.0615 0.0615 0.0615 0.0615 0.0615 49 0.2041 0.2041 0.2041 0.2041 0.2041 540 0.0609 0.0609 0.0609 0.0609 0.0609 50 0.2020 0.2020 0.2020 0.2020 0.2020 550 0.0604 0.0604 0.0604 0.0604 0.0604
Table 2.[MSER( ˆCip)]1/2for various Cip, and sample sizes (a) n= 1(1)50 and (b) n = 60(10)550 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 1.0000 1.0000 1.0000 1.0000 1.0000 60 0.1818 0.1818 0.1818 0.1818 0.1818 2 0.8660 0.8660 0.8660 0.8660 0.8660 70 0.1684 0.1684 0.1684 0.1684 0.1684 3 0.7454 0.7454 0.7454 0.7454 0.7454 80 0.1576 0.1576 0.1576 0.1576 0.1576 4 0.6614 0.6614 0.6614 0.6614 0.6614 90 0.1487 0.1487 0.1487 0.1487 0.1487 5 0.6000 0.6000 0.6000 0.6000 0.6000 100 0.1411 0.1411 0.1411 0.1411 0.1411 6 0.5528 0.5528 0.5528 0.5528 0.5528 110 0.1345 0.1345 0.1345 0.1345 0.1345 7 0.5151 0.5151 0.5151 0.5151 0.5151 120 0.1288 0.1288 0.1288 0.1288 0.1288 8 0.4841 0.4841 0.4841 0.4841 0.4841 130 0.1238 0.1238 0.1238 0.1238 0.1238 9 0.4581 0.4581 0.4581 0.4581 0.4581 140 0.1193 0.1193 0.1193 0.1193 0.1193 10 0.4359 0.4359 0.4359 0.4359 0.4359 150 0.1153 0.1153 0.1153 0.1153 0.1153 11 0.4166 0.4166 0.4166 0.4166 0.4166 160 0.1116 0.1116 0.1116 0.1116 0.1116 12 0.3997 0.3997 0.3997 0.3997 0.3997 170 0.1083 0.1083 0.1083 0.1083 0.1083 13 0.3846 0.3846 0.3846 0.3846 0.3846 180 0.1053 0.1053 0.1053 0.1053 0.1053 14 0.3712 0.3712 0.3712 0.3712 0.3712 190 0.1025 0.1025 0.1025 0.1025 0.1025 15 0.3590 0.3590 0.3590 0.3590 0.3590 200 0.0999 0.0999 0.0999 0.0999 0.0999 16 0.3480 0.3480 0.3480 0.3480 0.3480 210 0.0975 0.0975 0.0975 0.0975 0.0975 17 0.3379 0.3379 0.3379 0.3379 0.3379 220 0.0952 0.0952 0.0952 0.0952 0.0952 18 0.3287 0.3287 0.3287 0.3287 0.3287 230 0.0931 0.0931 0.0931 0.0931 0.0931 19 0.3201 0.3201 0.3201 0.3201 0.3201 240 0.0912 0.0912 0.0912 0.0912 0.0912 20 0.3123 0.3123 0.3123 0.3123 0.3123 250 0.0894 0.0894 0.0894 0.0894 0.0894 21 0.3049 0.3049 0.3049 0.3049 0.3049 260 0.0876 0.0876 0.0876 0.0876 0.0876 22 0.2981 0.2981 0.2981 0.2981 0.2981 270 0.0860 0.0860 0.0860 0.0860 0.0860 23 0.2917 0.2917 0.2917 0.2917 0.2917 280 0.0844 0.0844 0.0844 0.0844 0.0844 24 0.2857 0.2857 0.2857 0.2857 0.2857 290 0.0830 0.0830 0.0830 0.0830 0.0830 25 0.2800 0.2800 0.2800 0.2800 0.2800 300 0.0816 0.0816 0.0816 0.0816 0.0816 26 0.2747 0.2747 0.2747 0.2747 0.2747 310 0.0803 0.0803 0.0803 0.0803 0.0803 27 0.2696 0.2696 0.2696 0.2696 0.2696 320 0.0790 0.0790 0.0790 0.0790 0.0790 28 0.2649 0.2649 0.2649 0.2649 0.2649 330 0.0778 0.0778 0.0778 0.0778 0.0778 29 0.2603 0.2603 0.2603 0.2603 0.2603 340 0.0766 0.0766 0.0766 0.0766 0.0766 30 0.2560 0.2560 0.2560 0.2560 0.2560 350 0.0755 0.0755 0.0755 0.0755 0.0755 31 0.2519 0.2519 0.2519 0.2519 0.2519 360 0.0745 0.0745 0.0745 0.0745 0.0745 32 0.2480 0.2480 0.2480 0.2480 0.2480 370 0.0735 0.0735 0.0735 0.0735 0.0735 33 0.2443 0.2443 0.2443 0.2443 0.2443 380 0.0725 0.0725 0.0725 0.0725 0.0725 34 0.2407 0.2407 0.2407 0.2407 0.2407 390 0.0716 0.0716 0.0716 0.0716 0.0716 35 0.2373 0.2373 0.2373 0.2373 0.2373 400 0.0707 0.0707 0.0707 0.0707 0.0707 36 0.2341 0.2341 0.2341 0.2341 0.2341 410 0.0698 0.0698 0.0698 0.0698 0.0698 37 0.2309 0.2309 0.2309 0.2309 0.2309 420 0.0690 0.0690 0.0690 0.0690 0.0690 38 0.2279 0.2279 0.2279 0.2279 0.2279 430 0.0682 0.0682 0.0682 0.0682 0.0682 39 0.2250 0.2250 0.2250 0.2250 0.2250 440 0.0674 0.0674 0.0674 0.0674 0.0674 40 0.2222 0.2222 0.2222 0.2222 0.2222 450 0.0666 0.0666 0.0666 0.0666 0.0666 41 0.2195 0.2195 0.2195 0.2195 0.2195 460 0.0659 0.0659 0.0659 0.0659 0.0659 42 0.2169 0.2169 0.2169 0.2169 0.2169 470 0.0652 0.0652 0.0652 0.0652 0.0652 43 0.2144 0.2144 0.2144 0.2144 0.2144 480 0.0645 0.0645 0.0645 0.0645 0.0645 44 0.2120 0.2120 0.2120 0.2120 0.2120 490 0.0639 0.0639 0.0639 0.0639 0.0639 45 0.2096 0.2096 0.2096 0.2096 0.2096 500 0.0632 0.0632 0.0632 0.0632 0.0632 46 0.2074 0.2074 0.2074 0.2074 0.2074 510 0.0626 0.0626 0.0626 0.0626 0.0626 47 0.2052 0.2052 0.2052 0.2052 0.2052 520 0.0620 0.0620 0.0620 0.0620 0.0620 48 0.2031 0.2031 0.2031 0.2031 0.2031 530 0.0614 0.0614 0.0614 0.0614 0.0614 49 0.2010 0.2010 0.2010 0.2010 0.2010 540 0.0608 0.0608 0.0608 0.0608 0.0608 50 0.1990 0.1990 0.1990 0.1990 0.1990 550 0.0603 0.0603 0.0603 0.0603 0.0603
which is also a function of the sample size n only. Thus,[MSER( ˆCip)]1/2values are the same for all Cip values.
For short run applications (such as accepting a supplier providing short production runs in QS-9000 certification), the difference between the two relative errors is considered significant for sample sizes n ≤ 35, and we strongly recommend using the MLE ˆCip rather than the UMVUE ˆCip. For other applications with sample sizes n > 35, the difference between the two estimators is negligible (less than 0.52%).
3. ESTIMATING PROCESS INACCURACY To estimate the process inaccuracy, we consider the natural estimator ˆCia defined as the following, where
¯X = n
i=1Xi/n is the conventional estimator of the process mean µ. We note that the estimator ˆCia can also be written as a function of Cin:
ˆCia= ( ¯X − T )2 D2 = Cip n n ˆCia Cip = Cip n n( ¯X − T )2 σ2 If the process characteristic is normally distributed, then the estimator ˆCia is distributed as[Cip/n]χ12(δ), where χ12(δ) is a non-central chi-squared distribution with one degree of freedom and non-centrality parameter δ = n(µ − T )2/σ2. Therefore, the probability density function of ˆCia can be expressed as
g(y) =∞
k=0
[(ny)/(2Cip)]k+12 exp[−(ny)/(2Cip)]
y(k +1 2) ×(δ/2)kexp(−δ/2) (k + 1) , for y > 0
The rth moment, the expected value, the variance, and the mean squared error of ˆCia, therefore, can be calculated as E( ˆCiar) = C ip n r E[χ2 1(δ)]r =∞ k=0 2Cip n r (k + 1 2+ r) (k +1 2) ×(δ/2)(k + 1)kexp(−δ/2) E( ˆCia) = C ip n E[χ2 1(δ)] = C ip n (1 + δ) = Cip n +Cia Var( ˆCia) = C ip n 2 Var[χ2 1(δ)] = C ip n 2 (2 + 4δ) =4CipCia n + 2Cip2 n
MSE( ˆCia) = Var( ˆCia) + [E( ˆCia) − Cia]2 =4CipCia
n +
3Cip2
n2
Since ¯X is the MLE of µ, then by the invariance property of the MLE, the natural estimator ˆCia is the MLE of Cia. Noting that E( ˆCia) = Cia +
(Cip/n), and E( ˆCip) = Cip, the corrected estimator ˜Cia = ˆCia − ( ˆCip/n) must be unbiased for Cia. We can show that ˜Cia is the UMVUE of Cia, which is consistent. We can also show that √n( ˜Cia − Cia) converges to N(0, 4CipCia) in distribution, and ˜Cia is asymptotically efficient (see the Appendix for the proofs). Thus, in real-world applications using the UMVUE ˜Cia, which has all the desired statistical properties, as an estimate of Ciawould be reasonable. We note that the MLE ˆCia has smaller variance than the UMVUE ˜Cia. However, we can show that MSE( ˜Cia) = 4CipCia/n + [2/n(n − 1)](Cip)2, and so MSE( ˜Cia) − MSE( ˆCia) = [(3 − n)/n2(n − 1)](Cip)2, which is greater than 0 for n= 2, equal to 0 for n = 3, and less than 0 for n≥ 4. Therefore, the UMVUE ˜Cia has smaller mean squared error than the MLE ˆCia, and is more reliable for applications with n≥ 4.
Tables 3(a) and 3(b) display the relative error, [MSER( ˜Cia)]1/2, of the UMVUE ˜Cia for Cip = 1.00, 0.56, 0.44, 0.36, 0.25, and Cia = 2.25. The value of Cia is equivalent to Ca = 0.50. The relative errors,[MSER( ˜Cia)]1/2, for Cia = 5.06 and 0.56 are available from the authors. We note that if the process is perfectly centered, then Cia = 0.00 (equivalently,
Ca= 1.00). For example, for Cip= 1.00, Cia= 2.25, and n = 300 we have [MSER( ˜Cia)]1/2 = 0.0770. Thus, the average error (average relative deviation) of ˜Cia would be no greater than 7.70% of the true
Cia. Tables 4(a) and 4(b) display the relative error, [MSER( ˆCia)]1/2, of the MLE ˆCia for Cip = 1.00, 0.56, 0.44, 0.36, 0.25, and Cia = 2.25 (tables of [MSER( ˆCia)]1/2 for other values of Cia are available from the authors). We note that for n < 30, the difference between the two relative errors (percentage of deviations) is significant. However, for n > 30, the difference between the two is negligible (less than 0.3%), and using either of the two estimators is equally reliable.
Table 3.[MSER( ˜Cia)]1/2for various Cip, Cia= 2.25 and (a) n = 2(1)50 and (b) n = 60(10)550 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 ****** ****** ****** ****** ****** 60 0.1725 0.1292 0.1149 0.1034 0.0861 2 1.0423 0.7500 0.6588 0.5879 0.4843 70 0.1596 0.1196 0.1063 0.0957 0.0797 3 0.8114 0.5951 0.5257 0.4710 0.3902 80 0.1493 0.1119 0.0994 0.0895 0.0746 4 0.6909 0.5103 0.4517 0.4053 0.3364 90 0.1407 0.1055 0.0937 0.0844 0.0703 5 0.6126 0.4541 0.4024 0.3613 0.3002 100 0.1335 0.1001 0.0889 0.0800 0.0667 6 0.5563 0.4133 0.3665 0.3292 0.2737 110 0.1273 0.0954 0.0848 0.0763 0.0636 7 0.5132 0.3819 0.3387 0.3044 0.2531 120 0.1218 0.0913 0.0812 0.0731 0.0609 8 0.4788 0.3567 0.3165 0.2845 0.2366 130 0.1170 0.0877 0.0780 0.0702 0.0585 9 0.4506 0.3359 0.2981 0.2680 0.2230 140 0.1128 0.0846 0.0752 0.0676 0.0564 10 0.4268 0.3184 0.2826 0.2541 0.2115 150 0.1089 0.0817 0.0726 0.0653 0.0544 11 0.4065 0.3034 0.2693 0.2422 0.2016 160 0.1055 0.0791 0.0703 0.0633 0.0527 12 0.3888 0.2903 0.2578 0.2318 0.1929 170 0.1023 0.0767 0.0682 0.0614 0.0511 13 0.3732 0.2788 0.2475 0.2226 0.1853 180 0.0994 0.0746 0.0663 0.0596 0.0497 14 0.3594 0.2685 0.2385 0.2145 0.1786 190 0.0968 0.0726 0.0645 0.0581 0.0484 15 0.3470 0.2593 0.2303 0.2071 0.1725 200 0.0943 0.0707 0.0629 0.0566 0.0471 16 0.3358 0.2510 0.2230 0.2005 0.1670 210 0.0921 0.0690 0.0614 0.0552 0.0460 17 0.3256 0.2435 0.2163 0.1945 0.1620 220 0.0899 0.0674 0.0599 0.0539 0.0450 18 0.3163 0.2366 0.2101 0.1890 0.1574 230 0.0880 0.0660 0.0586 0.0528 0.0440 19 0.3078 0.2302 0.2045 0.1839 0.1532 240 0.0861 0.0646 0.0574 0.0516 0.0430 20 0.2999 0.2243 0.1993 0.1793 0.1493 250 0.0844 0.0633 0.0562 0.0506 0.0422 21 0.2926 0.2189 0.1945 0.1749 0.1457 260 0.0827 0.0620 0.0551 0.0496 0.0413 22 0.2858 0.2138 0.1900 0.1709 0.1423 270 0.0812 0.0609 0.0541 0.0487 0.0406 23 0.2794 0.2091 0.1858 0.1671 0.1392 280 0.0797 0.0598 0.0531 0.0478 0.0398 24 0.2735 0.2047 0.1818 0.1636 0.1362 290 0.0783 0.0587 0.0522 0.0470 0.0392 25 0.2679 0.2005 0.1781 0.1603 0.1335 300 0.0770 0.0577 0.0513 0.0462 0.0385 26 0.2626 0.1966 0.1747 0.1571 0.1309 310 0.0758 0.0568 0.0505 0.0454 0.0379 27 0.2577 0.1929 0.1714 0.1542 0.1284 320 0.0746 0.0559 0.0497 0.0447 0.0373 28 0.2530 0.1894 0.1683 0.1514 0.1261 330 0.0734 0.0551 0.0489 0.0440 0.0367 29 0.2486 0.1861 0.1654 0.1488 0.1239 340 0.0723 0.0542 0.0482 0.0434 0.0362 30 0.2444 0.1830 0.1626 0.1463 0.1218 350 0.0713 0.0535 0.0475 0.0428 0.0356 31 0.2404 0.1800 0.1599 0.1439 0.1198 360 0.0703 0.0527 0.0469 0.0422 0.0351 32 0.2365 0.1771 0.1574 0.1416 0.1180 370 0.0693 0.0520 0.0462 0.0416 0.0347 33 0.2329 0.1744 0.1550 0.1394 0.1162 380 0.0684 0.0513 0.0456 0.0410 0.0342 34 0.2294 0.1718 0.1527 0.1374 0.1144 390 0.0675 0.0506 0.0450 0.0405 0.0338 35 0.2261 0.1693 0.1505 0.1354 0.1128 400 0.0667 0.0500 0.0444 0.0400 0.0333 36 0.2229 0.1670 0.1484 0.1335 0.1112 410 0.0659 0.0494 0.0439 0.0395 0.0329 37 0.2199 0.1647 0.1463 0.1317 0.1097 420 0.0651 0.0488 0.0434 0.0390 0.0325 38 0.2169 0.1625 0.1444 0.1299 0.1082 430 0.0643 0.0482 0.0429 0.0386 0.0322 39 0.2141 0.1604 0.1425 0.1282 0.1068 440 0.0636 0.0477 0.0424 0.0381 0.0318 40 0.2114 0.1584 0.1407 0.1266 0.1055 450 0.0629 0.0471 0.0419 0.0377 0.0314 41 0.2088 0.1564 0.1390 0.1251 0.1042 460 0.0622 0.0466 0.0414 0.0373 0.0311 42 0.2063 0.1545 0.1373 0.1236 0.1029 470 0.0615 0.0461 0.0410 0.0369 0.0308 43 0.2039 0.1527 0.1357 0.1221 0.1017 480 0.0609 0.0456 0.0406 0.0365 0.0304 44 0.2015 0.1510 0.1342 0.1207 0.1006 490 0.0602 0.0452 0.0402 0.0361 0.0301 45 0.1993 0.1493 0.1327 0.1194 0.0994 500 0.0596 0.0447 0.0398 0.0358 0.0298 46 0.1971 0.1476 0.1312 0.1181 0.0984 510 0.0591 0.0443 0.0394 0.0354 0.0295 47 0.1950 0.1461 0.1298 0.1168 0.0973 520 0.0585 0.0439 0.0390 0.0351 0.0292 48 0.1929 0.1445 0.1284 0.1156 0.0963 530 0.0579 0.0434 0.0386 0.0348 0.0290 49 0.1909 0.1430 0.1271 0.1144 0.0953 540 0.0574 0.0430 0.0383 0.0344 0.0287 50 0.1890 0.1416 0.1258 0.1132 0.0943 550 0.0569 0.0426 0.0379 0.0341 0.0284
Table 4.[MSER( ˆCia)]1/2for various Cip, Cia= 2.25 and (a) n = 2(1)50 and (b) n = 60(10)550 Cip Cip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 1.5396 1.0897 0.9525 0.8466 0.6939 60 0.1726 0.1293 0.1149 0.1034 0.0861 2 1.0184 0.7395 0.6514 0.5824 0.4811 70 0.1597 0.1197 0.1064 0.0957 0.0797 3 0.8114 0.5951 0.5257 0.4710 0.3902 80 0.1494 0.1119 0.0995 0.0895 0.0746 4 0.6939 0.5116 0.4526 0.4060 0.3368 90 0.1408 0.1055 0.0938 0.0844 0.0703 5 0.6158 0.4555 0.4034 0.3620 0.3006 100 0.1336 0.1001 0.0890 0.0800 0.0667 6 0.5592 0.4146 0.3673 0.3298 0.2740 110 0.1273 0.0954 0.0848 0.0763 0.0636 7 0.5158 0.3830 0.3395 0.3050 0.2535 120 0.1219 0.0914 0.0812 0.0731 0.0609 8 0.4811 0.3577 0.3172 0.2850 0.2369 130 0.1171 0.0878 0.0780 0.0702 0.0585 9 0.4526 0.3368 0.2987 0.2684 0.2232 140 0.1128 0.0846 0.0752 0.0676 0.0564 10 0.4286 0.3192 0.2832 0.2545 0.2117 150 0.1090 0.0817 0.0726 0.0653 0.0544 11 0.4081 0.3041 0.2698 0.2525 0.2018 160 0.1055 0.0791 0.0703 0.0633 0.0527 12 0.3902 0.2909 0.2582 0.2321 0.1931 170 0.1024 0.0767 0.0682 0.0614 0.0511 13 0.3745 0.2793 0.2479 0.2229 0.1855 180 0.0995 0.0746 0.0663 0.0596 0.0497 14 0.3606 0.2690 0.2388 0.2147 0.1787 190 0.0968 0.0726 0.0645 0.0581 0.0484 15 0.3481 0.2598 0.2306 0.2074 0.1726 200 0.0944 0.0707 0.0629 0.0566 0.0472 16 0.3368 0.2515 0.2232 0.2007 0.1671 210 0.0921 0.0690 0.0614 0.0552 0.0460 17 0.3265 0.2439 0.2165 0.1947 0.1621 220 0.0900 0.0674 0.0599 0.0540 0.0450 18 0.3172 0.2369 0.2104 0.1892 0.1575 230 0.0880 0.0660 0.0586 0.0528 0.0440 19 0.3086 0.2305 0.2047 0.1841 0.1533 240 0.0861 0.0646 0.0574 0.0517 0.0430 20 0.3006 0.2247 0.1995 0.1794 0.1494 250 0.0844 0.0633 0.0562 0.0506 0.0422 21 0.2933 0.2192 0.1947 0.1751 0.1458 260 0.0827 0.0620 0.0551 0.0496 0.0414 22 0.2864 0.2141 0.1901 0.1710 0.1424 270 0.0812 0.0609 0.0541 0.0487 0.0406 23 0.2800 0.2094 0.1859 0.1672 0.1393 280 0.0797 0.0598 0.0531 0.0478 0.0398 24 0.2740 0.2049 0.1820 0.1637 0.1363 290 0.0783 0.0587 0.0522 0.0470 0.0392 25 0.2684 0.2007 0.1783 0.1604 0.1336 300 0.0770 0.0578 0.0513 0.0462 0.0385 26 0.2632 0.1968 0.1748 0.1573 0.1310 310 0.0758 0.0568 0.0505 0.0454 0.0379 27 0.2582 0.1931 0.1715 0.1543 0.1285 320 0.0746 0.0559 0.0497 0.0447 0.0373 28 0.2535 0.1896 0.1684 0.1515 0.1262 330 0.0734 0.0551 0.0489 0.0440 0.0367 29 0.2490 0.1863 0.1655 0.1489 0.1240 340 0.0723 0.0542 0.0482 0.0434 0.0362 30 0.2448 0.1831 0.1627 0.1464 0.1219 350 0.0713 0.0535 0.0475 0.0428 0.0356 31 0.2408 0.1801 0.1600 0.1440 0.1199 360 0.0703 0.0527 0.0469 0.0422 0.0351 32 0.2369 0.1773 0.1575 0.1417 0.1180 370 0.0693 0.0520 0.0462 0.0416 0.0347 33 0.2333 0.1746 0.1551 0.1395 0.1162 380 0.0684 0.0513 0.0456 0.0410 0.0342 34 0.2298 0.1720 0.1528 0.1374 0.1145 390 0.0675 0.0506 0.0450 0.0405 0.0338 35 0.2264 0.1695 0.1506 0.1355 0.1128 400 0.0667 0.0500 0.0445 0.0400 0.0333 36 0.2232 0.1671 0.1485 0.1336 0.1112 410 0.0659 0.0494 0.0439 0.0395 0.0329 37 0.2202 0.1648 0.1464 0.1317 0.1097 420 0.0651 0.0488 0.0434 0.0390 0.0325 38 0.2172 0.1626 0.1445 0.1300 0.1083 430 0.0643 0.0482 0.0429 0.0386 0.0322 39 0.2144 0.1605 0.1426 0.1283 0.1069 440 0.0636 0.0477 0.0424 0.0381 0.0318 40 0.2117 0.1585 0.1408 0.1267 0.1055 450 0.0629 0.0415 0.0419 0.0377 0.0314 41 0.2091 0.1565 0.1391 0.1251 0.1042 460 0.0622 0.0466 0.0415 0.0373 0.0311 42 0.2066 0.1546 0.1374 0.1236 0.1030 470 0.0615 0.0461 0.0410 0.0369 0.0308 43 0.2041 0.1528 0.1358 0.1222 0.1018 480 0.0609 0.0457 0.0406 0.0365 0.0304 44 0.2018 0.1511 0.1342 0.1208 0.1006 490 0.0603 0.0452 0.0402 0.0361 0.0301 45 0.1995 0.1494 0.1327 0.1194 0.0995 500 0.0596 0.0447 0.0398 0.0358 0.0298 46 0.1973 0.1477 0.1313 0.1181 0.0984 510 0.0591 0.0443 0.0394 0.0354 0.0295 47 0.1952 0.1462 0.1299 0.1168 0.0973 520 0.0585 0.0439 0.0390 0.0351 0.0292 48 0.1931 0.1446 0.1285 0.1156 0.0963 530 0.0579 0.0434 0.0386 0.0348 0.0290 49 0.1911 0.1431 0.1272 0.1144 0.0953 540 0.0574 0.0430 0.0383 0.0344 0.0287 50 0.1892 0.1417 0.1259 0.1133 0.0944 550 0.0569 0.0426 0.0379 0.0341 0.0284
4. ESTIMATING PROCESS INCAPABILITY To estimate the process incapability (a combined mea-sure of process imprecision and process inaccuracy), we consider the natural estimator ˆCpp defined as the following, where ¯X =ni=1Xi/n, which can also be written as a function of Cip:
ˆCpp= 1n n i=1 (Xi − ¯X)2 D2 + ( ¯X − T )2 D2 = Cnipn ˆCCpp ip = Cnip n i=1 (Xi− T )2 σ2
If the process characteristic is normally distributed, then the estimator ˆCppis distributed as[Cip/n]χn2(δ), where χn2(δ) is a non-central chi-squared distribution with n degrees of freedom and non-centrality parameter δ = n(µ − T )2/σ2 = nCia/Cip. Therefore, the probability density function of ˆCppcan be expressed as h(y) =∞ k=0 [(ny)/(2Cip)]k+(n/2)exp[−(ny)/(2Cip)] y(k + (n/2)) ×(δ/2)(k + 1)kexp(−δ/2) , for y > 0
The rth moment (hence the expected value, the variance, and the mean squared error) of ˆCpp, therefore can be calculated as follows:
E( ˆCppr ) = C ip n r E[χ2 n(δ)]r =∞ k=0 2Cip n r (k + (n/2) + r) (k + (n/2)) ×(δ/2)kexp(−δ/2) (k + 1) E( ˆCpp) = C ip n E[χ2 n(δ)] = Cip n (n + δ) = Cip+ Cia = Cpp Var( ˆCpp) = C ip n 2 Var[χ2 n(δ)] = C ip n 2 (2n + 4δ) =2Cipn (Cia+ Cpp)
MSE( ˆCpp) = Var( ˆCpp) + [E( ˆCpp) − Cpp]2 =2Cipn (Cia+ Cpp)
If the process characteristic follows the normal distribution, then we can show that ˆCpp is the MLE, which is also the UMVUE of Cpp. We can also show that ˆCppis consistent,√n( ˆCpp− Cpp) converges to N(0, 2CipCia+ 2CipCpp) in distribution, and ˆCpp is asymptotically efficient (see the Appendix for the proofs). Since the estimator has all the desired statistical properties, in practice using ˆCppto estimate process incapability would be reasonable.
5. DECISION MAKING
Under the normality assumption, n ˆCpp/(Cpp − Cia) is distributed as χ2
n(δ), a non-central chi-squared distribution with n degrees of freedom and non-centrality parameter δ= n(µ − T )2/σ2= nCia/Cip. Let W = W(X1, X2, . . . , Xn) be a statistic calculated from the sample data satisfying P{Cpp≤ W} = 1 − α, where the confidence level 1− α does not depend on
Cpp. Then, W is a 100(1−α)% upper confidence limit for Cpp. We note that
P{Cpp ≤ W} = P{Cpp− Cia≤ W − Cia} = P{1/(Cpp− Cia) ≥ 1/(W − Cia)} = P{n ˆCpp/( ˆCpp− Cia) ≥ n ˆCpp/(W −Cia)} = P{χ2 n(δ) ≥ n ˆCpp/(W − Cia)} = 1 − α Therefore, n ˆCpp/(W − Cia) = χn2(α, δ), where χ2
n(α, δ) is the (lower) αth percentile of the χn2(δ) distribution. A 100(1− α)% upper confidence limit on Cpp can be written in terms of ˆCpp as W = Cia+ [n ˆCpp/χ2
n(α, δ)].
Tables5(a),6(a), and7(a)give 90%, 95%, and 99% upper confidence limits for Cppunder µ = T with n given, and ˆCpp calculated from the sample data. On the other hand, ˆCpp = χn2(α, δ)(W − Cia)/n depends on W and Cia. In the special case when µ = T and
W equals the recommended maximum value for Cpp, the probability that Cpp ≤ W would be either 1 or 0 if Cppwere known. In practice, since Cppis unknown, we take a random sample of size n and calculate ˆCpp.
Suppose that a process is capable if ˆCpp ≤
χ2
n(α, δ)(C0− Cia)/n, where C0is the recommended maximum value, and we claim that the process is capable for at least 100(1−α)% of the time. Therefore,
Table 5. (a) The 90% upper confidence limits for Cppunder µ= T , with given ˆCpp. (b) The maximum value of ˆCppunder µ= T for which the process is capable (Cpp≤ C0) 90% of the time
(a) Sample size n ˆCpp 5 10 15 20 25 30 35 40 45 50 55 60 0.25 0.776 0.514 0.439 0.402 0.379 0.364 0.353 0.344 0.337 0.332 0.327 0.323 0.36 1.118 0.740 0.632 0.579 0.546 0.524 0.508 0.496 0.486 0.478 0.471 0.465 0.44 1.366 0.904 0.772 0.707 0.668 0.641 0.621 0.606 0.594 0.584 0.575 0.568 0.56 1.739 1.151 0.983 0.900 0.850 0.816 0.790 0.771 0.756 0.743 0.732 0.723 1.00 3.105 2.055 1.755 1.607 1.518 1.456 1.411 1.377 1.349 1.323 1.308 1.291 Sample size n ˆCpp 70 80 90 100 110 120 130 140 150 160 170 180 0.25 0.316 0.311 0.307 0.304 0.301 0.298 0.296 0.294 0.292 0.291 0.289 0.288 0.36 0.455 0.448 0.442 0.437 0.433 0.429 0.426 0.423 0.421 0.419 0.417 0.415 0.44 0.557 0.548 0.540 0.534 0.529 0.525 0.521 0.518 0.515 0.512 0.509 0.507 0.56 0.708 0.697 0.688 0.680 0.673 0.668 0.669 0.659 0.655 0.651 0.648 0.646 1.00 1.265 1.245 1.228 1.214 1.203 1.193 1.184 1.176 1.169 1.163 1.158 1.153 (b) Sample size n C0 5 10 15 20 25 30 35 40 45 50 55 60 0.25 0.081 0.122 0.142 0.156 0.165 0.172 0.177 0.182 0.185 0.188 0.191 0.194 0.36 0.116 0.175 0.205 0.224 0.237 0.247 0.255 0.261 0.267 0.271 0.275 0.279 0.44 0.142 0.214 0.251 0.274 0.290 0.302 0.312 0.320 0.326 0.332 0.336 0.341 0.56 0.180 0.272 0.319 0.348 0.369 0.385 0.397 0.407 0.415 0.422 0.428 0.434 1.00 0.322 0.487 0.570 0.622 0.659 0.687 0.708 0.726 0.741 0.754 0.765 0.774 Sample size n C0 70 80 90 100 110 120 130 140 150 160 170 180 0.25 0.198 0.201 0.204 0.206 0.208 0.210 0.211 0.213 0.214 0.215 0.216 0.217 0.36 0.285 0.289 0.293 0.296 0.299 0.302 0.304 0.306 0.308 0.309 0.311 0.312 0.44 0.348 0.354 0.358 0.362 0.366 0.369 0.372 0.374 0.376 0.378 0.380 0.382 0.56 0.443 0.450 0.456 0.461 0.466 0.470 0.473 0.476 0.479 0.481 0.484 0.486 1.00 0.790 0.803 0.814 0.824 0.832 0.839 0.845 0.850 0.855 0.860 0.864 0.868
the factor χn2(α, δ)(C0−Cia)/n is the maximum value of the estimated incapability index ˆCpp in order that the process is considered capable at least 100(1−
α)% of the time. Tables5(b),6(b), and7(b)give the maximum values of ˆCpp in the case with µ = T for the process to be considered capable (i.e., Cpp ≤ C0) 90%, 95%, and 99% of the time.
Suppose that the requirement for a process to be capable is that Cpp ≤ 1.00. We take a random sample of size n, and calculate ˆCpp. Using Table 6(b) based on the random sample of n= 30, for example, we obtain
C0= 0.616. Thus, if the calculated ˆCpp≤ 0.616, then we claim that the process is capable at least 95% of the time.
6. CONCLUSION
Greenwich and Jahr-Schaffrath [1] introduced the process incapability index Cpp = Cip + Cia, which provides an uncontaminated separation between information concerning the process precision (Cip) and process accuracy (Cia). In this note, we consider the three indices, and investigate the statistical properties of their natural estimators. For the three indices, we obtain their UMVUEs and MLEs. For each index, we compare the reliability of the two estimators based on their relative errors (square root of the relative mean squared error). In addition, we construct 90%, 95%, and 99% upper confidence limits,
Table 6. (a) The 95% upper confidence limits for Cppunder µ= T , with given ˆCpp. (b) The maximum value of ˆCppunder µ= T for which the process is capable (Cpp≤ C0) 95% of the time
(a) Sample size n ˆCpp 5 10 15 20 25 30 35 40 45 50 55 60 0.25 1.091 0.634 0.516 0.461 0.428 0.406 0.389 0.377 0.368 0.360 0.353 0.347 0.36 1.571 0.914 0.744 0.664 0.616 0.584 0.561 0.543 0.529 0.518 0.508 0.500 0.44 1.921 1.117 0.909 0.811 0.753 0.714 0.686 0.664 0.647 0.633 0.621 0.611 0.56 2.444 1.421 1.157 1.032 0.958 0.908 0.872 0.845 0.823 0.805 0.791 0.778 1.00 4.365 2.538 2.066 1.843 1.711 1.622 1.558 1.509 1.470 1.438 1.412 1.389 Sample size n ˆCpp 70 80 90 100 110 120 130 140 150 160 170 180 0.25 0.338 0.331 0.325 0.321 0.317 0.313 0.311 0.308 0.306 0.304 0.302 0.300 0.36 0.487 0.477 0.469 0.462 0.456 0.451 0.447 0.443 0.440 0.437 0.435 0.432 0.44 0.595 0.583 0.573 0.565 0.558 0.552 0.547 0.542 0.538 0.534 0.531 0.528 0.56 0.758 0.742 0.729 0.719 0.710 0.702 0.696 0.690 0.685 0.680 0.676 0.672 1.00 1.353 1.325 1.302 1.283 1.267 1.254 1.242 1.232 1.223 1.214 1.207 1.200 (b) Sample size n C0 5 10 15 20 25 30 35 40 45 50 55 60 0.25 0.057 0.099 0.121 0.136 0.146 0.154 0.160 0.166 0.170 0.174 0.177 0.180 0.36 0.082 0.142 0.174 0.195 0.210 0.222 0.231 0.239 0.245 0.250 0.255 0.259 0.44 0.101 0.173 0.213 0.239 0.257 0.271 0.282 0.292 0.299 0.306 0.312 0.317 0.56 0.128 0.221 0.271 0.304 0.327 0.345 0.359 0.371 0.381 0.389 0.397 0.403 1.00 0.229 0.394 0.484 0.543 0.584 0.616 0.642 0.663 0.680 0.695 0.708 0.720 Sample size n C0 70 80 90 100 110 120 130 140 150 160 170 180 0.25 0.185 0.189 0.192 0.195 0.197 0.199 0.201 0.203 0.204 0.206 0.207 0.208 0.36 0.266 0.272 0.277 0.281 0.284 0.287 0.290 0.292 0.294 0.296 0.298 0.300 0.44 0.325 0.332 0.338 0.343 0.347 0.351 0.354 0.357 0.360 0.362 0.365 0.367 0.56 0.414 0.423 0.430 0.436 0.442 0.447 0.451 0.455 0.458 0.461 0.464 0.467 1.00 0.739 0.755 0.768 0.779 0.789 0.798 0.805 0.812 0.818 0.823 0.829 0.833
and the maximum values of ˆCppfor which the process is capable. The results obtained in this paper are useful to the practitioners in choosing good estimators and making reliable decisions on judging process capability.
APPENDIX
Theorem 1. If the process characteristic is normally
distributed, then:
(a) ˆCipis the UMVUE ofCip; (b) ˆCipis consistent;
(c) √n( ˆCip − Cip) converges to N(0, 2Cip2) in
distribution;
(d) ˆCipis asymptotically efficient.
Proof. (a) Since Sn−12 is a sufficient and complete statistic for σ2, and the unbiased estimator ˆCip is a function Sn−12 of only, then by the Lehmann–Scheffe Theorem, ˆCipis the UMVUE. (b) For all ε > 0,
P{| ˆCip− Cip| > ε} < E( ˆCip− Cip)2/ε2 Since
E( ˆCip− Cip)2= Var( ˆCip) = 2Cip2/(n − 1) converges to zero, then ˆCip must be consistent. (c) Greenwich and Jahr-Schaffrath [1] showed that, under general conditions, √n( ˆCip − Cip) converges to N(0, σip2) in distribution, where σip2 = (µ4 −
Table 7. (a) The 99% upper confidence limits for Cppunder µ= T , with given ˆCpp. (b) The maximum value of ˆCppunder µ= T for which the process is capable (Cpp≤ C0) 99% of the time
(a) Sample size n ˆCpp 5 10 15 20 25 30 35 40 45 50 55 60 0.25 2.255 0.977 0.717 0.605 0.542 0.502 0.473 0.451 0.434 0.421 0.410 0.400 0.36 3.247 1.407 1.033 0.872 0.781 0.722 0.681 0.650 0.625 0.606 0.590 0.576 0.44 3.969 1.720 1.262 1.065 0.955 0.883 0.832 0.794 0.764 0.741 0.721 0.704 0.56 5.051 2.189 1.606 1.356 1.215 1.123 1.059 1.011 0.973 0.943 0.917 0.896 1.00 9.020 3.909 2.868 2.421 2.169 2.006 1.891 1.805 1.737 1.683 1.638 1.601 Sample size n ˆCpp 70 80 90 100 110 120 130 140 150 160 170 180 0.25 0.385 0.374 0.364 0.357 0.351 0.345 0.340 0.336 0.333 0.330 0.327 0.324 0.36 0.555 0.538 0.525 0.514 0.505 0.497 0.490 0.484 0.479 0.475 0.471 0.467 0.44 0.678 0.657 0.641 0.628 0.617 0.607 0.599 0.592 0.586 0.580 0.575 0.571 0.56 0.863 0.837 0.816 0.799 0.785 0.773 0.763 0.754 0.746 0.738 0.732 0.726 1.00 1.540 1.494 1.457 1.427 1.402 1.381 1.362 1.346 1.331 1.319 1.307 1.297 (b) Sample size n C0 5 10 15 20 25 30 35 40 45 50 55 60 0.25 0.028 0.064 0.087 0.103 0.115 0.125 0.132 0.139 0.144 0.149 0.153 0.156 0.36 0.040 0.092 0.126 0.149 0.166 0.179 0.190 0.199 0.207 0.214 0.220 0.225 0.44 0.049 0.113 0.153 0.182 0.203 0.219 0.233 0.244 0.253 0.261 0.269 0.275 0.56 0.062 0.143 0.195 0.231 0.258 0.279 0.296 0.310 0.322 0.333 0.342 0.350 1.00 0.111 0.256 0.349 0.413 0.461 0.498 0.529 0.554 0.576 0.594 0.610 0.625 Sample size n C0 70 80 90 100 110 120 130 140 150 160 170 180 0.25 0.162 0.167 0.172 0.175 0.178 0.181 0.184 0.186 0.188 0.190 0.191 0.193 0.36 0.234 0.241 0.247 0.252 0.257 0.261 0.264 0.268 0.270 0.273 0.275 0.278 0.44 0.286 0.294 0.302 0.308 0.314 0.319 0.323 0.327 0.330 0.334 0.337 0.339 0.56 0.364 0.375 0.384 0.392 0.399 0.406 0.411 0.416 0.421 0.425 0.428 0.432 1.00 0.649 0.669 0.686 0.701 0.713 0.724 0.734 0.743 0.751 0.758 0.765 0.771
distribution, µ4= 3σ4and Cip= (σ/D)2. (d) Under the normality assumption, the information matrix can be calculated as follows. Since the information lower bound is achieved, then ˆCip must be asymptotically efficient: I (θ) = I (µ, σ) = 1/σ2 0 0 1/(2σ4) , ∂C ip ∂µ ∂Cip ∂σ2 I−1(θ) n ∂Cip ∂µ ∂Cip ∂σ2 = 2Cip2 n
Theorem 2. If the process characteristic is normally
distributed, then:
(a) ˜Ciais the UMVUE ofCia; (b) ˜Ciais consistent;
(c) √n( ˜Cia − Cia) converges to N(0, 4CipCia) in
distribution;
(d) ˜Ciais asymptotically efficient.
Proof. (a) Noting that ¯X is a sufficient and complete
statistic for µ, and that the unbiased estimator ˜Cia is a function of ¯X only. By the Lehmann–Scheffe Theorem, ˜Ciais the UMVUE of Cia. (b) For any ε > 0,
P{| ˜Cia− Cia| > ε} < E( ˜Cia− Cia)2/ε2 Since
E( ˜Cia− Cia)2= 4CipCia/n + [2Cip2/(n 2− n)]
converges to zero, then ˜Cia must be consis-tent. (c) Greenwich and Jahr-Schaffrath [1] showed that under general conditions √n( ˆCia − Cia) con-verges to N(0, σia2) in distribution, where σia2 = 4(µ− T )2σ2/D4. Under the normality assumption, √n( ˆC
ia − Cia) must converge to N(0, 4CipCia) in distribution. Since ( ˜Cia − ˆCia) converges to zero in probability, then by Slutsky’s Theorem,
√ n( ˜Cia− Cia) = √ n( ˜Cia− ˆCia) + √ n( ˆCia− Cia) converges to N(0, 4CipCia) in distribution. (d) Under the normality assumption, the information matrix can be calculated as follows. Since the information lower bound is achieved, then the estimator ˜Cia must be asymptotically efficient: I (θ) = I (µ, σ) = 1/σ2 0 0 1/(2σ4) , ∂C ia ∂µ ∂Cia ∂σ2 I−1(θ) n ∂Cia ∂µ ∂Cia ∂σ2 = 4Cip2Cia n
Theorem 3. If the process characteristic is normally
distributed, then:
(a) ˆCppis the MLE ofCpp; (b) ˆCppis the UMVUE ofCpp; (c) ˆCppis consistent;
(d) √n( ˆCpp − Cpp) converges to N(0, 2CipCia + 2CipCpp) in distribution;
(e) ˆCppis asymptotically efficient.
Proof. (a) Since ( ¯X, Sn2) is the MLE of (µ, σ2), where S2
n = ni=1(Xi − ¯X)2/n, and ˆCpp = (Sn2/D2) + [( ¯X − T )2/D2], then by the invariance property of the MLE, ˆCpp is the MLE of Cpp. (b) We note that
( ¯X, S2
n) is sufficient and complete for (µ, σ2). Since the unbiased estimator ˆCpp is a function of ( ¯X, Sn2) only, then by the Lehmann–Scheffe theorem ˆCppis the UMVUE. (c) For all ε > 0, P{| ˆCpp− Cpp| > ε} < E( ˆCpp−Cpp)2/ε2. Since E( ˆCpp−Cpp)2= Var( ˆCpp) = 2Cip(Cia+ Cpp)/n converges to zero, then ˆCpp must be consistent. (d) Greenwich and Jahr-Schaffrath [1]
showed that under general conditions√n( ˆCpp− Cpp) converges to N(0, σpp2) in distribution, where
σ2
pp= [4(µ − T )2σ2/D4] + [4µ3(µ − T )/D4] + [(µ4− σ4)/D4]
Therefore, √n( ˆCpp − Cpp) converges to
N(0, 2CipCia+ 2CipCpp) in distribution if the process is normal. (e) Under the normality assumption, the information matrix can be calculated as follows. Since the information lower bound is achieved, then the estimator ˆCppmust be asymptotically efficient:
I (θ) = I (µ, σ) = 1/σ2 0 0 1/(2σ4) , ∂C pp ∂µ ∂Cpp ∂σ2 I−1(θ) n ∂Cpp ∂µ ∂Cpp ∂σ2 =2Cipn (Cia+ Cpp) REFERENCES
1. Greenwich M, Jahr-Schaffrath BL. A process incapability index. International Journal of Quality and Reliability Management 1995; 12(4):58–71.
2. Chan LK, Cheng SW, Spiring FA. A new measure of process capability: Cpm. Journal of Quality Technology 1988; 20(3):162–175.
3. Kane VE. Process capability indices. Journal of Quality Technology 1986; 18(1):41–52.
4. Pearn WL, Lin GH, Chen KS. Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory and Methods 1998; 27(4):985–1000.
Authors’ biographies:
W. L. Pearn is a Professor in the Department of
Industrial Engineering and Management, National Chiao-Tung University. He received his Ph.D. degree in operations research from the University of Maryland, College Park, MD, USA. He has published numerous papers in the areas of network optimization, scheduling, and process capability analysis.
G. H. Lin is a Lecturer in the Department of
Telecom-munications Engineering at the National Penghu Institute of Technology. He received his Ph.D. degree from the Department of Industrial Engineering and Management, National Chiao-Tung University. His research interests include applied statistics and quality management.
![Table 1. [MSE R ( ˆ C ip )] 1/2 for various C ip , and sample sizes (a) n = 2(1)50 and (b) n = 60(10)550 C ip C ip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 ****** ****** ****** ****** ****** 60 0.1841 0.1841 0.1841 0.1841 0.1841 2 1.4142 1.4](https://thumb-ap.123doks.com/thumbv2/9libinfo/7707141.145603/3.892.128.774.251.1059/table-mse-ˆ-c-various-sample-sizes-ip.webp)
![Table 2. [MSE R ( ˆ C ip )] 1/2 for various C ip , and sample sizes (a) n = 1(1)50 and (b) n = 60(10)550 C ip C ip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 1.0000 1.0000 1.0000 1.0000 1.0000 60 0.1818 0.1818 0.1818 0.1818 0.1818 2 0.8660 0](https://thumb-ap.123doks.com/thumbv2/9libinfo/7707141.145603/4.892.136.767.247.1057/table-mse-ˆ-c-various-sample-sizes-ip.webp)
![Table 3. [MSE R ( ˜ C ia )] 1/2 for various C ip , C ia = 2.25 and (a) n = 2(1)50 and (b) n = 60(10)550 C ip C ip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 ****** ****** ****** ****** ****** 60 0.1725 0.1292 0.1149 0.1034 0.0861 2 1.0423 0.75](https://thumb-ap.123doks.com/thumbv2/9libinfo/7707141.145603/6.892.124.780.245.1060/table-mse-r-c-ia-various-c-ip.webp)
![Table 4. [MSE R ( ˆ C ia )] 1/2 for various C ip , C ia = 2.25 and (a) n = 2(1)50 and (b) n = 60(10)550 C ip C ip n 1.00 0.56 0.44 0.36 0.25 n 1.00 0.56 0.44 0.36 0.25 1 1.5396 1.0897 0.9525 0.8466 0.6939 60 0.1726 0.1293 0.1149 0.1034 0.0861 2 1.0184 0.73](https://thumb-ap.123doks.com/thumbv2/9libinfo/7707141.145603/7.892.135.767.245.1059/table-mse-various-c-c-ip-c-ip.webp)
