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COMMUN. STATIST.—SIMULA., 31(3), 425–442 (2002) 1 2 3 4
CONTROL CHARTS AND SAMPLING PLANS
5 6 7 8 9 10
THE MEDIAN ABSOLUTE DEVIATIONS AND THEIR APPLICATIONS TO SHEWHART X CONTROL CHARTS
11 12 13
Chunjie Wu, Yi Zhao, and Zhaojun Wang*
14 15 16 17
School of Mathematics, Nankai University, Tianjin 300071, P.R. China
18 19 20
ABSTRACT
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Standard Shewhart X control charts with estimated control limits are widely used in practice. There are four ways to estimate the standard deviation generally, as which we call the average range, the average sample standard deviation, the pooled sample standard deviation and the average absolute deviation, respectively. We give three new estimators which based on median in order to estimate standard deviation in this paper, and we get their means through simulated computation. The simulated results are then used to discuss their properties when the data is from a "-contaminated normal distribution. At last, we simulate the in-control and outof-control average run length of Shewhart X control charts when the process standard deviation is estimated by seven different ways including three absolute deviations to median mentioned in the paper.
37 38 39 40 41
*Corresponding author. E-mail:
[email protected]
42
425 Copyright & 2002 by Marcel Dekker, Inc.
www.dekker.com
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Key Words: Contaminated data; Average Statistical process control; Simulation
run
length;
45 46 47
1. INTRODUCTION
48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
Statistical process control charts are popularly applied in industry. In particular, the standard Shewhart (SS) X control chart has been widely used in practice. For a process variable that is N(, 2 ), the process can be monitored by plotting the rational sample means on a SS X chart with 3 control limits, when the process mean and standard deviation are known. When and are unknown, we can estimate the control limits. Let Xij , j ¼ 1, 2 . . . , m; j ¼ 1, 2, . . . , n denote m independent historical iid samples of size n taken in control, that is Xij Nð, 2 Þ. Then the control limits can be estimated by d ¼ X þ 3 p^ffiffiffi , CL c ¼ X , LCL d ¼ X 3 p^ffiffiffi , UCL n n
ð1:1Þ
where X is an estimator for as follows m n 1X 1X X ¼ X m i¼1 n j¼1 ij
! ¼
m 1X X , m i¼1 i
ð1:2Þ
and ^ is one of the following four different estimators for (Chen (1), Montgomery (2), Wang (6)):
70 71 72
^ 1 ¼
R R þ R2 þ þ Rm ¼ 1 , d2 ðnÞ m d2 ðnÞ
ð1:3Þ
^ 2 ¼
1 S ¼ ðS þ S2 þ þ Sm Þ, c4 ðnÞ m c4 ðnÞ 1
ð1:4Þ
^ 3 ¼
Sp ¼ c4 ðmðn 1Þ þ 1Þ
73 74 75 76 77 78 79
1
2 m ðS1
2 þ S22 þ þ Sm Þ c4 ðmðn 1Þ þ 1Þ
1=2 ,
ð1:5Þ
80 m X n X
81 82 83 84
^ 4 ¼ D =gð1Þ n ¼
jXij X i j
i¼1 j¼1
mn gð1Þ n
,
ð1:6Þ
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SHEWHART X CONTROL CHARTS 85 86
427
where Ri is the range of ith rational subgroup or sample, and d2 is a function of the sample size n defined by
87 88
d2 ðnÞ ¼ EðUðnÞ Uð1Þ Þ,
ð1:7Þ
89 90 91 92 93 94 95 96 97
where UðnÞ and Uð1Þ are the largest and the smallest observations, respectively, in samples U1 , U2 , . . . , Un , which are P random samples from the standard normal distribution, Si2 ¼ ð1=ðn 1ÞÞ nj¼1 ðXij X i Þ2 , and c4 is a function of the sample size n defined by
2 c4 ðnÞ ¼ n1
1=2
ðn=2Þ : ððn 1Þ=2Þ
ð1:8Þ
98 99 100 101 102 103
Moreover, gð1Þ n is given by (see Wang (6)) 2ðn 1Þ gð1Þ n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 n 2nðn 1Þ
Z
þ1
pffiffiffiffiffiffiffiffiffiffiffi xð n 1xÞðxÞ dx,
ð1:9Þ
1
104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
where ðÞ and ðÞ are the cumulative distribution function (CDF) and probability density function (PDF) of Nð0, 1Þ respectively. In general, R , S , Sp and D are called the average range, the average standard deviation, the pooled sample standard deviation, and average absolute deviation of historical data, respectively. As we know, under the normality assumption the exact distribution of ^3 = is a scaled distribution, and the approximate probability density of ^1 and ^2 is given by Chen (1). Moreover, Chen (1) provided a unified presentation of three distributions above mentioned. The approximate distribution of ^ 4 = can be also approximated by a scaled distribution (see Wang (6)). In fact, there are also some other estimators for , such as the mean absolute deviation to mean and median absolute deviation to median (see Tukey (5), Rey (3)). They discussed some simple simulated properties of them when the data is from a "-contaminated normal distribution. In this paper, we’ll propose three statistics, which are used to estimate when the characteristic is from both a pure and a "-contaminated normal distribution. Under the normality assumption we can use them to estimate the standard deviation of process and discuss their fine properties in SS X charts through comparison them with other four estimators. Noticeably, we say the process is out of control when the data is "-contaminated. The simulated results in this paper are evaluated by IMSL library.
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Three new estimators for will be proposed in section 2, and their application to SS X will be given in section 3. The last section is concerned about the discussions.
130 131
2. THREE MEDIAN DEVIATIONS
132 133 134 135 136 137
Let Xij , i ¼ 1, . . . , m and j ¼ 1, . . . , n denote historical data of size n which are taken when the process is in-control, and suppose that iid Xij Nð, 2 Þ . Because of the robustness of median, we give three median statistics as follows:
138 139 140
D1 ¼
m 1X medianfjXij Xmed ðiÞ j, j ¼ 1, . . . , ng, m i¼1
ð2:1Þ
D2 ¼
m X n 1 X jX Xmed ðiÞ j, mn i¼1 j¼1 ij
ð2:2Þ
D3 ¼
m 1X medianfjXij X i j, j ¼ 1, . . . , ng, m i¼1
ð2:3Þ
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165
where Xmed ðiÞ is the median of the ith rational subgroup or sample. Statistics D1 , D2 , and D3 are called the median of absolute deviation to the median (MADM), the average absolute deviation to the median (AADM), and the median of average absolute deviation (MAAD), respectively. Obviously, they can also be used to estimate . We note that D2 is equal to D3 for the special case of n ¼ 4 (This is proved in the appendix). This fact can also be seen in Tables 1–5. However, it is not the case when n>4. Because the statistics D1 , D2 , and D3 are used as estimators for , the process standard deviation, it’s necessary to evaluate their expectations. However, it’s very difficult to obtain the analytic formula for the expectations of statistics D1 and D3 . Thus, we only derive the expectation of statistic D2 , which is given in the following proposition. Proposition. Let ðÞ and ðÞ be the cumulative distribution function and probability density function of standard normal distribution, Nð0, 1Þ, respectively, then the expectation of statistic D2 is given by Z þ1 n 2 X n! xðxÞ½1 ðxÞni ðxÞi1 dx, EðD2 Þ ¼ n i¼1þ½n=2 1 ðn iÞ!ði 1Þ!
166 167 168
ð2:4Þ where ½x denotes the integer part of x.
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SHEWHART X CONTROL CHARTS 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
429
Proof. Let Y1 , Y2 , . . . , Yn be n independent samples from standard normal distribution Nð0, 1Þ, and Ymed be the sample median. Let Yð1Þ Yð2Þ YðnÞ denote the order statistics of them. It’s easy to show that " # " # m X n n X 1 X ðiÞ Yi Ymed EðD2 Þ ¼ E X Xmed ¼ E mn i¼1 j¼1 ij n i¼1 " # ½n=2 n X X ¼ E ðYmed YðiÞ Þ þ ðYðiÞ Ymed Þ n i¼1 i¼½n=2þ1 " # ½n=2 n X X ¼ E YðiÞ YðiÞ : n i¼1 i¼1þ½n=2 For any fixed 1 i n, because the probability density function of the ith order statistic YðiÞ is as follows n! ðxÞðxÞi1 ð1 ðxÞÞni , ðn iÞ!ði 1Þ!
187 188
we can obtain the following result:
189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
EYðiÞ
Z þ1 n! ¼ yð yÞð yÞi1 ð1 ð yÞÞni dy ðn iÞ!ði 1Þ! 1 Z 1 n! ¼ xðxÞðxÞi1 ð1 ðxÞÞni dðxÞ ðn iÞ!ði 1Þ! þ1 Z þ1 n! xðxÞ½1 ðxÞi1 ðxÞni dx ¼ ðn iÞ!ði 1Þ! 1
n! ¼ ðn jÞ!ð j 1Þ! ¼ EYð jÞ ,
Z
þ1
xðxÞ½1 ðxÞnj ðxÞj1 dx 1
where j ¼ n þ 1 i. Thus we have the following result n 2 X EYðiÞ n i¼1þ½n=2 Z þ1 n 2 X n! xðxÞ½1 ðxÞni ðxÞi1 dx: ¼ n i¼1þ½n=2 1 ðn iÞ!ði 1Þ!
ED2 ¼
Then the proof of proposition is completed.
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Using the above proposition it’s easy to obtain an unbiased estimator for based on statistic D2 . Because the formula of the mean of D1 and D3 are difficult to obtain, simulation results of EðD1 Þ and EðD3 Þ are given in this paper (In practice, simulation methods are often used to evaluate the expectation of a statistic, see Ross (4)). Therefore, three unbiased estimators for can be obtained. Let ^ 5 , ^ 6 , and ^ 7 denote the unbiased estimators for based on D1 , D2 and D3 respectively. The simulated bias-adjusting factors for these three statistics are given in Table 1 (there are 100 000 times of replication). Because the sample median is more robust than the sample mean, we have a intuitive conclusion. That is, the unbiased estimators ^ 5 , ^ 6 , and ^ 7 based on D1 , D2 , and D3 , respectively, are more robust than the other four estimators given by (1.3)–(1.6), when the data is from the "-contaminated normal distribution. The CDF and PDF of the "-contaminated normal distribution considered in this paper are given by x x FðxÞ ¼ ð1 "Þ þ " , 0"1 ð2:5Þ 3 1 " x " x þ , 0 " 1: ð2:6Þ f ðxÞ ¼ 3 3
T1
232 233 234 235 236 237 238
We can easily obtain the population variance: 2 true ¼ ð1 þ 8"Þ 2 . To compare the biases among the above seven estimators for , the simulated results are given in Table 2 for several values of " and n. From Table 2 we observed that: (1)
239 240 241 242 243
(2)
244 245 246
(3)
247 248 249 250 251 252
(4)
T2
When the data is from the standard normal distribution, there are no significant differences among these seven estimators, moreover, the larger sample size n is, the more precise the estimators are. When the data is from the "-contaminated normal distribution, ^ 5 and ^ 7 are the worst estimators from the view point of bias, and as " and n increase, the bias of the estimators ^ 5 and ^ 7 increase. When the data is contaminated, ^ 1 , ^ 4 , ^ 5 , ^ 6 and ^ 7 do not converge to the true value as n increases. ^ 1 is slightly greater than true value, and ^4 , ^ 5 , ^ 6 and ^ 7 are smaller. The estimated values of based on ^4 are very similar to that based on ^ 6 .
As far as estimator is concerned, ^ 5 and ^ 7 are not good estimators for when the data is contaminated, but they are uniformly smaller than the
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D1
0.45303 0.49509 0.55346 0.56718 0.59338 0.59849 0.61374 0.61562 0.62502 0.62693
n
3 4 5 6 7 8 9 10 11 12
0.56420 0.66268 0.66329 0.70365 0.70371 0.72542 0.72572 0.73888 0.73937 0.74855
D2 0.71521 0.66268 0.62732 0.65448 0.66139 0.66179 0.65909 0.66348 0.66367 0.66557
D3
n 13 14 15 16 17 18 19 20 21 22
Table 1.
0.63337 0.63358 0.63809 0.63987 0.64327 0.64407 0.64681 0.64698 0.64925 0.64928
D1 0.74878 0.75523 0.75520 0.76039 0.76040 0.76449 0.76461 0.76766 0.76767 0.77015
D2 0.66506 0.66633 0.66548 0.66697 0.66712 0.66808 0.66782 0.66871 0.66836 0.66899
D3 23 24 25 26 27 28 29 30 50 100
n
The Bias-Adjusting Factors of D1 , D2 and D3
0.65225 0.65186 0.65320 0.65374 0.65506 0.65539 0.65633 0.65663 0.66377 0.66932
D1
0.77055 0.77265 0.77257 0.77448 0.77438 0.77593 0.77598 0.77732 0.78547 0.79173
D2
0.66940 0.66979 0.66959 0.67009 0.66995 0.67036 0.67000 0.67032 0.67193 0.67339
D3
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SHEWHART X CONTROL CHARTS 431
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WU, ZHAO, AND WANG
Table 2. Simulated Results of ^ for Several " and Sample Size n. In Each Combination of n and ", Each Value Are When ^1 ^7 is Used Respectively "
255 256
n
0.0
0.001
0.002
0.005
0.01
0.02
0.05
0.10
0.15
0.20
0.50
4
0.9953 0.9949 0.9945 0.9944 0.9958 0.9951 0.9951
1.0121 1.0122 1.0136 1.0123 1.0102 1.0126 1.0126
1.0039 1.0044 1.0082 1.0042 1.0014 1.0045 1.0045
1.0118 1.0112 1.0187 1.0098 1.0018 1.0081 1.0081
1.0201 1.0209 1.0340 1.0203 1.0092 1.0179 1.0179
1.0521 1.0511 1.0769 1.0482 1.0299 1.0435 1.0435
1.1350 1.1357 1.1851 1.1310 1.0782 1.1191 1.1191
1.2487 1.2479 1.3322 1.2387 1.1515 1.2181 1.2181
1.3764 1.3764 1.4842 1.3635 1.2354 1.3341 1.3341
1.6153 1.6124 1.7324 1.5950 1.4385 1.5587 1.5587
2.1549 2.1462 2.2458 2.1201 1.9606 2.0832 2.0832
5
0.9966 0.9969 0.9973 0.9972 1.0006 0.9977 0.9997
0.9967 0.9949 0.9952 0.9934 0.9927 0.9940 0.9888
1.0060 1.0067 1.0097 1.0072 1.0109 1.0065 1.0075
1.0166 1.0161 1.0245 1.0149 1.0034 1.0128 1.0108
1.0260 1.0245 1.0369 1.0213 1.0162 1.0195 1.0168
1.0579 1.0576 1.0809 1.0533 1.0357 1.0496 1.0497
1.1384 1.1350 1.1808 1.1226 1.0646 1.1130 1.1022
1.2703 1.2630 1.3457 1.2407 1.1205 1.2223 1.1901
1.4060 1.3954 1.4911 1.3640 1.2172 1.3426 1.3039
1.6476 1.6289 1.7350 1.5835 1.3824 1.5562 1.4876
2.1627 2.1332 2.2205 2.0809 1.8711 2.0594 1.9557
10
1.0037 1.0038 1.0031 1.0041 1.0052 1.0046 1.0034
1.0067 1.0059 1.0069 1.0051 1.0053 1.0051 1.0033
1.0013 1.0014 1.0032 1.0008 0.9960 1.0002 0.9981
1.0143 1.0118 1.0163 1.0082 1.0028 1.0078 1.0037
1.0403 1.0330 1.0424 1.0243 1.0100 1.0224 1.0132
1.0729 1.0605 1.0776 1.0455 1.0181 1.0426 1.0246
1.1787 1.1489 1.1878 1.1124 1.0485 1.1052 1.0659
1.3405 1.2846 1.3417 1.2205 1.1015 1.2081 1.1295
1.4893 1.4154 1.4799 1.3298 1.1585 1.3123 1.2039
1.7634 1.6580 1.7262 1.5436 1.3009 1.5225 1.3645
2.3132 2.1842 2.2350 2.0548 1.7484 2.0334 1.8219
20
0.9976 0.9980 0.9983 0.9983 1.0005 0.9983 0.9994 1.0005 1.0024 1.0024 1.0030 1.0035 1.0031 1.0034
1.0080 1.0038 1.0048 1.0015 0.9982 1.0012 0.9971 1.0078 1.0045 1.0048 1.0033 1.0020 1.0031 1.0021
1.0130 1.0086 1.0105 1.0057 1.0044 1.0054 1.0040 1.0195 1.0075 1.0081 1.0037 0.9999 1.0036 1.0001
1.0245 1.0155 1.0185 1.0097 1.0042 1.0091 1.0037 1.0410 1.0177 1.0192 1.0110 1.0059 1.0109 1.0062
1.0463 1.0299 1.0345 1.0193 1.0081 1.0186 1.0079 1.0893 1.0362 1.0395 1.0203 1.0066 1.0200 1.0077
1.0970 1.0626 1.0739 1.0398 1.0149 1.0381 1.0175 1.1637 1.0690 1.0746 1.0392 1.0155 1.0387 1.0160
1.2419 1.1584 1.1830 1.1044 1.0429 1.1005 1.0498 1.3825 1.1725 1.1838 1.1026 1.0417 1.1011 1.0428
1.4572 1.3092 1.3437 1.2119 1.0846 1.2044 1.1021 1.6717 1.3300 1.3466 1.2067 1.0854 1.2040 1.0899
1.6358 1.4437 1.4822 1.3173 1.1458 1.3079 1.1676 1.8891 1.4694 1.4872 1.3097 1.1344 1.3061 1.1419
1.9486 1.6937 1.7335 1.5242 1.2614 1.5116 1.2945 2.1960 1.7151 1.7317 1.5096 1.2424 1.5045 1.2563
2.4597 2.2061 2.2320 2.0263 1.6736 2.0139 1.7165 2.6158 2.2243 2.2351 2.0120 1.6305 2.0070 1.6487
1.0000 1.0004 1.0004 1.0008 1.0020 1.0007 1.0014
1.0152 1.0036 1.0038 1.0011 0.9980 1.0010 0.9978
1.0263 1.0083 1.0087 1.0047 1.0022 1.0046 1.0020
1.0682 1.0192 1.0202 1.0104 1.0058 1.0102 1.0054
1.1294 1.0370 1.0388 1.0199 1.0071 1.0196 1.0071
1.2477 1.0748 1.0780 1.0414 1.0161 1.0409 1.0164
1.5273 1.1768 1.1830 1.1014 1.0404 1.1006 1.0415
1.8515 1.3325 1.3408 1.2016 1.0812 1.2001 1.0845
2.0740 1.4746 1.4837 1.3032 1.1292 1.3012 1.1341
2.3506 1.7284 1.7369 1.5089 1.2394 1.5060 1.2466
2.6851 2.2268 2.2323 2.0017 1.6077 1.9989 1.6179
257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
50
281 282 283 284 285 286 287 288 289 290 291
100
(continued )
292 293 294
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SHEWHART X CONTROL CHARTS Table 2.
295 296 297 298 299
0.0
0.001
0.002
0.005
0.01
0.02
0.05
0.10
0.15
0.20
0.50
200
1.0018 1.0008 1.0008 1.0005 1.0001 1.0005 1.0004
1.0234 1.0045 1.0046 1.0025 1.0010 1.0024 1.0011
1.0447 1.0080 1.0083 1.0040 1.0008 1.0039 1.0011
1.1048 1.0202 1.0207 1.0108 1.0045 1.0107 1.0042
1.1959 1.0384 1.0394 1.0203 1.0076 1.0201 1.0082
1.3532 1.0737 1.0754 1.0394 1.0156 1.0391 1.0162
1.6993 1.1781 1.1812 1.0993 1.0380 1.0988 1.0386
2.0305 1.3376 1.3420 1.2017 1.0830 1.2009 1.0847
2.2256 1.4774 1.4820 1.2999 1.1259 1.2988 1.1285
2.4528 1.7256 1.7300 1.5002 1.2335 1.4989 1.2374
2.7412 2.2353 2.2379 2.0039 1.6052 2.0024 1.6107
True
1.0000
1.0040
1.0080
1.0198
1.0392
1.0770
1.1832
1.3416
1.4832
1.6124
2.2361
n
301 302 303 305
Continued "
300
304
433
306 307 308 309 310 311 312 313 314 315
true value, and they are smaller than others of those seven estimators. However, using ^ 5 and ^ 7 as the estimators for , the ARL of SS X control charts can be efficiently decreased (When the data is contaminated, we regard that the process is out of control). This conclusion can be observed in the next section. Using the simulation method we can also evaluate the standard deviation of these three estimators for . However, they are omitted here, because our interest only focus on their application to SS X control charts.
316 317 318 319
3. APPLICATIONS TO SS X CONTROL CHARTS
320
Chen (1) proposed some conclusions that are concerned about the effectiveness of the first three estimators for in terms of the ARL of SS X control charts when the control limits are estimated by historical data. Wang (6) considered the effectiveness of the first four estimators for under the same condition. Under the normality assumption, they got similar conclusions that the estimated control limits’ effect on ARL is noticeable when m and n are small. As Wang (6) pointed out, the performance of the estimator for , ^ , is better than other first three estimators in terms of ARL. To compare the effectiveness of 5 , 6 and 7 used in SS X control charts, let Yij , i ¼ 1, 2, 3, . . . , and j ¼ 1, 2, . . . , n denote current or future data, and assume the process mean and standard deviation are unknown. In this paper we only want to detect a shift in process mean. Let a denote the shift in the process mean we want to detect. Then the population distribution is as follows x a x a FðxÞ ¼ ð1 "Þ þ " , 0"1 ð3:1Þ 3
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
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From Eq. (1.1), we can evaluate the probability of ith standardized observation Yi falling outside the estimated control limits:
339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354
d or Y^ i > UCL d x , ^ Þ pðx , ^ Þ ¼ PrðY^ i < LCL UCLj
pffiffiffi x þ ð3^ = nÞ a pffiffiffi ¼ 1 ð1 "Þ = n
pffiffiffi x þ ð3^ = nÞ a p ffiffi ffi " 3= n
pffiffiffi x ð3^ = nÞ a pffiffiffi þ ð1 "Þ = n pffiffiffi
x ð3^ = nÞ a pffiffiffi þ " , 3= n from which we obtain the ARL for SS X charts ARL ¼ EðEðARL j x , ^ ÞÞ ¼ Eðx , ^ Þ
355 356 357 358 359 360 361 362 363 364 365 366
(1)
368 369 370 371 372 374 375 376 377 378
ð3:2Þ
In many practical situations, we don’t know whether the data is contaminated or not. In other words, we are not sure that whether the data is in control or not. In this situation, we always assume the data is from a normal distribution, and use the Shewhart control chart with the estimated control limits to detect the potential shift in process mean. Because it is difficult to calculate the expectation in Eq. (3.2), Monte-carlo methods are used to evaluate the ARL. The simulated results are given in Tables 3–8 (There T3 – T8 are 100 000 replications). From Tables 3–8, we observed that
367
373
1 : pðx , ^ Þ
(2)
When the sample is not contaminated and has a small or moderate shift in the process mean, the estimators ^1 , ^ 2 , . . . , ^ 7 show little difference in terms of the ARL (except when m ¼ 10, n ¼ 4, 5, where ^6 performs better than ^5 and ^7 ), while the sample is in control (hasn’t shifted and isn’t contaminated), the ARL based on D1 is uniformly better than the other six estimators. When the sample is contaminated, The ARL’s performance of ^5 is uniformly better than that of the other estimators. As the " increases, the superiority of ^5 becomes more obvious. We also note that there is little differences among ^4 , ^6 and ^7 and they are uniformly better than ^ 1 , ^ 2 , ^ 3 . From this point we also recognize that ^5 is the most robust one among the seven
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SHEWHART X CONTROL CHARTS 379 380
Table 3. ARL of the SS Charts with the Estimated Control Limits Based on the Seven Estimators ^1 ^7 , When the Data Is "-Contaminated (" ¼ 0:00)
381
"
382
a
0.00
383 384 385
m
4
386 387
10
858.3 836.7 775.4 888.2 3715.3 987.4 987.4
20
6.5 6.4 6.3 6.5 10.5 6.5 7.9
513.5 505.9 488.8 516.6 866.9 539.5 539.5
462.4 449.8 439.9 459.0 1037.3 460.9 635.1
369.3 358.1 356.4 363.3 445.3 364.4 417.5
8.0 8.0 7.9 8.0 9.4 8.2 8.2
30
453.5 450.3 441.5 458.1 635.1 473.2 473.2
419.4 414.6 411.0 421.7 669.0 421.6 517.1
367.1 360.6 359.8 363.8 410.6 364.5 396.1
50
412.1 410.3 406.1 414.3 502.8 422.6 422.6
395.9 392.7 390.7 395.1 507.0 394.9 436.5
100
388.2 387.7 385.7 389.4 424.9 394.6 394.6
200
383.2 382.5 381.6 382.9 404.7 387.4 387.4
396 398 399 400 401
404 405 406 407 409 410 411 412 413
416 417 418 419
n
10.6 10.5 10.3 10.7 17.3 11.0 11.0
394 395
415
n
407.2 379.9 377.2 392.3 647.7 395.2 546.6
393
414
n
594.9 559.7 539.7 580.9 3769.3 587.8 1375.7
391
408
2.0
5
390
402 403
1.0
4
389
397
0.0
10
388
392
435
5
10
4
5
10
2.0 1.9 1.9 1.9 2.0 1.9 2.0
1.3 1.3 1.3 1.3 1.3 1.3 1.3
1.1 1.1 1.1 1.1 1.2 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
5.2 5.2 5.2 5.2 6.2 5.2 5.6
1.9 1.9 1.8 1.9 1.9 1.9 1.9
1.2 1.2 1.2 1.2 1.3 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
7.4 7.4 7.3 7.4 8.2 7.5 7.5
5.0 5.0 5.0 5.0 5.5 5.0 5.2
1.8 1.8 1.8 1.8 1.8 1.8 1.8
1.2 1.2 1.2 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
365.4 361.4 360.7 363.5 394.6 364.6 380.8
6.9 6.9 6.9 6.9 7.3 7.0 7.0
4.8 4.8 4.8 4.8 5.1 4.8 4.9
1.8 1.8 1.8 1.8 1.8 1.8 1.8
1.2 1.2 1.2 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
381.1 380.3 379.3 382.1 433.4 381.9 404.1
367.3 365.8 365.6 367.0 382.3 367.3 375.1
6.6 6.6 6.6 6.6 6.8 6.6 6.6
4.6 4.6 4.6 4.6 4.8 4.6 4.7
1.8 1.8 1.8 1.8 1.8 1.8 1.8
1.2 1.2 1.2 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
374.8 374.3 374.0 375.2 401.6 374.4 384.3
368.0 367.5 367.5 368.3 375.9 368.9 371.3
6.4 6.4 6.4 6.4 6.6 6.5 6.5
4.6 4.6 4.6 4.6 4.7 4.6 4.6
1.8 1.8 1.8 1.8 1.8 1.8 1.8
1.2 1.2 1.2 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
420
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436 421 422
WU, ZHAO, AND WANG Table 4. ARL of the SS Charts with the Estimated Control Limits Based on the Sseven Estimators ^1 ^7 , When the Data Is "-Contaminated (" ¼ 0:01)
423
"
424
a
0.01
425 426
0.0
1.0
2.0
n
n
n
427
m
4
5
10
4
5
428 429
10
189.0 188.6 199.8 188.2 191.6 186.7 186.7
185.8 184.2 194.0 182.0 192.3 179.8 186.7
183.5 176.3 183.9 169.6 166.8 168.3 165.4
13.6 13.5 17.3 13.4 15.4 13.1 13.1
8.0 7.9 9.8 7.7 10.5 7.5 8.7
20
191.7 191.5 203.5 190.7 188.8 189.3 189.3
190.0 189.0 199.6 187.1 188.8 185.1 188.3
192.4 185.7 193.9 178.6 171.1 177.3 171.7
9.4 9.4 11.0 9.3 9.9 9.2 9.2
30
191.1 190.8 202.3 189.9 186.4 188.5 188.5
191.3 190.4 201.3 188.2 185.3 186.1 187.0
197.4 190.6 199.0 183.3 174.4 182.0 175.2
50
192.5 192.5 204.8 191.6 185.4 190.0 190.0
193.7 192.7 204.3 190.3 184.9 188.3 187.8
100
194.1 194.1 206.7 193.1 185.3 191.4 191.4
200
195.6 195.6 208.5 194.6 185.5 192.7 192.7
430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461
10
4
5
10
2.2 2.2 2.2 2.1 2.1 2.1 2.1
1.4 1.4 1.5 1.4 1.4 1.3 1.3
1.1 1.1 1.2 1.1 1.2 1.1 1.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0
6.1 6.0 6.7 6.0 6.5 5.9 6.2
2.0 2.0 2.1 2.0 1.9 2.0 1.9
1.3 1.3 1.3 1.3 1.3 1.3 1.3
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
8.3 8.3 9.3 8.3 8.6 8.2 8.2
5.6 5.6 6.1 5.5 5.7 5.4 5.6
2.0 2.0 2.0 1.9 1.9 1.9 1.9
1.3 1.3 1.3 1.3 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
199.5 192.9 201.8 185.6 175.0 184.2 176.7
7.8 7.7 8.5 7.7 7.6 7.6 7.6
5.4 5.4 5.7 5.3 5.3 5.2 5.3
2.0 1.9 2.0 1.9 1.9 1.9 1.9
1.2 1.2 1.3 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
194.9 194.0 205.6 191.5 183.8 189.5 187.9
202.3 195.9 204.8 188.5 176.5 187.0 178.7
7.3 7.3 7.9 7.3 7.1 7.2 7.2
5.1 5.1 5.4 5.1 5.0 5.0 5.0
1.9 1.9 2.0 1.9 1.8 1.9 1.8
1.2 1.2 1.2 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
196.1 195.3 207.3 192.7 183.6 190.6 188.7
203.9 197.8 206.6 190.4 178.5 189.1 180.6
7.2 7.2 7.7 7.1 6.9 7.1 7.1
5.1 5.1 5.3 5.0 4.8 5.0 4.9
1.9 1.9 1.9 1.9 1.8 1.9 1.8
1.2 1.2 1.2 1.2 1.2 1.2 1.2
1.1 1.1 1.1 1.1 1.1 1.1 1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0
462
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SHEWHART X CONTROL CHARTS 463 464
Table 5. ARL of the SS Charts with the Estimated Control Limits Based on the Seven Estimators ^1 ^7 , When the Data Is "-Contaminated (" ¼ 0.05)
465
"
466
a
0.05
467 468
0.0
1.0
2.0
n
n
n
469
m
4
5
470 471
10
72.8 72.8 83.8 71.8 64.0 69.5 69.5
74.0 73.1 83.0 70.6 62.8 68.7 67.3
4
5
79.6 73.7 79.8 67.4 57.5 66.2 59.5
18.5 18.5 25.1 18.1 15.1 16.7 16.7
13.4 13.0 17.7 11.9 10.7 11.1 11.3
20
72.9 72.9 82.5 72.0 64.2 70.1 70.1
74.0 73.4 81.8 71.3 63.1 69.7 68.1
80.0 74.8 80.5 68.9 59.2 67.8 61.4
14.6 14.5 20.1 14.1 11.5 13.2 13.2
30
72.9 72.8 81.8 72.0 64.4 70.2 70.2
74.2 73.5 81.8 71.5 62.9 70.0 68.0
80.6 75.5 81.3 69.8 60.0 68.7 62.2
50
73.1 73.1 82.0 72.3 64.5 70.5 70.5
74.3 73.7 81.7 71.7 63.4 70.3 68.3
100
73.4 73.4 82.0 72.6 64.9 70.8 70.8
200
73.5 73.5 82.1 72.8 65.1 71.0 71.0
472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503
437
10
10
4
5
10
4.1 3.4 4.1 2.9 2.4 2.8 2.5
1.8 1.8 2.6 1.8 1.7 1.7 1.7
1.3 1.3 1.6 1.3 1.3 1.3 1.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0
10.1 9.8 13.2 9.1 7.7 8.6 8.5
3.2 2.9 3.3 2.5 2.1 2.5 2.2
1.5 1.5 1.8 1.5 1.4 1.5 1.5
1.2 1.2 1.3 1.2 1.2 1.2 1.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0
12.9 12.9 17.6 12.5 10.2 11.8 11.8
9.1 8.9 11.9 8.3 6.9 7.9 7.7
3.1 2.8 3.1 2.5 2.1 2.4 2.2
1.5 1.5 1.7 1.5 1.4 1.4 1.4
1.2 1.2 1.3 1.2 1.2 1.2 1.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0
80.6 75.7 81.5 70.1 60.3 69.0 62.6
11.9 11.9 16.2 11.6 9.3 10.9 10.9
8.5 8.3 10.9 7.8 6.4 7.5 7.2
3.0 2.7 3.0 2.4 2.0 2.4 2.1
1.4 1.4 1.6 1.4 1.3 1.4 1.4
1.2 1.2 1.3 1.2 1.1 1.2 1.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0
74.5 73.9 81.8 72.0 63.5 70.5 68.5
80.8 76.0 81.7 70.4 60.7 69.4 63.0
11.0 11.0 14.6 10.8 8.6 10.2 10.2
7.9 7.8 10.0 7.3 5.9 7.0 6.7
2.9 2.6 2.9 2.3 2.0 2.3 2.1
1.4 1.4 1.5 1.4 1.3 1.4 1.4
1.2 1.2 1.2 1.2 1.1 1.2 1.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0
74.5 73.9 81.8 72.0 63.6 70.6 68.6
80.8 76.1 81.8 70.7 61.1 69.6 63.3
10.7 10.7 14.1 10.5 8.3 9.9 9.9
7.7 7.6 9.7 7.2 5.7 6.9 6.6
2.8 2.6 2.9 2.3 2.0 2.3 2.1
1.4 1.4 1.5 1.4 1.3 1.4 1.4
1.2 1.2 1.2 1.2 1.1 1.2 1.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0
504
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438 505 506
WU, ZHAO, AND WANG Table 6. ARL of the SS Charts with the Estimated Control Limits Based on the Seven Estimators ^1 ^7 , When the Data Is "-Contaminated (" ¼ 0:10)
507
"
508
a
0.10
509 510
0.0
1.0
2.0
n
n
n
511
m
4
5
512 513
10
51.1 51.1 63.3 49.9 40.9 47.0 47.0
51.9 50.8 61.3 47.9 39.3 46.0 44.1
4
5
57.6 50.7 56.4 44.2 34.7 43.0 36.8
21.2 21.2 28.2 20.5 15.4 18.8 18.8
17.1 16.5 22.0 15.0 11.2 13.9 13.1
20
49.1 49.1 59.2 48.1 40.2 46.0 46.0
50.2 49.3 57.9 47.1 38.7 45.5 43.5
56.6 50.5 56.1 44.5 35.2 43.4 37.3
18.8 18.8 25.4 18.1 13.1 16.6 16.6
30
48.3 48.3 57.4 47.4 39.9 45.4 45.4
49.7 48.9 57.2 46.8 38.4 45.3 43.1
56.5 50.6 56.0 44.7 35.5 43.6 37.6
50
48.0 48.0 56.7 47.1 39.7 45.2 45.2
49.2 48.5 56.4 46.4 38.3 45.0 42.9
100
47.8 47.7 56.2 46.9 39.7 45.1 45.1
200
47.6 47.6 55.9 46.8 39.6 45.0 45.0
514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545
10
10
4
5
10
7.5 5.7 7.2 4.2 2.8 4.0 3.1
2.8 2.8 4.2 2.7 2.1 2.4 2.4
1.8 1.8 2.4 1.7 1.5 1.6 1.6
1.1 1.0 1.1 1.0 1.0 1.0 1.0
14.6 14.1 19.1 12.9 9.1 12.0 11.1
5.9 4.6 5.8 3.6 2.5 3.4 2.7
2.1 2.1 2.8 2.0 1.7 1.9 1.9
1.5 1.5 1.8 1.4 1.3 1.4 1.4
1.0 1.0 1.0 1.0 1.0 1.0 1.0
17.5 17.5 23.9 16.9 12.1 15.5 15.5
13.8 13.3 18.3 12.1 8.3 11.3 10.3
5.6 4.4 5.5 3.4 2.4 3.3 2.6
1.9 1.9 2.5 1.9 1.6 1.8 1.8
1.5 1.4 1.7 1.4 1.3 1.4 1.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0
56.0 50.4 55.8 44.7 35.6 43.6 37.6
16.7 16.7 23.2 16.1 11.3 14.8 14.8
13.1 12.7 17.5 11.5 7.8 10.7 9.8
5.2 4.2 5.1 3.3 2.3 3.2 2.5
1.8 1.8 2.3 1.8 1.5 1.7 1.7
1.4 1.4 1.6 1.4 1.2 1.3 1.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0
49.0 48.3 56.0 46.3 38.2 44.9 42.8
55.7 50.2 55.6 44.6 35.6 43.6 37.7
15.9 15.9 22.2 15.3 10.7 14.1 14.1
12.5 12.0 16.7 10.9 7.3 10.2 9.2
4.9 4.0 4.9 3.2 2.3 3.1 2.5
1.8 1.8 2.2 1.7 1.5 1.7 1.7
1.4 1.4 1.6 1.3 1.2 1.3 1.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0
48.7 48.1 55.6 46.1 38.2 44.8 42.7
55.6 50.2 55.5 44.7 35.8 43.7 37.8
15.7 15.7 22.0 15.1 10.4 13.9 13.9
12.2 11.8 16.4 10.7 7.0 10.0 9.0
4.8 3.9 4.8 3.2 2.3 3.0 2.4
1.7 1.7 2.1 1.7 1.5 1.6 1.6
1.4 1.4 1.5 1.3 1.2 1.3 1.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0
546
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SHEWHART X CONTROL CHARTS 547 548
Table 7. ARL of the SS Charts with the Estimated Control Limits Based on the Seven Estimators ^1 ^7 , When the Data Is "-Contaminated (" ¼ 0:15)
549
"
550
a
0.15
551 552
0.0
1.0
2.0
n
n
n
553
m
4
5
554 555
10
45.0 44.8 58.1 43.3 33.8 40.1 40.1
45.9 44.3 55.7 40.9 31.5 38.8 36.3
4
5
53.1 44.1 50.2 36.6 26.5 35.3 28.7
22.7 22.7 30.1 21.9 16.0 20.0 20.0
20
42.1 42.0 53.0 40.8 32.4 38.3 38.3
43.3 42.2 51.7 39.5 30.5 37.8 35.3
51.2 43.5 49.3 36.5 26.6 35.3 28.8
30
41.1 41.0 51.0 39.9 31.7 37.6 37.6
42.5 41.5 50.4 38.9 29.9 37.3 34.7
50
40.5 40.4 49.9 39.4 31.4 37.3 37.3
100
200
556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587
439
10
10
4
5
10
19.5 18.7 24.3 16.9 11.8 15.7 14.4
11.0 8.2 10.1 5.9 3.4 5.5 3.9
4.0 4.0 5.9 3.8 2.8 3.4 3.4
2.6 2.5 3.4 2.2 1.7 2.0 2.0
1.2 1.1 1.1 1.1 1.0 1.1 1.0
20.8 20.7 27.2 20.0 14.2 18.4 18.4
17.7 17.1 22.1 15.5 10.2 14.4 13.0
9.4 7.0 8.8 5.1 2.9 4.7 3.3
2.9 2.9 4.3 2.8 2.1 2.5 2.5
2.0 2.0 2.6 1.8 1.5 1.7 1.6
1.1 1.1 1.1 1.1 1.0 1.0 1.0
50.7 43.3 49.0 36.6 26.8 35.4 28.9
20.0 19.9 26.2 19.2 13.5 17.7 17.7
17.1 16.5 21.6 14.9 9.6 13.9 12.3
9.0 6.7 8.5 4.8 2.8 4.5 3.2
2.7 2.6 3.8 2.5 1.9 2.3 2.3
1.9 1.8 2.3 1.7 1.4 1.6 1.5
1.1 1.1 1.1 1.0 1.0 1.0 1.0
41.9 41.0 49.6 38.6 29.7 37.0 34.5
50.2 43.1 48.8 36.5 26.7 35.3 28.9
19.6 19.5 25.8 18.8 12.9 17.2 17.2
16.7 16.2 21.4 14.6 9.2 13.6 12.0
8.5 6.4 8.1 4.6 2.7 4.4 3.1
2.5 2.5 3.4 2.4 1.8 2.2 2.2
1.8 1.7 2.2 1.6 1.3 1.6 1.5
1.1 1.1 1.1 1.0 1.0 1.0 1.0
40.0 39.9 49.1 39.0 31.2 36.9 36.9
41.5 40.6 48.8 38.2 29.5 36.7 34.2
49.7 42.8 48.4 36.3 26.7 35.2 28.8
19.2 19.1 25.4 18.4 12.5 16.9 16.9
16.4 15.8 21.1 14.2 8.7 13.2 11.6
8.1 6.1 7.7 4.5 2.6 4.2 3.0
2.3 2.3 3.2 2.3 1.7 2.1 2.1
1.7 1.7 2.1 1.6 1.3 1.5 1.5
1.1 1.1 1.1 1.0 1.0 1.0 1.0
39.8 39.8 48.8 38.8 31.0 36.8 36.8
41.2 40.4 48.5 38.1 29.4 36.6 34.0
49.6 42.7 48.4 36.4 26.8 35.3 28.9
19.1 19.1 25.4 18.3 12.4 16.8 16.8
16.2 15.6 21.0 14.1 8.5 13.1 11.4
7.9 6.0 7.6 4.4 2.6 4.1 2.9
2.3 2.3 3.0 2.2 1.7 2.0 2.0
1.7 1.7 2.0 1.6 1.3 1.5 1.4
1.1 1.1 1.1 1.0 1.0 1.0 1.0
588
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440 589 590
WU, ZHAO, AND WANG Table 8. ARL of the SS Charts with the Estimated Control Limits Based on the Seven Estimators ^1 ^7 , When the Data Is "-Contaminated
591
"
592
a
0.20
593 594
0.0
1.0
2.0
n
n
n
595
m
4
5
596 597
10
44.5 44.2 59.5 42.3 31.9 38.5 38.5
45.1 43.1 55.5 38.9 28.8 36.6 33.7
4
5
54.0 42.5 48.9 33.7 22.5 32.3 24.8
24.1 24.0 31.7 23.0 16.7 21.0 21.0
20
40.5 40.3 52.6 38.8 29.4 35.9 35.9
41.8 40.3 50.7 37.1 27.0 35.1 32.0
51.5 41.6 47.8 33.5 22.5 32.2 24.9
30
39.1 39.0 50.2 37.7 28.5 35.0 35.0
40.9 39.5 49.4 36.4 26.4 34.6 31.4
50
38.3 38.2 48.8 36.9 28.0 34.5 34.5
100
200
598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629
10
10
4
5
10
21.2 20.2 25.7 18.2 12.5 17.0 15.2
13.9 10.4 12.4 7.5 4.0 7.1 4.7
5.4 5.3 7.5 5.0 3.5 4.5 4.5
3.5 3.3 4.5 2.9 2.1 2.7 2.5
1.3 1.2 1.3 1.1 1.1 1.1 1.1
22.0 21.9 28.1 21.1 15.1 19.4 19.4
19.6 18.9 23.7 17.2 11.2 16.1 14.2
12.6 9.4 11.5 6.7 3.5 6.3 4.1
4.1 4.0 5.9 3.8 2.6 3.4 3.4
2.7 2.6 3.6 2.3 1.7 2.2 2.0
1.2 1.1 1.2 1.1 1.0 1.1 1.1
50.9 41.5 47.6 33.5 22.6 32.2 24.9
21.2 21.1 27.0 20.4 14.6 18.8 18.8
19.0 18.4 23.2 16.7 10.7 15.7 13.8
12.2 9.1 11.2 6.5 3.3 6.1 3.9
3.7 3.7 5.4 3.5 2.4 3.1 3.1
2.5 2.4 3.2 2.2 1.6 2.0 1.9
1.2 1.1 1.2 1.1 1.0 1.1 1.1
40.1 38.9 48.4 36.0 26.0 34.2 31.0
50.3 41.2 47.3 33.4 22.6 32.1 24.8
20.8 20.8 26.5 20.0 14.1 18.5 18.5
18.8 18.2 22.9 16.5 10.4 15.5 13.6
11.9 8.8 10.9 6.3 3.2 5.9 3.8
3.4 3.4 4.9 3.2 2.2 2.9 2.9
2.3 2.3 3.0 2.1 1.5 1.9 1.8
1.2 1.1 1.2 1.1 1.0 1.1 1.1
37.8 37.6 47.8 36.4 27.7 34.1 34.1
39.5 38.3 47.4 35.5 25.8 33.9 30.7
49.6 40.9 46.9 33.3 22.5 32.0 24.8
20.6 20.5 26.2 19.8 13.9 18.3 18.3
18.6 17.9 22.7 16.3 10.1 15.3 13.3
11.6 8.6 10.7 6.1 3.1 5.7 3.6
3.2 3.2 4.6 3.0 2.1 2.7 2.7
2.2 2.2 2.8 2.0 1.5 1.9 1.7
1.2 1.1 1.1 1.1 1.0 1.1 1.0
37.5 37.4 47.4 36.2 27.6 33.9 33.9
39.1 38.0 47.0 35.3 25.6 33.7 30.5
49.5 40.8 46.9 33.3 22.6 32.1 24.8
20.6 20.5 26.1 19.8 13.8 18.3 18.3
18.5 17.9 22.7 16.3 9.9 15.3 13.2
11.4 8.5 10.5 6.0 3.1 5.6 3.6
3.1 3.0 4.4 2.9 2.0 2.7 2.7
2.2 2.1 2.7 1.9 1.5 1.9 1.7
1.2 1.1 1.1 1.1 1.0 1.1 1.0
630
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(3)
635 636 637 638 639 640 641
(4)
441
estimators cited in this paper. Furthermore, ^7 , ^6 and ^4 are better than ^ 1 and ^ 2 , in addition, ^7 is superior to ^6 and ^4 , and ^3 is the worst of all. Note that when " is very small, ^ 6 is the best among the seven estimators in terms of ARL while ^ 3 and ^ 5 are the worst. The ARL is a monotone decreasing function of either of both of the variables m and n.
From these four points above mentioned, we suggest to use ^5 as the estimator for in SS X control charts when the sample is possibly contaminated.
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4. CONCLUSIONS AND DISCUSSIONS
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Three new unbiased estimators for are given in this paper. Their properties in SS X control charts with the estimated control limits are discussed. If the contaminated level in the historical data, ", is very small, ^ 6 based on the average absolute deviation to median is the best estimator among the seven estimators in terms of ARL. Otherwise, the estimator ^ 5 based on the median of the absolute deviation to median is the best estimator among the seven estimators in terms of ARL. Although the SS X chart discussed in this paper is used to detect the shift in process mean, it’s necessary to estimate the process variance 2 using historical data, when the variance is unknown. We can also discuss the estimated control limits’ effect on the performance of R control chart, Shewhart X chart with variable parameters, CUSUM chart, EWMA chart, and so on.
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APPENDIX The proof of D2 ¼ D3 for n ¼ 4 To prove D2 ¼ D3 when n ¼ 4, it’s necessary to prove D2 ¼ medianfjX1 X j, jX2 X j, jX3 X j, jX4 X jg ¼ ðX3 þ X4 X1 X2 Þ=4 ¼ D3 only, where X1 X2 X3 X4 . For simplicity, we define a1 ¼ ðX1 X Þ2 , a2 ¼ ðX2 X Þ2 , a3 ¼ ðX3 X Þ2 , and a4 ¼ ðX4 X Þ2 . It’s easy to prove a2 a1, a3 a1, a3 a4 and a2 a4. To determine the order of a1, a2, a3 and a4, we note that a4 a1 ¼ ð1=2ÞðX4 X1 Þ ½ðX4 þ X1 Þ ðX2 þ X3 Þ and a3 a2 ¼ ð1=2ÞðX3 X2 Þ½ðX2 þ X Þ ðX þ X4 Þ. p3ffiffiffiffiffi p1ffiffiffiffiffi If X1 þ X4 X2 þ X3 , then a3 a2 a1 a4, D2 ¼ ð a1 þ a2Þ=2 ¼ ðjX1 X j þ jX2 X jÞ=2. Moreover, from this condition it’s easy to know
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WU, ZHAO, AND WANG
X2 X , thus D2 ¼ ðX X1 þ X X2 Þ=2 ¼ ðX3 þ X4 X1 pXffiffiffiffiffi ¼ffiffiffiffiffi D3 . If 2 Þ=4p X1 þ X4 X2 þ X3 , then a2 a3 a4 a1, D2 ¼ ð a3 þ a4Þ=2 ¼ ðjX3 X j þ jX4 X jÞ=2. Moreover, from this condition it’s easy to know X3 X , thus D2 ¼ ðX3 X þ X4 X Þ=2 ¼ ðX3 þ X4 X1 X2 Þ=4 ¼ D3 . Then the proof of this conclusion is completed.
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ACKNOWLEDGMENTS The authors would like to thank some Ph.D. and M.S. students in our seminar for their valuable comments and suggestions. The authors wish to thank the editor and referees for their helpful comments.
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REFERENCES
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1. Chen, Gemai. The Mean and Standard Deviation of the Run Length Distribution of X Charts When Control Limits Are Estimated. Statistica Sinica 1997, 7, 789–798. 2. Montgomery, D.C. Introduction to Statistical Quality Control, 3rd Ed.; John Wiley: New York, 1997. 3. Rey, W.J.J. Introduction to Robust and Quasi-Robust Statistical Methods. Springer-Verlag, 1983. 4. Ross, S.M. A Course in Simulations. Macmilan Pub. Co., 1990. 5. Tukey, J.W. A Survey of Sampling from Contaminated Distributions. In Contributions to Probability and Statistics, Olkin, I., Ed.; Stanford University Press, 1960. 6. Wang, Z.J. The Average Absolute Deviation and Its Application to Shewhart X Control Charts. Submitted, 2000.
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