針對階段 I 樣本皆來自常態分配，部分樣本來自製程發生平均數 偏移的情況，我們提出核密度估計法來估計管制狀態下的常態分配之 平均數μ0，並發現當平均數偏移程度δ 較大時，估計表現尤佳。
事實上，核密度估計法並不受限於來自常態分配之樣本；而且，
失控狀態下的樣本不一定要和管制狀態下的樣本一樣來自同一分 配。另外，核密度估計法也可用於製程同時發生平均數偏移或變異數 改變之情況，只要混合樣本的分布情形在管制狀態下之分配平均值附 近呈現高峰皆適用。
我們曾經嘗試將核密度估計法套用於估計變異數的時候，但因為 樣本變異數服從 Gamma 分配，其密度函數最高點不會對應到平均 數，所以估計效果很差。未來可以繼續研究新的估計方法，希望其對 樣本不只來自兩個不同的分配之情況皆適用。
參 考 文 獻
1. Champ, C. W. and Chou, S. P. (2003). “Comparison of Standard and Individual Limits Phase I Shewhart X , R, and S Charts”. Quality and Reliability Engineering International 19, pp. 161-170.
2. Chen, G. (1997). “The Mean and Standard Deviation of the Run Length Distribution of X Charts When Control Limits Are Estimated”. Statistica Sinica 7, pp. 789-798.
3. Ghosh, B. K., Reynolds, M. R., and Hui, Y. V. (1981). “Shewhart X -Charts with Estimated Process Variance”. Communications in Statistics- Theory and Methods 18, pp. 1797-1822.
4. Jensen, W. A., Jones-Farmer, L. A., Champ, C. W., and Woodall, W.
H. (2006). “Effects of Parameter Estimation on Control Chart
Properties: A Literature Review”. Journal of Quality Technology 38, pp. 349-364.
5. Montgomery, D. C. (2005). Introduction to Statistical Quality Control, 5th ed. John Wiley, New York.
6. Quesenberry, C. P. (1993). “The Effect of Sample Size on Estimated Limits forX andX Control Charts”. Journal of Quality Technology 25, pp. 237-247.
7. Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product. D. Van Nostrand, New York.
8. Simonoff, J. S. (1996). Smoothing Methods in Statistics.
Springer-Verlag, New York.
9. Woodall, W. H. and Montgomery, D. C. (1999). “Research Issues and
表一：資料為混合樣本，用E X( ) 3 /± σ n當管制上下限之α 和1−β的理論值
α 理論值
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027 0.0028 0.05 0.0027 0.0027 0.0029 0.0031 0.0034 0.0038 0.0043 0.1 0.0027 0.0029 0.0034 0.0043 0.0056 0.0075 0.0100 0.15 0.0027 0.0031 0.0043 0.0065 0.0100 0.0154 0.0231 0.2 0.0027 0.0034 0.0056 0.0100 0.0177 0.0299 0.0486 1−β理論值
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0027 0.0292 0.2158 0.6257 0.9233 0.9944 0.9999 0.05 0.0027 0.0263 0.1906 0.5739 0.8941 0.9896 0.9996 0.1 0.0027 0.0231 0.1617 0.5075 0.8473 0.9789 0.9988 0.15 0.0027 0.0202 0.1358 0.4408 0.7885 0.9601 0.9966 0.2 0.0027 0.0177 0.1129 0.3757 0.7183 0.9295 0.9910
表二：資料為混合樣本，使用直接估計法估計參數，10000 個模擬μˆ0、σˆ、α 和1−β的平均值及標準差
ˆ0
μ 平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0001( 0.0448 ) 0.0052 ( 0.0443 ) 0.0103 ( 0.0450 ) 0.0154 ( 0.0448 ) 0.0201 ( 0.0448 ) 0.0247 ( 0.0444 ) 0.0293 ( 0.0452 ) 0.05 -0.0004( 0.0443 ) 0.0243 ( 0.0447 ) 0.0499 ( 0.0449 ) 0.0752 ( 0.0446 ) 0.1002 ( 0.0449 ) 0.1250 ( 0.0452 ) 0.1505 ( 0.0451 ) 0.1 0.0000( 0.0444 ) 0.0498 ( 0.0446 ) 0.0997 ( 0.0443 ) 0.1499 ( 0.0446 ) 0.1990 ( 0.0445 ) 0.2494 ( 0.0446 ) 0.2999 ( 0.0452 ) 0.15 0.0006( 0.0448 ) 0.0749 ( 0.0448 ) 0.1498 ( 0.0446 ) 0.2248 ( 0.0444 ) 0.2995 ( 0.0442 ) 0.3746 ( 0.0447 ) 0.4499 ( 0.0447 ) 0.2 0.0001( 0.0446 ) 0.0995 ( 0.0445 ) 0.1997 ( 0.0446 ) 0.3004 ( 0.0443 ) 0.3998 ( 0.0450 ) 0.5005 ( 0.0449 ) 0.6001 ( 0.0449 ) σˆ平均值(σˆ標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.9993 ( 0.0351 ) 1.0002 ( 0.0355 ) 0.9997 ( 0.0357 ) 1.0005 ( 0.0356 ) 0.9999 ( 0.0350 ) 1.0003 ( 0.0353 ) 1.0001 ( 0.0359 ) 0.05 0.9994 ( 0.0355 ) 0.9997 ( 0.0355 ) 1.0001 ( 0.0353 ) 0.9997 ( 0.0353 ) 0.9999 ( 0.0352 ) 1.0000 ( 0.0350 ) 0.9989 ( 0.0353 ) 0.1 0.9998 ( 0.0356 ) 1.0000 ( 0.0357 ) 0.9991 ( 0.0350 ) 0.9997 ( 0.0354 ) 1.0000 ( 0.0357 ) 0.9997 ( 0.0356 ) 1.0002 ( 0.0357 ) 0.15 1.0003 ( 0.0351 ) 0.9995 ( 0.0350 ) 0.9998 ( 0.0354 ) 1.0002 ( 0.0352 ) 1.0000 ( 0.0353 ) 0.9993 ( 0.0350 ) 0.9999 ( 0.0354 )
α 平均值(α 標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0028 ( 0.0054 ) 0.0028 ( 0.0054 ) 0.0027 ( 0.0053 ) 0.0027 ( 0.0053 ) 0.0027 ( 0.0053 ) 0.0028 ( 0.0054 ) 0.0027 ( 0.0052 ) 0.05 0.0027 ( 0.0053 ) 0.0028 ( 0.0055 ) 0.0029 ( 0.0057 ) 0.0031 ( 0.0058 ) 0.0034 ( 0.0060 ) 0.0038 ( 0.0064 ) 0.0043 ( 0.0068 ) 0.1 0.0028 ( 0.0053 ) 0.0029 ( 0.0058 ) 0.0034 ( 0.0062 ) 0.0044 ( 0.0070 ) 0.0057 ( 0.0080 ) 0.0074 ( 0.0091 ) 0.0100 ( 0.0107 ) 0.15 0.0027 ( 0.0053 ) 0.0031 ( 0.0061 ) 0.0043 ( 0.0071 ) 0.0065 ( 0.0087 ) 0.0101 ( 0.0109 ) 0.0157 ( 0.0135 ) 0.0234 ( 0.0165 ) 0.2 0.0027 ( 0.0053 ) 0.0033 ( 0.0064 ) 0.0056 ( 0.0085 ) 0.0101 ( 0.0113 ) 0.0177 ( 0.0151 ) 0.0302 ( 0.0191 ) 0.0489 ( 0.0245 ) 1−β平均值(1−β標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 NA ( NA ) 0.0291 ( 0.1681 ) 0.2121 ( 0.4088 ) 0.6256 ( 0.4840 ) 0.9198 ( 0.2716 ) 0.9935 ( 0.0804 ) 0.9999 ( 0.0100 ) 0.05 NA ( NA ) 0.0279 ( 0.0738 ) 0.1925 ( 0.1776 ) 0.5764 ( 0.2220 ) 0.8936 ( 0.1396 ) 0.9893 ( 0.0456 ) 0.9997 ( 0.0080 ) 0.1 NA ( NA ) 0.0237 ( 0.0479 ) 0.1643 ( 0.1184 ) 0.5104 ( 0.1590 ) 0.8504 ( 0.1127 ) 0.9804 ( 0.0439 ) 0.9985 ( 0.0121 ) 0.15 NA ( NA ) 0.0198 ( 0.0357 ) 0.1360 ( 0.0894 ) 0.4392 ( 0.1288 ) 0.7890 ( 0.1051 ) 0.9598 ( 0.0504 ) 0.9965 ( 0.0152 ) 0.2 NA ( NA ) 0.0174 ( 0.0292 ) 0.1136 ( 0.0714 ) 0.3771 ( 0.1086 ) 0.7177 ( 0.1024 ) 0.9304 ( 0.0574 ) 0.9912 ( 0.0210 )
表三：資料為混合樣本，使用重複篩選法估計參數，10000 個模擬μˆ0、σˆ、α 和1−β的平均值及標準差
ˆ0
μ 平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 -0.0006( 0.0454 ) 0.0045 ( 0.0449 ) 0.0071 ( 0.0457 ) 0.0039 ( 0.0456 ) 0.0018 ( 0.0455 ) 0.0004 ( 0.0464 ) 0.0004 ( 0.0461 ) 0.05 -0.0006( 0.0452 ) 0.0241 ( 0.0460 ) 0.0355 ( 0.0478 ) 0.0238 ( 0.0503 ) 0.0068 ( 0.0482 ) 0.0030 ( 0.0476 ) 0.0026 ( 0.0470 ) 0.1 0.0000( 0.0453 ) 0.0482 ( 0.0460 ) 0.0749 ( 0.0496 ) 0.0511 ( 0.0567 ) 0.0149 ( 0.0513 ) 0.0081 ( 0.0485 ) 0.0103 ( 0.0487 ) 0.15 -0.0004( 0.0450 ) 0.0726 ( 0.0462 ) 0.1186 ( 0.0520 ) 0.0873 ( 0.0636 ) 0.0285 ( 0.0559 ) 0.0170 ( 0.0510 ) 0.0243 ( 0.0505 ) 0.2 -0.0001( 0.0459 ) 0.0974 ( 0.0463 ) 0.1654 ( 0.0528 ) 0.1340 ( 0.0731 ) 0.0464 ( 0.0619 ) 0.0326 ( 0.0539 ) 0.0464 ( 0.0542 ) σˆ平均值(σˆ標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.9997 ( 0.0356 ) 1.0007 ( 0.0354 ) 0.9996 ( 0.0360 ) 1.0004 ( 0.0358 ) 0.9998 ( 0.0356 ) 0.9999 ( 0.0353 ) 0.9998 ( 0.0355 ) 0.05 1.0002 ( 0.0353 ) 1.0003 ( 0.0358 ) 0.9996 ( 0.0359 ) 0.9996 ( 0.0365 ) 0.9998 ( 0.0367 ) 0.9997 ( 0.0363 ) 0.9999 ( 0.0365 ) 0.1 0.9994 ( 0.0353 ) 0.9997 ( 0.0360 ) 1.0000 ( 0.0358 ) 1.0007 ( 0.0368 ) 1.0009 ( 0.0373 ) 1.0001 ( 0.0369 ) 1.0002 ( 0.0380 ) 0.15 0.9997 ( 0.0357 ) 1.0003 ( 0.0357 ) 1.0007 ( 0.0355 ) 1.0003 ( 0.0377 ) 1.0001 ( 0.0383 ) 1.0002 ( 0.0384 ) 0.9997 ( 0.0393 )
α 平均值(α 標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0028 ( 0.0054 ) 0.0027 ( 0.0055 ) 0.0029 ( 0.0056 ) 0.0029 ( 0.0056 ) 0.0029 ( 0.0056 ) 0.0029 ( 0.0056 ) 0.0031 ( 0.0056 ) 0.05 0.0028 ( 0.0054 ) 0.0028 ( 0.0056 ) 0.0031 ( 0.0058 ) 0.0034 ( 0.0061 ) 0.0040 ( 0.0065 ) 0.0046 ( 0.0070 ) 0.0052 ( 0.0074 ) 0.1 0.0027 ( 0.0053 ) 0.0030 ( 0.0058 ) 0.0035 ( 0.0063 ) 0.0048 ( 0.0074 ) 0.0066 ( 0.0086 ) 0.0085 ( 0.0098 ) 0.0112 ( 0.0112 ) 0.15 0.0027 ( 0.0054 ) 0.0031 ( 0.0061 ) 0.0044 ( 0.0074 ) 0.0071 ( 0.0093 ) 0.0110 ( 0.0115 ) 0.0165 ( 0.0140 ) 0.0242 ( 0.0171 ) 0.2 0.0028 ( 0.0053 ) 0.0034 ( 0.0067 ) 0.0058 ( 0.0086 ) 0.0105 ( 0.0116 ) 0.0186 ( 0.0152 ) 0.0313 ( 0.0198 ) 0.0494 ( 0.0245 ) 1−β平均值(1−β標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 NA ( NA ) 0.0313 ( 0.1741 ) 0.2207 ( 0.4147 ) 0.6245 ( 0.4843 ) 0.9244 ( 0.2644 ) 0.9951 ( 0.0698 ) 1.0000 ( 0.0000 ) 0.05 NA ( NA ) 0.0265 ( 0.0735 ) 0.1977 ( 0.1874 ) 0.6061 ( 0.2309 ) 0.9214 ( 0.1250 ) 0.9939 ( 0.0353 ) 0.9998 ( 0.0057 ) 0.1 NA ( NA ) 0.0233 ( 0.0483 ) 0.1744 ( 0.1317 ) 0.5832 ( 0.1811 ) 0.9184 ( 0.0925 ) 0.9938 ( 0.0253 ) 0.9998 ( 0.0047 ) 0.15 NA ( NA ) 0.0210 ( 0.0377 ) 0.1482 ( 0.1040 ) 0.5526 ( 0.1635 ) 0.9131 ( 0.0824 ) 0.9934 ( 0.0213 ) 0.9998 ( 0.0033 ) 0.2 NA ( NA ) 0.0179 ( 0.0304 ) 0.1283 ( 0.0871 ) 0.5112 ( 0.1580 ) 0.9041 ( 0.0795 ) 0.9926 ( 0.0199 ) 0.9998 ( 0.0032 )
表四：資料為混合樣本，使用核密度估計法估計參數，代入不同的帶寬h，10000 個模擬μˆ0的平均值及標準差
(a) h = 0.1，μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0011( 0.1573 ) 0.0051 ( 0.1568 ) 0.0001 ( 0.1610 ) -0.0009( 0.1586) -0.0012(0.1564 ) 0.0041( 0.1575 ) 0.0024( 0.1592 ) 0.05 -0.0003( 0.1567 ) 0.0194 ( 0.1621 ) 0.0040 ( 0.1624 ) 0.0008( 0.1617 ) 0.0007( 0.1617 ) 0.0005( 0.1597 ) -0.0025(0.1599 ) 0.1 0.0011( 0.1616 ) 0.0354 ( 0.1660 ) 0.0148 ( 0.1682 ) -0.0009( 0.1625) 0.0008( 0.1642 ) -0.0039(0.1625 ) 0.0013( 0.1629 ) 0.15 0.0008( 0.1586 ) 0.0523 ( 0.1696 ) 0.0249 ( 0.1759 ) 0.0042( 0.1652 ) -0.0021(0.1643 ) 0.0002( 0.1648 ) 0.0014( 0.1644 ) 0.2 0.0026( 0.1579 ) 0.0740 ( 0.1750 ) 0.0385 ( 0.1837 ) 0.0053( 0.1736 ) 0.0008( 0.1689 ) -0.0023(0.1678 ) 0.0012( 0.1688 ) (b) h = 0.3，μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 -0.0013( 0.0732 ) 0.0034 ( 0.0739 ) 0.0027 ( 0.0739 ) -0.0002( 0.0742) 0.0000( 0.0736 ) -0.0003( 0.0724) 0.0003( 0.0738 ) 0.05 0.0011( 0.0737 ) 0.0191 ( 0.0761 ) 0.0104 ( 0.0772 ) 0.0006( 0.0759 ) -0.0004( 0.0751) 0.0012( 0.0746 ) 0.0003( 0.0749 ) 0.1 -0.0004( 0.0737 ) 0.0351 ( 0.0781 ) 0.0208 ( 0.0817 ) 0.0046( 0.0791 ) 0.0003( 0.0774 ) -0.0002( 0.0775) 0.0008( 0.0773 ) 0.15 0.0001( 0.0732 ) 0.0553 ( 0.0814 ) 0.0356 ( 0.0895 ) 0.0056( 0.0828 ) 0.0006( 0.0792 ) 0.0000( 0.0800 ) 0.0002( 0.0801 )
(c) h = 0.5，μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 -0.0006( 0.0530 ) 0.0040 ( 0.0532 ) 0.0041 ( 0.0535 ) 0.0012 ( 0.0537 ) 0.0001 ( 0.0533 ) 0.0000( 0.0529 ) 0.0004( 0.0534 ) 0.05 0.0003( 0.0534 ) 0.0211 ( 0.0543 ) 0.0183 ( 0.0562 ) 0.0062 ( 0.0557 ) 0.0010 ( 0.0546 ) -0.0008( 0.0542) 0.0001( 0.0541 ) 0.1 -0.0001( 0.0532 ) 0.0407 ( 0.0543 ) 0.0378 ( 0.0589 ) 0.0148 ( 0.0585 ) 0.0029 ( 0.0562 ) -0.0004( 0.0558) 0.0012( 0.0554 ) 0.15 -0.0003( 0.0527 ) 0.0620 ( 0.0560 ) 0.0624 ( 0.0637 ) 0.0229 ( 0.0625 ) 0.0044 ( 0.0580 ) -0.0006( 0.0581) 0.0002( 0.0578 ) 0.2 -0.0003( 0.0532 ) 0.0862 ( 0.0572 ) 0.0917 ( 0.0683 ) 0.0347 ( 0.0667 ) 0.0056 ( 0.0617 ) 0.0010( 0.0592 ) -0.0007( 0.0593) (d) h = 0.7，μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0013( 0.0479 ) 0.0042 ( 0.0483 ) 0.0056 ( 0.0491 ) 0.0029 ( 0.0489 ) 0.0006 ( 0.0480 ) 0.0006 ( 0.0482 ) 0.0004( 0.0483 ) 0.05 -0.0002( 0.0475 ) 0.0212 ( 0.0489 ) 0.0263 ( 0.0501 ) 0.0166 ( 0.0510 ) 0.0057 ( 0.0504 ) 0.0022 ( 0.0494 ) -0.0001( 0.0497) 0.1 0.0002( 0.0486 ) 0.0436 ( 0.0491 ) 0.0549 ( 0.0519 ) 0.0344 ( 0.0539 ) 0.0134 ( 0.0523 ) 0.0020 ( 0.0508 ) 0.0009( 0.0512 ) 0.15 -0.0001( 0.0485 ) 0.0665 ( 0.0492 ) 0.0892 ( 0.0550 ) 0.0569 ( 0.0580 ) 0.0202 ( 0.0553 ) 0.0055 ( 0.0533 ) 0.0011( 0.0526 ) 0.2 0.0001( 0.0481 ) 0.0912 ( 0.0496 ) 0.1260 ( 0.0569 ) 0.0852 ( 0.0627 ) 0.0310 ( 0.0585 ) 0.0075 ( 0.0555 ) 0.0009( 0.0543 )
表五：資料為混合樣本，使用核密度估計法估計參數，10000 個模擬μˆ0、σˆ、α 和1−β的平均值及標準差
ˆ0
μ 平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 -0.0006( 0.0530 ) 0.0040 ( 0.0532 ) 0.0041 ( 0.0535 ) 0.0012 ( 0.0537 ) 0.0001 ( 0.0533 ) 0.0000 ( 0.0529 ) 0.0004( 0.0534 ) 0.05 0.0003( 0.0534 ) 0.0211 ( 0.0543 ) 0.0183 ( 0.0562 ) 0.0062 ( 0.0557 ) 0.0010 ( 0.0546 ) 0.0008 ( 0.0542 ) 0.0001( 0.0541 ) 0.1 -0.0001( 0.0532 ) 0.0407 ( 0.0543 ) 0.0378 ( 0.0589 ) 0.0148 ( 0.0585 ) 0.0029 ( 0.0562 ) 0.0004 ( 0.0558 ) 0.0012( 0.0554 ) 0.15 -0.0003( 0.0527 ) 0.0620 ( 0.0560 ) 0.0624 ( 0.0637 ) 0.0229 ( 0.0625 ) 0.0044 ( 0.0580 ) 0.0006 ( 0.0581 ) 0.0002( 0.0578 ) 0.2 -0.0003( 0.0532 ) 0.0862 ( 0.0572 ) 0.0917 ( 0.0683 ) 0.0347 ( 0.0667 ) 0.0056 ( 0.0617 ) 0.0010 ( 0.0592 ) -0.0007( 0.0593 ) σˆ平均值(σˆ標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 1.0000 ( 0.0353 ) 0.9999 ( 0.0349 ) 0.9999 ( 0.0356 ) 0.9994 ( 0.0359 ) 0.9996 ( 0.0351 ) 0.9998 ( 0.0351 ) 1.0003 ( 0.0355 ) 0.05 0.9999 ( 0.0355 ) 1.0002 ( 0.0352 ) 0.9996 ( 0.0358 ) 1.0005 ( 0.0353 ) 0.9996 ( 0.0349 ) 0.9995 ( 0.0358 ) 1.0000 ( 0.0354 ) 0.1 1.0001 ( 0.0355 ) 0.9995 ( 0.0354 ) 0.9995 ( 0.0356 ) 0.9999 ( 0.0353 ) 1.0002 ( 0.0351 ) 0.9999 ( 0.0357 ) 1.0001 ( 0.0351 ) 0.15 1.0008 ( 0.0354 ) 1.0005 ( 0.0355 ) 1.0003 ( 0.0351 ) 0.9994 ( 0.0349 ) 1.0001 ( 0.0351 ) 1.0001 ( 0.0350 ) 0.9997 ( 0.0353 )
α 平均值(α 標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 0.0030 ( 0.0056 ) 0.0030 ( 0.0056 ) 0.0030 ( 0.0055 ) 0.0031 ( 0.0057 ) 0.0030 ( 0.0057 ) 0.0031 ( 0.0057 ) 0.0031 ( 0.0057 ) 0.05 0.0030 ( 0.0056 ) 0.0030 ( 0.0057 ) 0.0031 ( 0.0058 ) 0.0030 ( 0.0057 ) 0.0031 ( 0.0059 ) 0.0030 ( 0.0058 ) 0.0031 ( 0.0057 ) 0.1 0.0029 ( 0.0055 ) 0.0032 ( 0.0060 ) 0.0032 ( 0.0061 ) 0.0031 ( 0.0059 ) 0.0030 ( 0.0059 ) 0.0031 ( 0.0059 ) 0.0030 ( 0.0060 ) 0.15 0.0030 ( 0.0056 ) 0.0033 ( 0.0064 ) 0.0035 ( 0.0065 ) 0.0032 ( 0.0062 ) 0.0030 ( 0.0061 ) 0.0030 ( 0.0061 ) 0.0031 ( 0.0061 ) 0.2 0.0031 ( 0.0056 ) 0.0037 ( 0.0069 ) 0.0039 ( 0.0072 ) 0.0032 ( 0.0064 ) 0.0031 ( 0.0064 ) 0.0031 ( 0.0063 ) 0.0031 ( 0.0064 ) 1−β平均值(1−β標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.01 NA ( NA ) 0.0322 ( 0.1765 ) 0.2241 ( 0.4170 ) 0.6464 ( 0.4781 ) 0.9257 ( 0.2623 ) 0.9932 ( 0.0822 ) 0.9999 ( 0.0100 ) 0.05 NA ( NA ) 0.0284 ( 0.0750 ) 0.2161 ( 0.1920 ) 0.6287 ( 0.2241 ) 0.9253 ( 0.1197 ) 0.9945 ( 0.0332 ) 0.9998 ( 0.0066 ) 0.1 NA ( NA ) 0.0269 ( 0.0517 ) 0.2016 ( 0.1379 ) 0.6256 ( 0.1678 ) 0.9260 ( 0.0858 ) 0.9947 ( 0.0229 ) 0.9999 ( 0.0026 ) 0.15 NA ( NA ) 0.0232 ( 0.0403 ) 0.1886 ( 0.1150 ) 0.6199 ( 0.1447 ) 0.9248 ( 0.0726 ) 0.9943 ( 0.0198 ) 0.9998 ( 0.0032 ) 0.2 NA ( NA ) 0.0208 ( 0.0331 ) 0.1716 ( 0.1007 ) 0.6070 ( 0.1353 ) 0.9246 ( 0.0648 ) 0.9948 ( 0.0162 ) 0.9999 ( 0.0025 )
表六：階段 I 為( , )δ p =(0, 0.05)時的情況，階段 II 的 ARL 和 SDRL 之平均估計量
直接估計法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 437.4318 ( 5.02 ) 47.7263 ( 0.49 ) 5.2368 ( 0.03 ) 1.6511 ( 0.00 ) 1.0918 ( 0.00 ) 1.0071 ( 0.00 ) 1.0002 ( 0.00 )
436.9313 47.2216 4.7026 1.0235 0.303 0.0764 0.0122
40 399.0741 ( 2.52 ) 39.7920 ( 0.24 ) 4.8484 ( 0.02 ) 1.6086 ( 0.00 ) 1.0839 ( 0.00 ) 1.0059 ( 0.00 ) 1.0002 ( 0.00 )
398.5737 39.2878 4.3157 0.9828 0.2947 0.0731 0.0112
100 377.6774 ( 1.41 ) 35.5088 ( 0.13 ) 4.6108 ( 0.01 ) 1.5800 ( 0.00 ) 1.0785 ( 0.00 ) 1.0052 ( 0.00 ) 1.0001 ( 0.00 )
377.177 35.0048 4.0787 0.9546 0.2882 0.0707 0.0106
300 374.5815 ( 0.78 ) 34.2616 ( 0.07 ) 4.5473 ( 0.01 ) 1.5731 ( 0.00 ) 1.0771 ( 0.00 ) 1.0050 ( 0.00 ) 1.0001 ( 0.00 )
374.0812 33.7578 4.0157 0.9486 0.2872 0.0702 0.0104
500 371.2904 ( 0.59 ) 33.8035 ( 0.05 ) 4.5176 ( 0.00 ) 1.5691 ( 0.00 ) 1.0764 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
370.7901 33.2996 3.9861 0.9445 0.2861 0.0698 0.0103
1000 371.4106 ( 0.41 ) 33.6322 ( 0.03 ) 4.5091 ( 0.00 ) 1.5682 ( 0.00 ) 1.0762 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
370.9102 33.1283 3.9776 0.9437 0.286 0.0697 0.0103
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
重複篩選法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 442.5107 ( 4.92 ) 48.7719 ( 0.52 ) 5.2899 ( 0.03 ) 1.6573 ( 0.00 ) 1.0929 ( 0.00 ) 1.0072 ( 0.00 ) 1.0002 ( 0.00 )
442.0101 48.2672 4.7558 1.03 0.3046 0.0769 0.0124
40 397.0788 ( 2.62 ) 39.9481 ( 0.26 ) 4.8530 ( 0.02 ) 1.6088 ( 0.00 ) 1.0839 ( 0.00 ) 1.0059 ( 0.00 ) 1.0002 ( 0.00 )
396.5783 39.4439 4.3202 0.9828 0.2945 0.0731 0.0112
100 379.1390 ( 1.43 ) 35.7250 ( 0.13 ) 4.6267 ( 0.01 ) 1.5823 ( 0.00 ) 1.0789 ( 0.00 ) 1.0052 ( 0.00 ) 1.0001 ( 0.00 )
378.6386 35.221 4.0947 0.9572 0.289 0.0709 0.0106
300 372.0144 ( 0.75 ) 34.0803 ( 0.07 ) 4.5334 ( 0.01 ) 1.5710 ( 0.00 ) 1.0767 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
371.514 33.5764 4.0018 0.9463 0.2865 0.07 0.0103
500 371.4313 ( 0.58 ) 33.8825 ( 0.05 ) 4.5240 ( 0.00 ) 1.5701 ( 0.00 ) 1.0765 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
370.9309 33.3786 3.9925 0.9456 0.2865 0.0699 0.0103
1000 371.2092 ( 0.41 ) 33.6323 ( 0.04 ) 4.5089 ( 0.00 ) 1.5682 ( 0.00 ) 1.0762 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
370.7088 33.1285 3.9774 0.9437 0.286 0.0697 0.0103
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
369.8980 32.8970 3.9639 0.9420 0.2856 0.0696 0.0102
核密度估計法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 415.1536 ( 4.65 ) 49.9419 ( 0.59 ) 5.3223 ( 0.03 ) 1.6574 ( 0.00 ) 1.0931 ( 0.00 ) 1.0073 ( 0.00 ) 1.0002 ( 0.00 )
414.653 49.437 4.7873 1.0281 0.3029 0.0765 0.0124
40 381.8413 ( 2.49 ) 41.1554 ( 0.30 ) 4.9134 ( 0.02 ) 1.6154 ( 0.00 ) 1.0852 ( 0.00 ) 1.0061 ( 0.00 ) 1.0002 ( 0.00 )
381.3408 40.651 4.3803 0.9889 0.2957 0.0736 0.0114
100 371.9531 ( 1.37 ) 36.1617 ( 0.14 ) 4.6530 ( 0.01 ) 1.5857 ( 0.00 ) 1.0795 ( 0.00 ) 1.0053 ( 0.00 ) 1.0001 ( 0.00 )
371.4527 35.6577 4.121 0.9606 0.2898 0.0712 0.0107
300 369.7127 ( 0.76 ) 34.3263 ( 0.07 ) 4.5492 ( 0.01 ) 1.5732 ( 0.00 ) 1.0771 ( 0.00 ) 1.0050 ( 0.00 ) 1.0001 ( 0.00 )
369.2124 33.8224 4.0176 0.9485 0.2871 0.0702 0.0104
500 369.4954 ( 0.58 ) 33.8325 ( 0.06 ) 4.5182 ( 0.00 ) 1.5691 ( 0.00 ) 1.0764 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
368.9951 33.3286 3.9866 0.9443 0.286 0.0698 0.0103
1000 370.1080 ( 0.41 ) 33.6813 ( 0.04 ) 4.5117 ( 0.00 ) 1.5685 ( 0.00 ) 1.0762 ( 0.00 ) 1.0049 ( 0.00 ) 1.0001 ( 0.00 )
369.6077 33.1775 3.9803 0.944 0.2861 0.0698 0.0103
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
表七：階段 I 為( , )δ p =(1, 0.2)時的情況，階段 II 的 ARL 和 SDRL 之平均估計量
直接估計法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 263.1608 ( 3.93 ) 152.4961 ( 1.9 ) 11.0356 ( 0.08 ) 2.3430 ( 0.01 ) 1.2102 ( 0.00 ) 1.0217 ( 0.00 ) 1.0010 ( 0.00 )
262.6598 151.9943 10.5186 1.7624 0.4899 0.1382 0.027
40 215.1833 ( 1.58 ) 122.1872 ( 0.9 ) 9.7936 ( 0.04 ) 2.2416 ( 0.00 ) 1.1939 ( 0.00 ) 1.0188 ( 0.00 ) 1.0007 ( 0.00 )
214.6824 121.6857 9.2775 1.6624 0.4738 0.133 0.0252
100 191.6602 ( 0.8 ) 106.9659 ( 0.43 ) 9.1616 ( 0.02 ) 2.1864 ( 0.00 ) 1.1847 ( 0.00 ) 1.0172 ( 0.00 ) 1.0006 ( 0.00 )
191.1594 106.4645 8.6461 1.6083 0.4649 0.13 0.0241
300 182.0988 ( 0.43 ) 102.3454 ( 0.23 ) 8.9783 ( 0.01 ) 2.1716 ( 0.00 ) 1.1822 ( 0.00 ) 1.0167 ( 0.00 ) 1.0006 ( 0.00 )
181.5981 101.8441 8.4632 1.5944 0.4631 0.1295 0.0238
500 181.0128 ( 0.33 ) 101.2713 ( 0.18 ) 8.9311 ( 0.01 ) 2.1673 ( 0.00 ) 1.1814 ( 0.00 ) 1.0165 ( 0.00 ) 1.0006 ( 0.00 )
180.5121 100.77 8.416 1.5901 0.4624 0.1292 0.0237
1000 179.6288 ( 0.23 ) 100.4270 ( 0.12 ) 8.8945 ( 0.01 ) 2.1639 ( 0.00 ) 1.1809 ( 0.00 ) 1.0164 ( 0.00 ) 1.0006 ( 0.00 )
179.1281 99.9257 8.3795 1.5868 0.4618 0.129 0.0237
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
369.8980 32.8970 3.9639 0.9420 0.2856 0.0696 0.0102
重複篩選法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 284.8826 ( 3.17 ) 135.7128 ( 1.85 ) 10.0634 ( 0.08 ) 2.2224 ( 0.01 ) 1.1894 ( 0.00 ) 1.0192 ( 0.00 ) 1.0009 ( 0.00 )
284.3817 135.2105 9.543 1.6328 0.4562 0.127 0.0244
40 250.4702 ( 1.81 ) 105.6967 ( 0.85 ) 8.8390 ( 0.04 ) 2.1230 ( 0.00 ) 1.1732 ( 0.00 ) 1.0163 ( 0.00 ) 1.0006 ( 0.00 )
249.9695 105.1949 8.3204 1.5365 0.4416 0.1222 0.0226
100 231.1157 ( 0.98 ) 90.8754 ( 0.42 ) 8.2007 ( 0.02 ) 2.0654 ( 0.00 ) 1.1633 ( 0.00 ) 1.0146 ( 0.00 ) 1.0005 ( 0.00 )
230.6151 90.3737 7.683 1.4803 0.4321 0.1189 0.0214
300 220.4009 ( 0.53 ) 84.5327 ( 0.21 ) 7.9187 ( 0.01 ) 2.0384 ( 0.00 ) 1.1586 ( 0.00 ) 1.0138 ( 0.00 ) 1.0004 ( 0.00 )
219.9003 84.0311 7.4014 1.4539 0.4274 0.1172 0.0208
500 217.7829 ( 0.40 ) 83.6597 ( 0.17 ) 7.8827 ( 0.01 ) 2.0353 ( 0.00 ) 1.1580 ( 0.00 ) 1.0137 ( 0.00 ) 1.0004 ( 0.00 )
217.2823 83.1582 7.3654 1.451 0.427 0.1171 0.0208
1000 216.2028 ( 0.28 ) 82.7584 ( 0.11 ) 7.8429 ( 0.01 ) 2.0316 ( 0.00 ) 1.1574 ( 0.00 ) 1.0135 ( 0.00 ) 1.0004 ( 0.00 )
215.7022 82.2568 7.3257 1.4474 0.4264 0.1169 0.0207
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
核密度估計法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 349.8392 ( 4.15 ) 96.1688 ( 1.55 ) 8.0391 ( 0.09 ) 1.9737 ( 0.01 ) 1.1466 ( 0.00 ) 1.0140 ( 0.00 ) 1.0006 ( 0.00 )
349.3384 95.6653 7.5123 1.3666 0.3881 0.1045 0.019
40 321.1608 ( 2.34 ) 73.1144 ( 0.66 ) 6.9154 ( 0.04 ) 1.8785 ( 0.00 ) 1.1307 ( 0.00 ) 1.0113 ( 0.00 ) 1.0004 ( 0.00 )
320.6603 72.6115 6.3905 1.2742 0.3728 0.0992 0.0172
100 312.1983 ( 1.33 ) 60.8974 ( 0.30 ) 6.3465 ( 0.02 ) 1.8230 ( 0.00 ) 1.1207 ( 0.00 ) 1.0096 ( 0.00 ) 1.0003 ( 0.00 )
311.6979 60.3949 5.823 1.2209 0.3633 0.0956 0.016
300 309.4428 ( 0.75 ) 56.3047 ( 0.15 ) 6.1246 ( 0.01 ) 1.8000 ( 0.00 ) 1.1165 ( 0.00 ) 1.0090 ( 0.00 ) 1.0002 ( 0.00 )
308.9424 55.8023 5.6016 1.1986 0.3591 0.094 0.0154
500 306.9473 ( 0.58 ) 55.4987 ( 0.12 ) 6.0857 ( 0.01 ) 1.7960 ( 0.00 ) 1.1157 ( 0.00 ) 1.0088 ( 0.00 ) 1.0002 ( 0.00 )
306.4469 54.9963 5.5628 1.1948 0.3584 0.0938 0.0153
1000 305.9376 ( 0.41 ) 54.8012 ( 0.08 ) 6.0509 ( 0.01 ) 1.7922 ( 0.00 ) 1.1150 ( 0.00 ) 1.0087 ( 0.00 ) 1.0002 ( 0.00 )
305.4372 54.2989 5.5281 1.1911 0.3577 0.0935 0.0153
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
369.8980 32.8970 3.9639 0.9420 0.2856 0.0696 0.0102
表八：階段 I 為( , )δ p =(2, 0.1)時的情況，階段 II 的 ARL 和 SDRL 之平均估計量
直接估計法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 259.0842 ( 3.21 ) 151.1937 ( 1.76 ) 11.0137 ( 0.08 ) 2.3431 ( 0.01 ) 1.2103 ( 0.00 ) 1.0217 ( 0.00 ) 1.001 ( 0.00 )
258.5832 150.692 10.4968 1.7628 0.4904 0.1383 0.0271
40 212.9041 ( 1.50 ) 120.7065 ( 0.87 ) 9.7308 ( 0.04 ) 2.2351 ( 0.00 ) 1.1928 ( 0.00 ) 1.0186 ( 0.00 ) 1.0007 ( 0.00 )
212.4032 120.205 9.2147 1.6558 0.4724 0.1325 0.025
100 192.0376 ( 0.83 ) 108.0714 ( 0.44 ) 9.2233 ( 0.02 ) 2.1938 ( 0.00 ) 1.1860 ( 0.00 ) 1.0173 ( 0.00 ) 1.0006 ( 0.00 )
191.5368 107.5701 8.7079 1.6161 0.4668 0.1307 0.0243
300 182.7799 ( 0.43 ) 102.2426 ( 0.23 ) 8.9714 ( 0.01 ) 2.1708 ( 0.00 ) 1.1820 ( 0.00 ) 1.0166 ( 0.00 ) 1.0006 ( 0.00 )
182.2792 101.7413 8.4563 1.5934 0.4629 0.1294 0.0238
500 180.9933 ( 0.33 ) 101.2463 ( 0.18 ) 8.9300 ( 0.01 ) 2.1672 ( 0.00 ) 1.1814 ( 0.00 ) 1.0165 ( 0.00 ) 1.0006 ( 0.00 )
180.4926 100.745 8.4149 1.59 0.4624 0.1292 0.0237
1000 179.1272 ( 0.23 ) 100.4570 ( 0.12 ) 8.8964 ( 0.01 ) 2.1641 ( 0.00 ) 1.1809 ( 0.00 ) 1.0164 ( 0.00 ) 1.0006 ( 0.00 )
178.6265 99.9557 8.3814 1.587 0.4619 0.129 0.0237
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
重複篩選法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 428.2139 ( 4.82 ) 59.8503 ( 0.8 ) 5.8924 ( 0.04 ) 1.7287 ( 0.00 ) 1.1052 ( 0.00 ) 1.0087 ( 0.00 ) 1.0003 ( 0.00 )
427.7133 59.3458 5.3598 1.1057 0.3235 0.0832 0.0139
40 389.7870 ( 2.71 ) 45.8138 ( 0.38 ) 5.2174 ( 0.02 ) 1.6573 ( 0.00 ) 1.0924 ( 0.00 ) 1.0069 ( 0.00 ) 1.0002 ( 0.00 )
389.2865 45.3098 4.686 1.0351 0.3087 0.0779 0.0123
100 374.8046 ( 1.47 ) 39.5816 ( 0.16 ) 4.9049 ( 0.01 ) 1.6226 ( 0.00 ) 1.0859 ( 0.00 ) 1.0059 ( 0.00 ) 1.0001 ( 0.00 )
374.3042 39.0779 4.3746 1.0019 0.3019 0.0752 0.0115
300 369.8475 ( 0.79 ) 37.0851 ( 0.08 ) 4.7623 ( 0.01 ) 1.6052 ( 0.00 ) 1.0826 ( 0.00 ) 1.0055 ( 0.00 ) 1.0001 ( 0.00 )
369.3471 36.5815 4.2323 0.9845 0.2979 0.0737 0.0111
500 368.7747 ( 0.62 ) 36.6685 ( 0.06 ) 4.7383 ( 0.00 ) 1.6022 ( 0.00 ) 1.0820 ( 0.00 ) 1.0054 ( 0.00 ) 1.0001 ( 0.00 )
368.2744 36.1649 4.2083 0.9816 0.2972 0.0734 0.011
1000 368.5056 ( 0.43 ) 36.2806 ( 0.04 ) 4.7152 ( 0.00 ) 1.5993 ( 0.00 ) 1.0815 ( 0.00 ) 1.0054 ( 0.00 ) 1.0001 ( 0.00 )
368.0052 35.777 4.1853 0.9787 0.2965 0.0732 0.0109
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
369.8980 32.8970 3.9639 0.9420 0.2856 0.0696 0.0102
核密度估計法：
δ%
m 0.0 0.5 1.0 1.5 2.0 2.5 3.0
20 416.1985 ( 4.66 ) 55.7820 ( 0.84 ) 5.6292 ( 0.04 ) 1.6928 ( 0.00 ) 1.0992 ( 0.00 ) 1.0081 ( 0.00 ) 1.0003 ( 0.00 )
415.6979 55.2771 5.0947 1.0654 0.3121 0.0796 0.0131
40 373.5297 ( 2.47 ) 41.9736 ( 0.31 ) 4.9572 ( 0.02 ) 1.6205 ( 0.00 ) 1.0862 ( 0.00 ) 1.0062 ( 0.00 ) 1.0002 ( 0.00 )
373.0292 41.4692 4.424 0.9939 0.2968 0.074 0.0115
100 369.9462 ( 1.39 ) 36.8038 ( 0.15 ) 4.6942 ( 0.01 ) 1.5912 ( 0.00 ) 1.0805 ( 0.00 ) 1.0054 ( 0.00 ) 1.0001 ( 0.00 )
369.4459 36.2998 4.1623 0.9665 0.2914 0.0718 0.0108
300 368.8357 ( 0.75 ) 34.8694 ( 0.08 ) 4.5906 ( 0.01 ) 1.5793 ( 0.00 ) 1.0782 ( 0.00 ) 1.0051 ( 0.00 ) 1.0001 ( 0.00 )
368.3353 34.3655 4.0592 0.9554 0.2892 0.0708 0.0105
500 369.5282 ( 0.59 ) 34.4325 ( 0.06 ) 4.5646 ( 0.00 ) 1.5760 ( 0.00 ) 1.0776 ( 0.00 ) 1.0050 ( 0.00 ) 1.0001 ( 0.00 )
369.0278 33.9287 4.0333 0.9521 0.2884 0.0705 0.0104
1000 370.0263 ( 0.41 ) 34.1212 ( 0.04 ) 4.5461 ( 0.00 ) 1.5737 ( 0.00 ) 1.0771 ( 0.00 ) 1.005 ( 0.00 ) 1.0001 ( 0.00 )
369.526 33.6174 4.0149 0.9499 0.2878 0.0703 0.0104
∞ 370.3983 33.4008 4.4953 1.5665 1.0758 1.0048 1.0001
表九：對σ 已知和未知的情況，分別建議的
L
逼近值(a)當σ已知，h = 0.5
m 30 50 100 300 500 1000
L 3.07 3.04 3.02 3.01 3 3
當σ 已知，h0 =1.059σ′ ′n−1/ 5
m 30 50 100 300 500 1000
L 3.22 3.17 3.12 3.07 3.06 3.04
(b)當 m = 100，σ 未知，h0 =1.059σ′ ′n−1/ 5
n 2 3 4 5 6 7 8 9 10
L 231.5 19.04 9.18 6.62 5.53 4.94 4.58 4.33 4.15 n 11 12 13 14 15 16 17 18 19 L 4.02 3.92 3.84 3.77 3.72 3.67 3.63 3.59 3.57 n 20 30 40 50 60 70 80 90 100 L 3.54 3.38 3.31 3.27 3.24 3.23 3.21 3.2 3.19
表十：樣本來自 t 分配，使用核密度估計法估計參數，10000 個模擬μˆ0的平均值及標準差
ν = 5的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 0.0005 ( 0.0691 ) 0.0227 ( 0.0722 ) 0.0140 ( 0.0726 ) 0.0033 ( 0.0728 ) 0.0001 ( 0.0715 ) -0.0008( 0.0716) 0.0006 ( 0.0714 ) 0.1 0.0003 ( 0.0696 ) 0.0471 ( 0.0744 ) 0.0325 ( 0.0775 ) 0.0069 ( 0.0742 ) 0.0008 ( 0.0737 ) 0.0002 ( 0.0738 ) -0.0001( 0.0742) 0.15 0.0005 ( 0.0706 ) 0.0746 ( 0.0769 ) 0.0520 ( 0.0848 ) 0.0119 ( 0.0783 ) 0.0007 ( 0.0764 ) -0.0006( 0.0760) 0.0003 ( 0.0756 ) 0.2 0.0012 ( 0.0702 ) 0.1022 ( 0.0794 ) 0.0762 ( 0.0899 ) 0.0177 ( 0.0823 ) 0.0014 ( 0.0789 ) -0.0001( 0.0782) -0.0011( 0.0776) ν = 10 的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 -0.0002( 0.0608 ) 0.0203 ( 0.0615 ) 0.0158 ( 0.0642 ) 0.0056 ( 0.0629 ) 0.0019 ( 0.0621 ) 0.0002 ( 0.0618 ) -0.0003( 0.0622) 0.1 -0.0007( 0.0605 ) 0.0429 ( 0.0632 ) 0.0368 ( 0.0672 ) 0.0119 ( 0.0659 ) 0.0017 ( 0.0649 ) -0.0007( 0.0634) -0.0003( 0.0643) 0.15 -0.0010( 0.0607 ) 0.0678 ( 0.0643 ) 0.0592 ( 0.0730 ) 0.0183 ( 0.0694 ) 0.0026 ( 0.0669 ) -0.0004( 0.0652) 0.0003 ( 0.0655 ) 0.2 -0.0002( 0.0606 ) 0.0941 ( 0.0665 ) 0.0875 ( 0.0787 ) 0.0265 ( 0.0730 ) 0.0046 ( 0.0688 ) 0.0008 ( 0.0679 ) 0.0000 ( 0.0674 )
ν = 20 的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 -0.0005( 0.0569 ) 0.0200 ( 0.0574 ) 0.0174 ( 0.0602 ) 0.0071 ( 0.0590 ) 0.0009 ( 0.0576 ) 0.0003 ( 0.0583 ) 0.0002 ( 0.0581 ) 0.1 -0.0001( 0.0567 ) 0.0417 ( 0.0594 ) 0.0382 ( 0.0638 ) 0.0128 ( 0.0623 ) 0.0026 ( 0.0599 ) 0.0005 ( 0.0601 ) 0.0019 ( 0.0598 ) 0.15 -0.0003( 0.0566 ) 0.0656 ( 0.0599 ) 0.0608 ( 0.0671 ) 0.0216 ( 0.0660 ) 0.0039 ( 0.0619 ) 0.0009 ( 0.0613 ) 0.0000 ( 0.0607 ) 0.2 -0.0010( 0.0564 ) 0.0899 ( 0.0612 ) 0.0910 ( 0.0728 ) 0.0304 ( 0.0692 ) 0.0054 ( 0.0642 ) 0.0009 ( 0.0630 ) -0.0002( 0.0626) ν = 30 的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 0.0004 ( 0.0554 ) 0.0200 ( 0.0557 ) 0.0173 ( 0.0585 ) 0.0058 ( 0.0574 ) 0.0007 ( 0.0574 ) -0.0008( 0.0569) -0.0008( 0.0564) 0.1 0.0001 ( 0.0555 ) 0.0413 ( 0.0579 ) 0.0381 ( 0.0616 ) 0.0133 ( 0.0607 ) 0.0019 ( 0.0580 ) 0.0007 ( 0.0590 ) -0.0002( 0.0583) 0.15 -0.0002( 0.0552 ) 0.0643 ( 0.0584 ) 0.0625 ( 0.0659 ) 0.0212 ( 0.0644 ) 0.0031 ( 0.0602 ) 0.0000 ( 0.0607 ) -0.0009( 0.0594) 0.2 -0.0006( 0.0555 ) 0.0884 ( 0.0593 ) 0.0908 ( 0.0697 ) 0.0321 ( 0.0680 ) 0.0054 ( 0.0628 ) 0.0002 ( 0.0616 ) -0.0004( 0.0623)
表十一：樣本來自 logistic 分配，使用核密度估計法估計參數，10000 個模擬μˆ0的平均值及標準差
β= 0.01的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 0.0000 ( 0.0020 ) 0.0003 ( 0.0020 ) 0.0003 ( 0.0020 ) 0.0006 ( 0.0022 ) 0.0019 ( 0.0029 ) 0.0012 ( 0.0037 ) 0.0013 ( 0.0030 ) 0.1 0.0000 ( 0.0020 ) 0.0007 ( 0.0020 ) 0.0011 ( 0.0020 ) 0.0020 ( 0.0022 ) 0.0039 ( 0.0028 ) 0.0038 ( 0.0037 ) 0.0035 ( 0.0030 ) 0.15 0.0000 ( 0.0020 ) 0.0012 ( 0.0020 ) 0.0020 ( 0.0020 ) 0.0032 ( 0.0022 ) 0.0055 ( 0.0025 ) 0.0069 ( 0.0035 ) 0.0063 ( 0.0035 ) 0.2 0.0000 ( 0.0020 ) 0.0016 ( 0.0020 ) 0.0030 ( 0.0020 ) 0.0045 ( 0.0021 ) 0.0069 ( 0.0023 ) 0.0095 ( 0.0029 ) 0.0103 ( 0.0037 ) β= 0.1的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 0.0000 ( 0.0085 ) 0.0045 ( 0.0085 ) 0.0084 ( 0.0087 ) 0.0120 ( 0.0088 ) 0.0146 ( 0.0087 ) 0.0159 ( 0.0089 ) 0.0161 ( 0.0090 ) 0.1 -0.0001( 0.0085 ) 0.0091 ( 0.0085 ) 0.0173 ( 0.0087 ) 0.0244 ( 0.0089 ) 0.0300 ( 0.0089 ) 0.0332 ( 0.0091 ) 0.0343 ( 0.0094 ) 0.15 -0.0002( 0.0086 ) 0.0134 ( 0.0087 ) 0.0260 ( 0.0087 ) 0.0374 ( 0.0089 ) 0.0461 ( 0.0091 ) 0.0521 ( 0.0092 ) 0.0546 ( 0.0095 ) 0.2 0.0000 ( 0.0085 ) 0.0179 ( 0.0087 ) 0.0351 ( 0.0085 ) 0.0506 ( 0.0089 ) 0.0633 ( 0.0090 ) 0.0724 ( 0.0093 ) 0.0772 ( 0.0096 )
β= 0.5的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 -0.0007( 0.0447 ) 0.0180 ( 0.0453 ) 0.0174 ( 0.0471 ) 0.0077 ( 0.0470 ) 0.0009 ( 0.0465 ) -0.0005 ( 0.046 ) 0.0001 ( 0.0465 ) 0.1 -0.0003( 0.0453 ) 0.0381 ( 0.0458 ) 0.0380 ( 0.0502 ) 0.0168 ( 0.0498 ) 0.0040 ( 0.0483 ) 0.0001 ( 0.0474 ) 0.0000 ( 0.0467 ) 0.15 -0.0002( 0.0450 ) 0.0576 ( 0.0468 ) 0.0617 ( 0.0530 ) 0.0272 ( 0.0531 ) 0.0063 ( 0.0509 ) -0.0001( 0.0489) 0.0001 ( 0.0490 ) 0.2 0.0001 ( 0.0453 ) 0.0787 ( 0.0484 ) 0.0900 ( 0.0568 ) 0.0395 ( 0.0566 ) 0.0089 ( 0.0517 ) 0.0020 ( 0.0497 ) 0.0005 ( 0.0505 ) β= 1的μˆ0平均值(μˆ0標準差)：
δ
p 0 0.5 1 1.5 2 2.5 3
0.05 0.0002 ( 0.1331 ) 0.0266 ( 0.1396 ) 0.0173 ( 0.1405 ) 0.0036 ( 0.1383 ) 0.0000 ( 0.1379 ) -0.0024( 0.1377) 0.0004 ( 0.1386 ) 0.1 0.0023 ( 0.1341 ) 0.0610 ( 0.1447 ) 0.0307 ( 0.1495 ) 0.0019 ( 0.1442 ) 0.0001 ( 0.1425 ) 0.0002 ( 0.1413 ) 0.0020 ( 0.1419 ) 0.15 -0.0001( 0.1338 ) 0.0972 ( 0.1505 ) 0.0523 ( 0.1617 ) 0.0121 ( 0.1479 ) 0.0044 ( 0.1459 ) -0.0016( 0.1451) 0.0028 ( 0.1455 ) 0.2 -0.0020( 0.1319 ) 0.1371 ( 0.1563 ) 0.0780 ( 0.1759 ) 0.0145 ( 0.1548 ) 0.0000 ( 0.1519 ) 0.0004 ( 0.1519 ) 0.0009 ( 0.1504 )
黑色：直接估計法 藍色：重複篩選法 紅色：核密度估計法
黑色：直接估計法 藍色：重複篩選法 紅色：核密度估計法
圖二：對三種不同的估計方法，10000 個模擬μˆ0標準差之折線圖
黑色：直接估計法 藍色：重複篩選法 紅色：核密度估計法
黑色：直接估計法 藍色：重複篩選法 紅色：核密度估計法
圖四：對三種不同的估計方法，10000 個模擬α 標準差之折線圖
黑色：直接估計法 藍色：重複篩選法 紅色：核密度估計法
黑色：直接估計法 藍色：重複篩選法 紅色：核密度估計法
圖六：對三種不同的估計方法，10000 個模擬1−β標準差之折線圖
phase I delta=0, p=0.05; phase II delta=0
phase I delta=0, p=0.05; phase II delta=0.5
0
phase I delta=0, p=0.05; phase II delta=2
1.065
phase I delta=1, p=0.2; phase II delta=0
phase I delta=1, p=0.2; phase II delta=0.5
0
phase I delta=1, p=0.2; phase II delta=2
1
phase I delta=2, p=0.1; phase II delta=0
phase I delta=2, p=0.1; phase II delta=0.5
0
phase I delta=2, p=0.1; phase II delta=2
1