Cumulative Sum Chart Modeled under the Presence of Outliers
Abstract
1. Introduction
2. Overview of CUSUM Charts with Estimated Parameters
3. Variability in the CUSUM Chart Performance
3.1. Effect of Estimation on the Two-Sided CUSUM Chart Performance
3.2. Effect of Estimation on Two-Sided CUSUM Control Limits
4. The Outliers and CUSUM Chart with Estimated Parameters
5. Performance of the Tukey CUSUM Control Chart
5.1. Performance Comparison with Respect to m
5.2. Performance Comparison with Respect to α
5.3. Performance Comparison with Respect to w
6. Illustrative Example
7. Practical Application
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
CUSUM | Cumulative Sum |
ARL | Average Run Length |
SDRL | Standard Deviation of Run Length |
RSS | Received Signal Strength |
RL | Run Length |
AR | Activity Recognition |
AReM | Assembly Reliability Evaluation Method |
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Shewhart Chart | CUSUM | Robust CUSUM | Current Study | |
---|---|---|---|---|
Year | 1931 | 1954 | 2013 | 2020 |
Author(s) | Shewhart W.A. | Page E.S. | Nazir et al. | Abbas et al. |
Parameter of Interest | Process Mean | Process Mean | Process Mean | Process Mean |
Plotting Statistic | Sample Mean | Cumulative sum of sample mean | Cumulative sum of robust estimators like sample median etc. | Cumulative sum of sample mean |
Advantages |
|
|
|
|
Disadvantages |
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|
k | h | RL Properties | δ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 0.25 | 0.50 | 0.75 | 1.00 | 1.50 | 2.00 | 3.00 | 4.00 | 5.00 | |||
0.25 | 6.8516 | ARL | 200.000 | 63.588 | 24.229 | 14.076 | 9.862 | 6.195 | 4.558 | 3.060 | 2.328 | 2.013 |
SDRL | 187.82 | 51.58 | 14.73 | 6.66 | 3.90 | 1.92 | 1.20 | 0.66 | 0.48 | 0.22 | ||
0.50 | 4.1713 | ARL | 199.997 | 83.100 | 28.438 | 13.921 | 8.724 | 4.918 | 3.456 | 2.259 | 1.772 | 1.372 |
SDRL | 195.13 | 77.61 | 23.27 | 9.35 | 4.84 | 2.04 | 1.19 | 0.59 | 0.47 | 0.26 | ||
0.75 | 2.9332 | ARL | 200.008 | 102.897 | 37.458 | 16.792 | 9.446 | 4.641 | 3.063 | 1.898 | 1.381 | 1.094 |
SDRL | 197.68 | 99.68 | 34.29 | 13.74 | 6.68 | 2.43 | 1.29 | 0.64 | 0.5 | 0.29 | ||
1.00 | 2.2137 | ARL | 200.006 | 119.968 | 48.841 | 21.705 | 11.406 | 4.864 | 2.956 | 1.695 | 1.219 | 1.037 |
SDRL | 198.41 | 117.92 | 46.83 | 19.58 | 9.35 | 3.10 | 1.50 | 0.68 | 0.42 | 0.19 |
m | k | h | RL Properties | δ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 0.25 | 0.50 | 0.75 | 1.00 | 1.50 | 2.00 | 3.00 | 4.00 | 5.00 | ||||
10 | 0.25 | 7.609 | ARL | 199.383 | 104.343 | 33.241 | 16.709 | 11.229 | 6.884 | 5.019 | 3.343 | 2.551 | 2.093 |
SDRL | 215.236 | 139.682 | 37.559 | 10.384 | 5.039 | 2.223 | 1.331 | 0.700 | 0.538 | 0.299 | |||
0.50 | 4.447 | ARL | 200.059 | 117.118 | 39.461 | 16.697 | 9.759 | 5.280 | 3.668 | 2.370 | 1.865 | 1.479 | |
SDRL | 211.211 | 147.580 | 51.001 | 15.205 | 6.407 | 2.326 | 1.287 | 0.622 | 0.441 | 0.501 | |||
0.75 | 3.070 | ARL | 199.955 | 129.525 | 49.912 | 20.360 | 10.664 | 4.935 | 3.205 | 1.971 | 1.438 | 1.122 | |
SDRL | 206.777 | 152.891 | 63.945 | 21.998 | 9.066 | 2.794 | 1.395 | 0.654 | 0.513 | 0.327 | |||
1.00 | 2.290 | ARL | 199.845 | 139.734 | 61.592 | 26.110 | 12.972 | 5.174 | 3.073 | 1.742 | 1.247 | 1.045 | |
SDRL | 204.832 | 158.023 | 75.770 | 29.932 | 12.769 | 3.601 | 1.628 | 0.699 | 0.442 | 0.207 | |||
50 | 0.25 | 7.071 | ARL | 199.989 | 72.686 | 25.880 | 14.694 | 10.214 | 6.387 | 4.689 | 3.141 | 2.389 | 2.033 |
SDRL | 191.101 | 70.530 | 17.200 | 7.228 | 4.108 | 1.985 | 1.228 | 0.669 | 0.505 | 0.230 | |||
0.50 | 4.246 | ARL | 200.328 | 91.515 | 30.404 | 14.476 | 8.958 | 5.010 | 3.510 | 2.289 | 1.799 | 1.401 | |
SDRL | 196.721 | 94.950 | 27.139 | 10.218 | 5.097 | 2.101 | 1.208 | 0.595 | 0.462 | 0.491 | |||
0.75 | 2.968 | ARL | 200.300 | 109.736 | 39.790 | 17.480 | 9.687 | 4.709 | 3.097 | 1.916 | 1.396 | 1.100 | |
SDRL | 198.429 | 113.866 | 39.054 | 15.037 | 7.057 | 2.504 | 1.309 | 0.643 | 0.502 | 0.300 | |||
1.00 | 2.233 | ARL | 200.129 | 125.358 | 51.433 | 22.548 | 11.722 | 4.928 | 2.983 | 1.706 | 1.226 | 1.039 | |
SDRL | 199.137 | 128.594 | 52.200 | 21.314 | 9.944 | 3.184 | 1.526 | 0.683 | 0.427 | 0.193 | |||
100 | 0.25 | 6.970 | ARL | 199.946 | 68.273 | 25.057 | 14.395 | 10.048 | 6.297 | 4.627 | 3.104 | 2.360 | 2.024 |
SDRL | 188.664 | 60.973 | 15.890 | 6.949 | 4.005 | 1.950 | 1.212 | 0.665 | 0.496 | 0.223 | |||
0.50 | 4.212 | ARL | 200.430 | 87.512 | 29.501 | 14.200 | 8.846 | 4.968 | 3.487 | 2.275 | 1.786 | 1.388 | |
SDRL | 196.061 | 86.556 | 25.151 | 9.773 | 4.973 | 2.072 | 1.200 | 0.593 | 0.467 | 0.488 | |||
0.75 | 2.952 | ARL | 200.415 | 106.371 | 38.649 | 17.134 | 9.561 | 4.677 | 3.082 | 1.907 | 1.389 | 1.097 | |
SDRL | 197.742 | 107.160 | 36.670 | 14.361 | 6.835 | 2.470 | 1.300 | 0.641 | 0.500 | 0.297 | |||
1.00 | 2.224 | ARL | 200.613 | 122.849 | 50.144 | 22.134 | 11.580 | 4.902 | 2.970 | 1.701 | 1.223 | 1.038 | |
SDRL | 199.037 | 123.517 | 49.436 | 20.463 | 9.676 | 3.147 | 1.517 | 0.681 | 0.425 | 0.192 | |||
500 | 0.25 | 6.878 | ARL | 200.257 | 64.528 | 24.400 | 14.146 | 9.902 | 6.219 | 4.575 | 3.069 | 2.334 | 2.015 |
SDRL | 187.975 | 53.602 | 14.933 | 6.715 | 3.917 | 1.927 | 1.201 | 0.662 | 0.486 | 0.220 | |||
0.50 | 4.180 | ARL | 200.348 | 83.934 | 28.700 | 13.981 | 8.748 | 4.925 | 3.464 | 2.262 | 1.775 | 1.374 | |
SDRL | 195.114 | 79.434 | 23.596 | 9.440 | 4.861 | 2.047 | 1.190 | 0.589 | 0.470 | 0.485 | |||
0.75 | 2.938 | ARL | 200.330 | 103.981 | 37.728 | 16.880 | 9.480 | 4.650 | 3.067 | 1.900 | 1.383 | 1.095 | |
SDRL | 197.702 | 101.518 | 34.763 | 13.883 | 6.700 | 2.445 | 1.293 | 0.640 | 0.498 | 0.294 | |||
1.00 | 2.215 | ARL | 199.622 | 120.377 | 49.084 | 21.814 | 11.427 | 4.867 | 2.957 | 1.695 | 1.221 | 1.037 | |
SDRL | 197.769 | 119.228 | 47.308 | 19.817 | 9.394 | 3.105 | 1.503 | 0.678 | 0.423 | 0.189 | |||
1000 | 0.25 | 6.870 | ARL | 200.435 | 64.137 | 24.327 | 14.114 | 9.893 | 6.214 | 4.567 | 3.065 | 2.332 | 2.014 |
SDRL | 187.977 | 52.723 | 14.812 | 6.680 | 3.920 | 1.921 | 1.197 | 0.661 | 0.485 | 0.220 | |||
0.50 | 4.174 | ARL | 199.921 | 83.373 | 28.488 | 13.939 | 8.729 | 4.923 | 3.460 | 2.260 | 1.773 | 1.372 | |
SDRL | 195.128 | 78.425 | 23.332 | 9.391 | 4.837 | 2.049 | 1.189 | 0.589 | 0.471 | 0.484 | |||
0.75 | 2.936 | ARL | 200.261 | 103.481 | 37.557 | 16.832 | 9.456 | 4.644 | 3.066 | 1.900 | 1.382 | 1.095 | |
SDRL | 197.588 | 100.857 | 34.560 | 13.811 | 6.670 | 2.434 | 1.292 | 0.641 | 0.498 | 0.293 | |||
1.00 | 2.215 | ARL | 200.364 | 120.208 | 48.942 | 21.746 | 11.422 | 4.870 | 2.959 | 1.695 | 1.220 | 1.038 | |
SDRL | 198.748 | 118.622 | 46.945 | 19.748 | 9.394 | 3.107 | 1.505 | 0.679 | 0.423 | 0.190 |
m | k | h | RL Properties | δ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 0.25 | 0.50 | 0.75 | 1.00 | 1.50 | 2.00 | 3.00 | 4.00 | 5.00 | ||||
10 | 0.25 | 6.363 | ARL | 200.405 | 60.102 | 22.762 | 13.209 | 9.262 | 5.821 | 4.284 | 2.881 | 2.228 | 1.928 |
SDRL | 307.837 | 58.830 | 15.242 | 6.830 | 4.001 | 1.990 | 1.258 | 0.712 | 0.473 | 0.347 | |||
0.50 | 3.769 | ARL | 200.186 | 79.654 | 26.646 | 12.915 | 8.080 | 4.554 | 3.205 | 2.103 | 1.611 | 1.256 | |
SDRL | 373.977 | 116.713 | 27.155 | 9.947 | 5.004 | 2.089 | 1.210 | 0.626 | 0.521 | 0.437 | |||
0.75 | 2.610 | ARL | 200.504 | 101.344 | 35.910 | 15.787 | 8.789 | 4.289 | 2.826 | 1.741 | 1.279 | 1.063 | |
SDRL | 404.643 | 185.346 | 50.475 | 16.502 | 7.323 | 2.518 | 1.321 | 0.662 | 0.458 | 0.244 | |||
1.00 | 1.947 | ARL | 199.994 | 119.320 | 47.933 | 20.909 | 10.796 | 4.528 | 2.734 | 1.568 | 1.162 | 1.026 | |
SDRL | 419.821 | 241.979 | 82.668 | 27.810 | 11.547 | 3.348 | 1.557 | 0.671 | 0.377 | 0.159 | |||
50 | 0.25 | 6.750 | ARL | 200.033 | 62.844 | 23.946 | 13.890 | 9.733 | 6.116 | 4.504 | 3.021 | 2.307 | 2.000 |
SDRL | 208.518 | 53.158 | 14.855 | 6.701 | 3.919 | 1.933 | 1.212 | 0.675 | 0.478 | 0.245 | |||
0.50 | 4.087 | ARL | 200.596 | 82.357 | 28.048 | 13.731 | 8.599 | 4.841 | 3.402 | 2.228 | 1.737 | 1.344 | |
SDRL | 229.818 | 84.129 | 23.969 | 9.497 | 4.876 | 2.054 | 1.191 | 0.596 | 0.486 | 0.476 | |||
0.75 | 2.863 | ARL | 200.545 | 102.188 | 37.047 | 16.574 | 9.301 | 4.560 | 3.012 | 1.863 | 1.358 | 1.086 | |
SDRL | 240.560 | 113.083 | 36.671 | 14.217 | 6.771 | 2.453 | 1.297 | 0.646 | 0.491 | 0.281 | |||
1.00 | 2.155 | ARL | 200.400 | 119.428 | 48.460 | 21.504 | 11.254 | 4.786 | 2.907 | 1.666 | 1.206 | 1.034 | |
SDRL | 246.062 | 139.068 | 51.874 | 20.928 | 9.711 | 3.141 | 1.514 | 0.678 | 0.413 | 0.182 | |||
100 | 0.25 | 6.794 | ARL | 199.294 | 63.155 | 24.052 | 13.966 | 9.786 | 6.154 | 4.526 | 3.037 | 2.316 | 2.006 |
SDRL | 197.077 | 52.484 | 14.746 | 6.675 | 3.906 | 1.927 | 1.206 | 0.668 | 0.480 | 0.232 | |||
0.50 | 4.131 | ARL | 200.578 | 82.766 | 28.225 | 13.823 | 8.652 | 4.884 | 3.431 | 2.244 | 1.755 | 1.359 | |
SDRL | 212.753 | 80.763 | 23.573 | 9.417 | 4.841 | 2.051 | 1.189 | 0.593 | 0.479 | 0.480 | |||
0.75 | 2.898 | ARL | 200.473 | 102.694 | 37.300 | 16.685 | 9.375 | 4.607 | 3.037 | 1.880 | 1.370 | 1.090 | |
SDRL | 218.562 | 106.845 | 35.484 | 13.983 | 6.724 | 2.450 | 1.294 | 0.643 | 0.494 | 0.287 | |||
1.00 | 2.183 | ARL | 199.677 | 119.568 | 48.559 | 21.547 | 11.325 | 4.824 | 2.929 | 1.680 | 1.212 | 1.035 | |
SDRL | 220.710 | 127.965 | 49.291 | 20.189 | 9.534 | 3.126 | 1.506 | 0.678 | 0.417 | 0.185 | |||
500 | 0.25 | 6.848 | ARL | 200.572 | 63.552 | 24.226 | 14.080 | 9.862 | 6.193 | 4.557 | 3.059 | 2.328 | 2.013 |
SDRL | 190.749 | 51.951 | 14.746 | 6.675 | 3.904 | 1.923 | 1.200 | 0.663 | 0.483 | 0.222 | |||
0.50 | 4.165 | ARL | 200.223 | 83.177 | 28.415 | 13.913 | 8.707 | 4.914 | 3.453 | 2.257 | 1.769 | 1.369 | |
SDRL | 198.681 | 78.455 | 23.301 | 9.377 | 4.831 | 2.049 | 1.189 | 0.590 | 0.473 | 0.483 | |||
0.75 | 2.928 | ARL | 200.675 | 103.041 | 37.404 | 16.783 | 9.438 | 4.635 | 3.060 | 1.894 | 1.380 | 1.093 | |
SDRL | 201.922 | 101.276 | 34.557 | 13.773 | 6.672 | 2.436 | 1.293 | 0.641 | 0.497 | 0.291 | |||
1.00 | 2.209 | ARL | 200.376 | 120.019 | 48.878 | 21.737 | 11.392 | 4.860 | 2.951 | 1.693 | 1.219 | 1.036 | |
SDRL | 203.135 | 120.335 | 47.389 | 19.798 | 9.403 | 3.108 | 1.500 | 0.678 | 0.421 | 0.187 | |||
1000 | 0.25 | 6.851 | ARL | 200.377 | 63.648 | 24.239 | 14.082 | 9.862 | 6.196 | 4.558 | 3.059 | 2.328 | 2.013 |
SDRL | 189.279 | 52.107 | 14.755 | 6.672 | 3.897 | 1.920 | 1.200 | 0.662 | 0.484 | 0.221 | |||
0.50 | 4.171 | ARL | 201.035 | 83.254 | 28.515 | 13.940 | 8.726 | 4.918 | 3.455 | 2.258 | 1.772 | 1.372 | |
SDRL | 197.398 | 78.294 | 23.316 | 9.384 | 4.844 | 2.045 | 1.186 | 0.589 | 0.471 | 0.484 | |||
0.75 | 2.933 | ARL | 200.892 | 103.327 | 37.531 | 16.820 | 9.435 | 4.644 | 3.063 | 1.898 | 1.381 | 1.094 | |
SDRL | 200.377 | 100.860 | 34.491 | 13.805 | 6.655 | 2.440 | 1.291 | 0.640 | 0.497 | 0.292 | |||
1.00 | 2.210 | ARL | 199.787 | 120.041 | 48.756 | 21.688 | 11.387 | 4.859 | 2.956 | 1.694 | 1.218 | 1.037 | |
SDRL | 199.806 | 119.231 | 47.036 | 19.668 | 9.374 | 3.104 | 1.506 | 0.678 | 0.421 | 0.189 |
m | k | h | RL Properties | δ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 0.25 | 0.50 | 0.75 | 1.00 | 1.50 | 2.00 | 3.00 | 4.00 | 5.00 | ||||
10 | 0.25 | 7.059 | ARL | 199.839 | 102.365 | 31.358 | 15.661 | 10.522 | 6.458 | 4.716 | 3.143 | 2.408 | 2.036 |
SDRL | 343.854 | 200.559 | 42.962 | 10.656 | 5.112 | 2.294 | 1.399 | 0.762 | 0.538 | 0.334 | |||
0.50 | 4.010 | ARL | 200.741 | 115.384 | 37.440 | 15.548 | 9.038 | 4.874 | 3.392 | 2.204 | 1.700 | 1.333 | |
SDRL | 403.561 | 253.069 | 72.922 | 17.311 | 6.674 | 2.360 | 1.308 | 0.649 | 0.515 | 0.473 | |||
0.75 | 2.719 | ARL | 199.462 | 128.278 | 48.250 | 19.298 | 9.906 | 4.551 | 2.945 | 1.801 | 1.319 | 1.080 | |
SDRL | 428.983 | 293.013 | 104.853 | 28.936 | 10.291 | 2.891 | 1.420 | 0.681 | 0.480 | 0.272 | |||
1.00 | 2.010 | ARL | 200.078 | 139.638 | 60.751 | 25.381 | 12.370 | 4.810 | 2.837 | 1.607 | 1.181 | 1.031 | |
SDRL | 444.178 | 321.987 | 137.963 | 46.283 | 17.357 | 3.932 | 1.684 | 0.694 | 0.396 | 0.174 | |||
50 | 0.25 | 6.917 | ARL | 199.699 | 67.771 | 24.906 | 14.312 | 9.989 | 6.255 | 4.601 | 3.085 | 2.348 | 2.019 |
SDRL | 198.389 | 61.666 | 15.955 | 6.957 | 4.018 | 1.962 | 1.222 | 0.671 | 0.493 | 0.233 | |||
0.50 | 4.169 | ARL | 200.336 | 87.245 | 29.240 | 14.097 | 8.774 | 4.924 | 3.462 | 2.259 | 1.769 | 1.373 | |
SDRL | 212.799 | 90.310 | 25.530 | 9.843 | 4.987 | 2.078 | 1.203 | 0.595 | 0.475 | 0.484 | |||
0.75 | 2.913 | ARL | 199.648 | 105.684 | 38.333 | 16.994 | 9.493 | 4.631 | 3.053 | 1.889 | 1.376 | 1.093 | |
SDRL | 217.876 | 113.732 | 37.851 | 14.602 | 6.933 | 2.478 | 1.304 | 0.646 | 0.496 | 0.291 | |||
1.00 | 2.193 | ARL | 199.956 | 122.288 | 49.919 | 22.003 | 11.479 | 4.851 | 2.946 | 1.685 | 1.215 | 1.037 | |
SDRL | 221.744 | 133.825 | 52.081 | 21.110 | 9.823 | 3.163 | 1.520 | 0.680 | 0.420 | 0.188 | |||
100 | 0.25 | 6.917 | ARL | 199.699 | 67.771 | 24.906 | 14.312 | 9.989 | 6.255 | 4.601 | 3.085 | 2.348 | 2.019 |
SDRL | 198.389 | 61.666 | 15.955 | 6.957 | 4.018 | 1.962 | 1.222 | 0.671 | 0.493 | 0.233 | |||
0.50 | 4.169 | ARL | 200.336 | 87.245 | 29.240 | 14.097 | 8.774 | 4.924 | 3.462 | 2.259 | 1.769 | 1.373 | |
SDRL | 212.799 | 90.310 | 25.530 | 9.843 | 4.987 | 2.078 | 1.203 | 0.595 | 0.475 | 0.484 | |||
0.75 | 2.913 | ARL | 199.648 | 105.684 | 38.333 | 16.994 | 9.493 | 4.631 | 3.053 | 1.889 | 1.376 | 1.093 | |
SDRL | 217.876 | 113.732 | 37.851 | 14.602 | 6.933 | 2.478 | 1.304 | 0.646 | 0.496 | 0.291 | |||
1.00 | 2.193 | ARL | 199.956 | 122.288 | 49.919 | 22.003 | 11.479 | 4.851 | 2.946 | 1.685 | 1.215 | 1.037 | |
SDRL | 221.744 | 133.825 | 52.081 | 21.110 | 9.823 | 3.163 | 1.520 | 0.680 | 0.420 | 0.188 | |||
500 | 0.25 | 6.868 | ARL | 199.857 | 64.355 | 24.398 | 14.122 | 9.885 | 6.210 | 4.566 | 3.067 | 2.332 | 2.014 |
SDRL | 189.936 | 53.531 | 15.008 | 6.715 | 3.920 | 1.926 | 1.201 | 0.663 | 0.485 | 0.222 | |||
0.50 | 4.171 | ARL | 200.044 | 83.865 | 28.607 | 13.953 | 8.734 | 4.921 | 3.457 | 2.260 | 1.770 | 1.372 | |
SDRL | 198.638 | 80.168 | 23.684 | 9.432 | 4.865 | 2.049 | 1.190 | 0.589 | 0.472 | 0.484 | |||
0.75 | 2.930 | ARL | 200.139 | 103.405 | 37.595 | 16.856 | 9.462 | 4.637 | 3.063 | 1.896 | 1.380 | 1.094 | |
SDRL | 201.232 | 102.432 | 34.956 | 13.942 | 6.707 | 2.443 | 1.294 | 0.641 | 0.497 | 0.292 | |||
1.00 | 2.209 | ARL | 199.771 | 120.239 | 48.995 | 21.793 | 11.431 | 4.859 | 2.953 | 1.694 | 1.218 | 1.037 | |
SDRL | 202.617 | 120.900 | 47.734 | 19.947 | 9.460 | 3.105 | 1.505 | 0.679 | 0.421 | 0.189 | |||
1000 | 0.25 | 6.860 | ARL | 199.887 | 63.963 | 24.302 | 14.097 | 9.870 | 6.205 | 4.563 | 3.063 | 2.330 | 2.014 |
SDRL | 188.749 | 52.600 | 14.842 | 6.692 | 3.908 | 1.924 | 1.201 | 0.661 | 0.484 | 0.221 | |||
0.50 | 4.172 | ARL | 200.175 | 83.432 | 28.542 | 13.943 | 8.728 | 4.917 | 3.458 | 2.260 | 1.772 | 1.372 | |
SDRL | 196.712 | 78.731 | 23.482 | 9.400 | 4.849 | 2.046 | 1.190 | 0.589 | 0.472 | 0.484 | |||
0.75 | 2.933 | ARL | 200.433 | 103.187 | 37.606 | 16.838 | 9.465 | 4.642 | 3.063 | 1.898 | 1.382 | 1.094 | |
SDRL | 199.533 | 101.221 | 34.688 | 13.848 | 6.697 | 2.439 | 1.292 | 0.640 | 0.498 | 0.292 | |||
1.00 | 2.213 | ARL | 200.351 | 120.455 | 49.060 | 21.776 | 11.414 | 4.866 | 2.958 | 1.695 | 1.219 | 1.037 | |
SDRL | 201.344 | 120.017 | 47.496 | 19.808 | 9.407 | 3.109 | 1.505 | 0.679 | 0.422 | 0.189 |
µ | σ | m | k | |||||||
---|---|---|---|---|---|---|---|---|---|---|
0.25 | 0.50 | 0.75 | 1.00 | 1.25 | 1.50 | 1.75 | 2.00 | |||
known | known | - | 6.852 | 4.171 | 2.933 | 2.214 | 1.741 | 1.387 | 1.089 | 0.819 |
estimated | known | 10 | 7.609 | 4.447 | 3.070 | 2.290 | 1.789 | 1.423 | 1.120 | 0.845 |
20 | 7.306 | 4.331 | 3.010 | 2.257 | 1.768 | 1.407 | 1.106 | 0.832 | ||
30 | 7.199 | 4.290 | 2.986 | 2.244 | 1.760 | 1.400 | 1.100 | 0.829 | ||
50 | 7.071 | 4.246 | 2.968 | 2.233 | 1.753 | 1.395 | 1.096 | 0.826 | ||
100 | 6.970 | 4.212 | 2.952 | 2.224 | 1.747 | 1.390 | 1.092 | 0.822 | ||
500 | 6.878 | 4.180 | 2.938 | 2.215 | 1.742 | 1.388 | 1.090 | 0.820 | ||
1000 | 6.870 | 4.174 | 2.936 | 2.215 | 1.741 | 1.387 | 1.090 | 0.819 | ||
known | estimated | 10 | 6.363 | 3.769 | 2.610 | 1.947 | 1.518 | 1.195 | 0.916 | 0.650 |
20 | 6.598 | 3.957 | 2.758 | 2.070 | 1.620 | 1.281 | 0.994 | 0.729 | ||
30 | 6.689 | 4.030 | 2.816 | 2.117 | 1.660 | 1.315 | 1.025 | 0.758 | ||
50 | 6.750 | 4.087 | 2.863 | 2.155 | 1.690 | 1.344 | 1.050 | 0.781 | ||
100 | 6.794 | 4.131 | 2.898 | 2.183 | 1.716 | 1.365 | 1.070 | 0.799 | ||
500 | 6.848 | 4.165 | 2.928 | 2.209 | 1.736 | 1.382 | 1.086 | 0.814 | ||
1000 | 6.851 | 4.171 | 2.933 | 2.210 | 1.739 | 1.386 | 1.088 | 0.818 | ||
estimated | estimated | 10 | 7.059 | 4.010 | 2.719 | 2.010 | 1.557 | 1.223 | 0.938 | 0.672 |
20 | 7.051 | 4.110 | 2.827 | 2.107 | 1.645 | 1.299 | 1.009 | 0.743 | ||
30 | 7.019 | 4.140 | 2.866 | 2.144 | 1.675 | 1.328 | 1.035 | 0.768 | ||
50 | 6.973 | 4.159 | 2.896 | 2.171 | 1.701 | 1.352 | 1.057 | 0.788 | ||
100 | 6.917 | 4.169 | 2.913 | 2.193 | 1.721 | 1.370 | 1.074 | 0.804 | ||
500 | 6.868 | 4.171 | 2.930 | 2.209 | 1.737 | 1.384 | 1.086 | 0.815 | ||
1000 | 6.860 | 4.172 | 2.933 | 2.213 | 1.740 | 1.386 | 1.088 | 0.818 |
m | k | h | RL Properties | w = 1 | w = 2 | w = 3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
α | α | α | ||||||||||||||||
0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | ||||
10 | 0.25 | 7.609 | ARL | 199 | 198 | 196 | 194 | 192 | 199 | 195 | 190 | 182 | 178 | 199 | 192 | 183 | 173 | 163 |
SDRL | 215 | 214 | 213 | 212 | 212 | 215 | 213 | 211 | 206 | 205 | 215 | 213 | 208 | 204 | 198 | |||
0.50 | 4.447 | ARL | 200 | 199 | 198 | 196 | 195 | 200 | 195 | 191 | 187 | 180 | 200 | 194 | 185 | 177 | 167 | |
SDRL | 211 | 210 | 211 | 210 | 209 | 211 | 209 | 206 | 206 | 200 | 211 | 210 | 204 | 201 | 196 | |||
0.75 | 3.070 | ARL | 200 | 199 | 199 | 198 | 196 | 200 | 198 | 194 | 189 | 184 | 200 | 193 | 187 | 179 | 171 | |
SDRL | 207 | 207 | 207 | 206 | 207 | 207 | 206 | 204 | 203 | 198 | 207 | 203 | 202 | 198 | 193 | |||
1.00 | 2.290 | ARL | 200 | 199 | 198 | 197 | 196 | 200 | 197 | 195 | 189 | 186 | 200 | 195 | 189 | 182 | 176 | |
SDRL | 205 | 203 | 204 | 202 | 202 | 205 | 202 | 202 | 198 | 196 | 205 | 201 | 199 | 195 | 191 | |||
50 | 0.25 | 7.071 | ARL | 200 | 198 | 197 | 194 | 192 | 200 | 197 | 190 | 181 | 169 | 200 | 192 | 179 | 164 | 147 |
SDRL | 191 | 189 | 190 | 187 | 186 | 191 | 189 | 184 | 177 | 169 | 191 | 187 | 177 | 166 | 154 | |||
0.50 | 4.246 | ARL | 200 | 199 | 198 | 196 | 194 | 200 | 197 | 192 | 186 | 178 | 200 | 194 | 186 | 172 | 158 | |
SDRL | 197 | 195 | 195 | 194 | 192 | 197 | 195 | 191 | 186 | 179 | 197 | 193 | 185 | 176 | 165 | |||
0.75 | 2.968 | ARL | 200 | 199 | 199 | 198 | 195 | 200 | 198 | 195 | 189 | 184 | 200 | 196 | 188 | 179 | 167 | |
SDRL | 198 | 197 | 198 | 197 | 195 | 198 | 198 | 195 | 190 | 185 | 198 | 195 | 189 | 182 | 172 | |||
1.00 | 2.233 | ARL | 200 | 200 | 199 | 199 | 197 | 200 | 198 | 195 | 192 | 188 | 200 | 197 | 191 | 183 | 174 | |
SDRL | 199 | 199 | 198 | 198 | 196 | 199 | 197 | 195 | 191 | 190 | 199 | 196 | 192 | 186 | 177 | |||
100 | 0.25 | 6.97 | ARL | 200 | 199 | 197 | 195 | 191 | 200 | 196 | 189 | 180 | 168 | 200 | 193 | 178 | 161 | 141 |
SDRL | 189 | 188 | 186 | 185 | 180 | 189 | 187 | 179 | 172 | 163 | 189 | 183 | 172 | 159 | 142 | |||
0.50 | 4.212 | ARL | 200 | 200 | 200 | 196 | 195 | 200 | 198 | 193 | 187 | 177 | 200 | 196 | 185 | 173 | 156 | |
SDRL | 196 | 196 | 196 | 192 | 192 | 196 | 193 | 189 | 185 | 176 | 196 | 192 | 183 | 173 | 158 | |||
0.75 | 2.952 | ARL | 200 | 201 | 199 | 197 | 196 | 200 | 200 | 194 | 191 | 185 | 200 | 196 | 189 | 178 | 168 | |
SDRL | 198 | 199 | 197 | 195 | 195 | 198 | 196 | 192 | 189 | 184 | 198 | 195 | 189 | 178 | 168 | |||
1.00 | 2.224 | ARL | 201 | 199 | 201 | 199 | 197 | 201 | 199 | 196 | 194 | 188 | 201 | 199 | 192 | 185 | 173 | |
SDRL | 199 | 198 | 200 | 197 | 197 | 199 | 197 | 195 | 193 | 188 | 199 | 198 | 192 | 185 | 174 | |||
500 | 0.25 | 6.878 | ARL | 200 | 199 | 197 | 195 | 189 | 200 | 196 | 189 | 177 | 164 | 200 | 194 | 179 | 156 | 134 |
SDRL | 188 | 187 | 187 | 183 | 178 | 188 | 184 | 177 | 166 | 154 | 188 | 181 | 168 | 146 | 126 | |||
0.50 | 4.18 | ARL | 200 | 201 | 199 | 197 | 193 | 200 | 199 | 192 | 186 | 175 | 200 | 196 | 185 | 171 | 153 | |
SDRL | 195 | 195 | 194 | 192 | 189 | 195 | 193 | 188 | 181 | 172 | 195 | 191 | 181 | 167 | 151 | |||
0.75 | 2.938 | ARL | 200 | 201 | 199 | 197 | 195 | 200 | 200 | 195 | 190 | 183 | 200 | 198 | 191 | 180 | 166 | |
SDRL | 198 | 198 | 196 | 195 | 193 | 198 | 197 | 193 | 187 | 180 | 198 | 195 | 190 | 179 | 164 | |||
1.00 | 2.215 | ARL | 200 | 200 | 199 | 199 | 196 | 200 | 198 | 197 | 192 | 188 | 200 | 198 | 193 | 185 | 174 | |
SDRL | 198 | 199 | 197 | 197 | 193 | 198 | 196 | 196 | 190 | 185 | 198 | 196 | 191 | 182 | 173 | |||
1000 | 0.25 | 6.870 | ARL | 200 | 201 | 198 | 193 | 190 | 200 | 198 | 190 | 178 | 163 | 200 | 194 | 178 | 156 | 134 |
SDRL | 188 | 189 | 186 | 181 | 179 | 188 | 186 | 178 | 166 | 153 | 188 | 182 | 167 | 144 | 124 | |||
0.50 | 4.174 | ARL | 200 | 200 | 198 | 197 | 194 | 200 | 198 | 193 | 186 | 176 | 200 | 196 | 185 | 171 | 153 | |
SDRL | 195 | 195 | 193 | 191 | 189 | 195 | 193 | 188 | 181 | 171 | 195 | 190 | 180 | 167 | 149 | |||
0.75 | 2.936 | ARL | 200 | 199 | 199 | 197 | 196 | 200 | 200 | 197 | 191 | 183 | 200 | 198 | 191 | 179 | 166 | |
SDRL | 198 | 195 | 197 | 194 | 194 | 198 | 198 | 194 | 189 | 180 | 198 | 195 | 189 | 175 | 165 | |||
1.00 | 2.215 | ARL | 200 | 199 | 200 | 198 | 198 | 200 | 200 | 198 | 192 | 188 | 200 | 198 | 192 | 185 | 173 | |
SDRL | 199 | 199 | 199 | 196 | 197 | 199 | 198 | 195 | 190 | 186 | 199 | 196 | 191 | 183 | 172 |
m | k | h | RL Properties | w = 1 | w = 2 | w = 3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
α | α | α | ||||||||||||||||
0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | ||||
10 | 0.25 | 6.363 | ARL | 200 | 232 | 271 | 308 | 369 | 200 | 1252 | 2827 | 5180 | 7632 | 200 | 5766 | 13,029 | 22,028 | 33,346 |
SDRL | 308 | 720 | 1271 | 1442 | 2202 | 308 | 25,588 | 41,781 | 59,472 | 72,550 | 308 | 66,775 | 101,057 | 131,849 | 163,026 | |||
0.50 | 3.769 | ARL | 200 | 253 | 320 | 402 | 498 | 200 | 2374 | 5025 | 8677 | 13,606 | 200 | 8574 | 19,938 | 32,755 | 49,639 | |
SDRL | 374 | 1216 | 1839 | 2660 | 3384 | 374 | 39,571 | 59,416 | 79,131 | 100,612 | 374 | 82,482 | 127,093 | 162,747 | 200,013 | |||
0.75 | 2.610 | ARL | 201 | 266 | 374 | 471 | 569 | 201 | 2614 | 6115 | 10,306 | 15,478 | 201 | 10,440 | 23,698 | 39,039 | 56,370 | |
SDRL | 405 | 1397 | 2665 | 3349 | 3891 | 405 | 41,686 | 66,631 | 86,598 | 107,915 | 405 | 92,015 | 139,038 | 178,484 | 213,382 | |||
1.00 | 1.947 | ARL | 200 | 289 | 386 | 484 | 599 | 200 | 3152 | 6751 | 10,527 | 16,605 | 200 | 10,322 | 24,092 | 40,720 | 59,187 | |
SDRL | 420 | 1851 | 2745 | 3539 | 4090 | 420 | 46,733 | 69,185 | 87,750 | 111,809 | 420 | 91,131 | 140,118 | 181,993 | 218,900 | |||
50 | 0.25 | 6.750 | ARL | 200 | 213 | 229 | 245 | 262 | 200 | 249 | 313 | 399 | 522 | 200 | 320 | 598 | 1020 | 2048 |
SDRL | 209 | 225 | 249 | 271 | 298 | 209 | 304 | 455 | 892 | 1381 | 209 | 958 | 6844 | 9008 | 19,563 | |||
0.50 | 4.087 | ARL | 201 | 220 | 240 | 263 | 287 | 201 | 274 | 377 | 550 | 767 | 201 | 413 | 1043 | 2332 | 5481 | |
SDRL | 230 | 259 | 294 | 328 | 379 | 230 | 453 | 1093 | 4144 | 4185 | 230 | 2854 | 11,610 | 24,597 | 45,072 | |||
0.75 | 2.863 | ARL | 201 | 221 | 245 | 272 | 299 | 201 | 288 | 425 | 614 | 970 | 201 | 499 | 1284 | 3489 | 7734 | |
SDRL | 241 | 274 | 314 | 379 | 417 | 241 | 655 | 1509 | 2443 | 7384 | 241 | 5406 | 14,261 | 32,417 | 56,026 | |||
1.00 | 2.155 | ARL | 200 | 223 | 248 | 275 | 309 | 200 | 295 | 453 | 713 | 1113 | 200 | 517 | 1622 | 3965 | 9219 | |
SDRL | 246 | 285 | 340 | 383 | 450 | 246 | 628 | 3646 | 5086 | 8738 | 246 | 5297 | 19,603 | 35,460 | 63,990 | |||
100 | 0.25 | 6.794 | ARL | 199 | 212 | 226 | 242 | 257 | 199 | 240 | 294 | 360 | 440 | 199 | 284 | 416 | 621 | 984 |
SDRL | 197 | 210 | 229 | 246 | 266 | 197 | 250 | 329 | 427 | 573 | 197 | 334 | 636 | 1439 | 2903 | |||
0.50 | 4.131 | ARL | 201 | 218 | 237 | 258 | 281 | 201 | 258 | 335 | 437 | 582 | 201 | 333 | 555 | 1005 | 1883 | |
SDRL | 213 | 235 | 258 | 284 | 315 | 213 | 301 | 428 | 616 | 953 | 213 | 1159 | 1079 | 5010 | 9240 | |||
0.75 | 2.898 | ARL | 200 | 221 | 243 | 265 | 292 | 200 | 267 | 358 | 494 | 671 | 200 | 351 | 649 | 1290 | 2600 | |
SDRL | 219 | 244 | 276 | 302 | 338 | 219 | 330 | 495 | 799 | 1297 | 219 | 586 | 2578 | 7319 | 14,579 | |||
1.00 | 2.183 | ARL | 200 | 221 | 242 | 270 | 295 | 200 | 269 | 376 | 517 | 718 | 200 | 360 | 700 | 1417 | 3124 | |
SDRL | 221 | 249 | 277 | 321 | 349 | 221 | 340 | 555 | 948 | 1354 | 221 | 595 | 2335 | 6141 | 18,020 | |||
500 | 0.25 | 6.848 | ARL | 201 | 213 | 227 | 242 | 257 | 201 | 238 | 284 | 340 | 411 | 201 | 272 | 375 | 523 | 735 |
SDRL | 191 | 203 | 218 | 232 | 248 | 191 | 228 | 278 | 337 | 412 | 191 | 267 | 381 | 550 | 807 | |||
0.50 | 4.165 | ARL | 200 | 218 | 237 | 255 | 276 | 200 | 251 | 317 | 399 | 510 | 200 | 298 | 454 | 702 | 1098 | |
SDRL | 199 | 216 | 236 | 256 | 280 | 199 | 253 | 326 | 416 | 543 | 199 | 306 | 492 | 796 | 1317 | |||
0.75 | 2.928 | ARL | 201 | 218 | 240 | 261 | 286 | 201 | 257 | 333 | 435 | 565 | 201 | 315 | 499 | 814 | 1350 | |
SDRL | 202 | 221 | 243 | 265 | 292 | 202 | 264 | 347 | 463 | 613 | 202 | 330 | 554 | 963 | 1698 | |||
1.00 | 2.209 | ARL | 200 | 219 | 241 | 264 | 291 | 200 | 261 | 343 | 455 | 597 | 200 | 321 | 527 | 882 | 1509 | |
SDRL | 203 | 222 | 247 | 272 | 298 | 203 | 269 | 361 | 488 | 652 | 203 | 341 | 595 | 1052 | 1921 | |||
1000 | 0.25 | 6.851 | ARL | 200 | 214 | 227 | 240 | 256 | 200 | 237 | 284 | 339 | 405 | 200 | 269 | 368 | 514 | 717 |
SDRL | 189 | 203 | 216 | 228 | 244 | 189 | 227 | 273 | 330 | 401 | 189 | 260 | 362 | 517 | 740 | |||
0.50 | 4.171 | ARL | 201 | 219 | 235 | 254 | 277 | 201 | 251 | 313 | 396 | 501 | 201 | 295 | 445 | 681 | 1058 | |
SDRL | 197 | 216 | 233 | 251 | 277 | 197 | 249 | 314 | 402 | 512 | 197 | 298 | 457 | 726 | 1138 | |||
0.75 | 2.933 | ARL | 201 | 220 | 240 | 262 | 285 | 201 | 258 | 332 | 432 | 558 | 201 | 311 | 490 | 784 | 1280 | |
SDRL | 200 | 220 | 240 | 263 | 288 | 200 | 259 | 338 | 442 | 578 | 200 | 318 | 516 | 840 | 1420 | |||
1.00 | 2.210 | ARL | 200 | 219 | 242 | 265 | 289 | 200 | 261 | 340 | 446 | 586 | 200 | 317 | 511 | 844 | 1408 | |
SDRL | 200 | 221 | 244 | 267 | 290 | 200 | 264 | 349 | 460 | 612 | 200 | 329 | 542 | 911 | 1580 |
m | k | h | RL Properties | w = 1 | w = 2 | w = 3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
α | α | α | ||||||||||||||||
0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | ||||
10 | 0.25 | 7.059 | ARL | 200 | 220 | 238 | 257 | 264 | 200 | 289 | 420 | 506 | 522 | 200 | 415 | 667 | 834 | 1018 |
SDRL | 344 | 579 | 802 | 1325 | 880 | 344 | 2389 | 8400 | 8836 | 7702 | 344 | 6970 | 12,597 | 13,937 | 15,538 | |||
0.50 | 4.010 | ARL | 201 | 244 | 315 | 333 | 390 | 201 | 691 | 1221 | 2017 | 2556 | 201 | 2465 | 4835 | 7722 | 10,719 | |
SDRL | 404 | 1375 | 5270 | 4092 | 5614 | 404 | 15,507 | 21,367 | 30,537 | 34,157 | 404 | 38,100 | 54,564 | 70,640 | 84,741 | |||
0.75 | 2.719 | ARL | 199 | 291 | 340 | 409 | 497 | 199 | 1402 | 2839 | 4258 | 6234 | 199 | 4856 | 10,452 | 18,230 | 25,507 | |
SDRL | 429 | 5091 | 5290 | 5868 | 8207 | 429 | 27,821 | 40,327 | 50,689 | 62,254 | 429 | 58,258 | 87,142 | 116,950 | 139,084 | |||
1.00 | 2.010 | ARL | 200 | 280 | 396 | 468 | 642 | 200 | 1919 | 3740 | 6103 | 8942 | 200 | 7038 | 15,601 | 24,673 | 35,470 | |
SDRL | 444 | 2471 | 7494 | 7088 | 11,812 | 444 | 33,960 | 48,185 | 62,320 | 77,490 | 444 | 72,692 | 109,882 | 138,233 | 165,823 | |||
50 | 0.25 | 6.917 | ARL | 201 | 212 | 227 | 233 | 242 | 201 | 238 | 268 | 297 | 319 | 201 | 265 | 323 | 366 | 391 |
SDRL | 212 | 228 | 250 | 259 | 270 | 212 | 276 | 331 | 381 | 429 | 212 | 355 | 474 | 560 | 616 | |||
0.50 | 4.169 | ARL | 201 | 218 | 236 | 253 | 271 | 201 | 260 | 326 | 405 | 482 | 201 | 333 | 495 | 715 | 977 | |
SDRL | 231 | 259 | 290 | 312 | 345 | 231 | 371 | 528 | 684 | 894 | 231 | 678 | 1371 | 2239 | 3977 | |||
0.75 | 2.913 | ARL | 200 | 222 | 242 | 265 | 289 | 200 | 276 | 372 | 497 | 642 | 200 | 393 | 719 | 1281 | 2057 | |
SDRL | 240 | 282 | 313 | 352 | 392 | 240 | 454 | 738 | 1374 | 1769 | 240 | 1885 | 4811 | 9559 | 14,183 | |||
1.00 | 2.193 | ARL | 199 | 221 | 247 | 273 | 297 | 199 | 282 | 402 | 554 | 770 | 199 | 435 | 925 | 1826 | 3613 | |
SDRL | 244 | 281 | 333 | 382 | 419 | 244 | 488 | 928 | 1704 | 3350 | 244 | 1775 | 7761 | 14,799 | 28,661 | |||
100 | 0.25 | 6.917 | ARL | 200 | 211 | 223 | 232 | 241 | 200 | 234 | 266 | 291 | 310 | 200 | 262 | 313 | 350 | 368 |
SDRL | 198 | 210 | 225 | 237 | 248 | 198 | 242 | 285 | 318 | 345 | 198 | 287 | 356 | 409 | 440 | |||
0.50 | 4.169 | ARL | 200 | 217 | 234 | 252 | 268 | 200 | 252 | 311 | 375 | 441 | 200 | 302 | 434 | 591 | 767 | |
SDRL | 213 | 231 | 256 | 277 | 299 | 213 | 285 | 378 | 470 | 570 | 213 | 388 | 606 | 906 | 1233 | |||
0.75 | 2.913 | ARL | 200 | 219 | 238 | 261 | 280 | 200 | 260 | 337 | 430 | 546 | 200 | 330 | 526 | 816 | 1212 | |
SDRL | 218 | 243 | 266 | 298 | 325 | 218 | 311 | 434 | 584 | 811 | 218 | 472 | 895 | 1682 | 2721 | |||
1.00 | 2.193 | ARL | 200 | 222 | 242 | 268 | 290 | 200 | 269 | 358 | 474 | 619 | 200 | 347 | 596 | 1016 | 1662 | |
SDRL | 222 | 252 | 278 | 311 | 339 | 222 | 336 | 489 | 703 | 994 | 222 | 586 | 1330 | 2710 | 4963 | |||
500 | 0.25 | 6.868 | ARL | 200 | 210 | 222 | 232 | 240 | 200 | 233 | 263 | 286 | 302 | 200 | 260 | 311 | 340 | 352 |
SDRL | 190 | 201 | 212 | 222 | 231 | 190 | 224 | 255 | 280 | 295 | 190 | 253 | 306 | 341 | 352 | |||
0.50 | 4.171 | ARL | 200 | 215 | 232 | 248 | 264 | 200 | 247 | 302 | 359 | 414 | 200 | 288 | 402 | 528 | 662 | |
SDRL | 199 | 216 | 231 | 247 | 265 | 199 | 248 | 307 | 369 | 429 | 199 | 297 | 420 | 561 | 712 | |||
0.75 | 2.93 | ARL | 200 | 218 | 237 | 258 | 277 | 200 | 256 | 323 | 403 | 496 | 200 | 307 | 459 | 670 | 947 | |
SDRL | 201 | 220 | 240 | 264 | 281 | 201 | 262 | 336 | 424 | 530 | 201 | 320 | 499 | 747 | 1071 | |||
1.00 | 2.209 | ARL | 200 | 219 | 240 | 262 | 285 | 200 | 260 | 335 | 430 | 545 | 200 | 317 | 495 | 762 | 1164 | |
SDRL | 203 | 224 | 244 | 269 | 296 | 203 | 269 | 350 | 456 | 590 | 203 | 335 | 545 | 865 | 1382 | |||
1000 | 0.25 | 6.860 | ARL | 200 | 212 | 223 | 232 | 239 | 200 | 234 | 264 | 287 | 302 | 200 | 258 | 310 | 339 | 346 |
SDRL | 189 | 200 | 211 | 222 | 228 | 189 | 224 | 254 | 275 | 293 | 189 | 248 | 302 | 329 | 336 | |||
0.50 | 4.172 | ARL | 200 | 215 | 232 | 248 | 263 | 200 | 247 | 300 | 357 | 416 | 200 | 289 | 399 | 523 | 653 | |
SDRL | 197 | 213 | 230 | 245 | 261 | 197 | 244 | 301 | 359 | 418 | 197 | 289 | 406 | 534 | 675 | |||
0.75 | 2.933 | ARL | 200 | 218 | 238 | 257 | 277 | 200 | 256 | 322 | 398 | 491 | 200 | 304 | 453 | 659 | 927 | |
SDRL | 200 | 219 | 239 | 258 | 278 | 200 | 257 | 326 | 409 | 507 | 200 | 309 | 466 | 690 | 986 | |||
1.00 | 2.213 | ARL | 200 | 218 | 240 | 262 | 285 | 200 | 259 | 333 | 425 | 542 | 200 | 313 | 489 | 752 | 1124 | |
SDRL | 201 | 218 | 240 | 265 | 289 | 201 | 264 | 338 | 436 | 562 | 201 | 320 | 511 | 801 | 1219 |
1 | 2 | 3 | |||||
---|---|---|---|---|---|---|---|
0 | 0 | −0.00007 | 0 | 0.0001 | 0 | 0.00006 | |
1 | 0.97097 | 1 | 0.97101 | 1 | 0.971 | ||
0.01 | 0.01 | 0.0051 | 0.02 | 0.00564 | 0.03 | 0.00539 | |
1.0299 | 0.97727 | 1.1196 | 0.97881 | 1.2691 | 0.97878 | ||
0.02 | 0.02 | 0.01064 | 0.04 | 0.01127 | 0.06 | 0.01083 | |
1.0596 | 0.98373 | 1.2384 | 0.9866 | 1.5364 | 0.98683 | ||
0.03 | 0.03 | 0.01624 | 0.06 | 0.01713 | 0.09 | 0.01664 | |
1.0891 | 0.99016 | 1.3564 | 0.99458 | 1.8019 | 0.99498 | ||
0.04 | 0.04 | 0.02158 | 0.08 | 0.02323 | 0.12 | 0.0223 | |
1.1184 | 0.99666 | 1.4736 | 1.00311 | 2.0656 | 1.00392 | ||
0.05 | 0.05 | 0.027 | 0.1 | 0.02928 | 0.15 | 0.0284 | |
1.1475 | 1.00323 | 1.59 | 1.01156 | 2.3275 | 1.01339 | ||
0.06 | 0.06 | 0.03274 | 0.12 | 0.03568 | 0.18 | 0.03429 | |
1.1764 | 1.00988 | 1.7056 | 1.02088 | 2.5876 | 1.02238 | ||
0.07 | 0.07 | 0.03847 | 0.14 | 0.04203 | 0.21 | 0.04068 | |
1.2051 | 1.01709 | 1.8204 | 1.02994 | 2.8459 | 1.03251 | ||
0.08 | 0.08 | 0.04424 | 0.16 | 0.04859 | 0.24 | 0.04752 | |
1.2336 | 1.02381 | 1.9344 | 1.03942 | 3.1024 | 1.04268 | ||
0.09 | 0.09 | 0.05017 | 0.18 | 0.05536 | 0.27 | 0.0539 | |
1.2619 | 1.03107 | 2.0476 | 1.04928 | 3.3571 | 1.05381 | ||
0.1 | 0.1 | 0.05596 | 0.2 | 0.06209 | 0.3 | 0.06083 | |
1.29 | 1.03811 | 2.16 | 1.05968 | 3.61 | 1.06487 |
Phase-I Subgroups of RSS Values Chest-Left Ankle Mote | |||||||||||||||
No | RSS | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 17.50 | 21.50 | 17.75 | 20.50 | 14.25 | 18.300 | 2.847 | ||||||||
2 | 11.80 | 21.50 | 23.33 | 18.50 | 18.00 | 18.626 | 4.399 | ||||||||
3 | 18.00 | 17.50 | 15.33 | 16.25 | 18.75 | 17.166 | 1.372 | ||||||||
4 | 12.50 | 17.50 | 16.00 | 19.25 | 13.00 | 15.650 | 2.892 | ||||||||
5 | 22.00 | 20.25 | 16.25 | 15.33 | 17.75 | 18.316 | 2.776 | ||||||||
6 | 16.50 | 24.00 | 22.75 | 20.75 | 20.75 | 20.950 | 2.847 | ||||||||
7 | 24.50 | 13.25 | 18.25 | 23.00 | 19.00 | 19.600 | 4.418 | ||||||||
8 | 13.00 | 15.33 | 17.33 | 12.33 | 22.50 | 16.098 | 4.089 | ||||||||
9 | 20.50 | 14.50 | 16.75 | 17.25 | 13.33 | 16.466 | 2.769 | ||||||||
10 | 17.50 | 17.25 | 13.75 | 17.67 | 14.67 | 16.168 | 1.823 | ||||||||
11 | 14.00 | 13.00 | 19.00 | 16.00 | 11.25 | 14.650 | 2.977 | ||||||||
12 | 18.00 | 17.00 | 17.50 | 24.00 | 16.00 | 18.500 | 3.162 | ||||||||
13 | 14.75 | 12.50 | 17.75 | 10.67 | 20.25 | 15.184 | 3.874 | ||||||||
14 | 18.00 | 11.75 | 12.00 | 19.75 | 12.75 | 14.850 | 3.744 | ||||||||
15 | 18.00 | 16.67 | 17.67 | 12.80 | 19.25 | 16.878 | 2.459 | ||||||||
16 | 19.75 | 24.00 | 15.67 | 21.33 | 19.67 | 20.084 | 3.027 | ||||||||
17 | 26.75 | 17.75 | 22.75 | 16.00 | 11.75 | 19.000 | 5.858 | ||||||||
18 | 11.25 | 19.67 | 23.75 | 15.75 | 17.00 | 17.484 | 4.641 | ||||||||
19 | 19.25 | 16.50 | 15.25 | 19.25 | 16.25 | 17.300 | 1.841 | ||||||||
20 | 6.00 | 13.50 | 15.00 | 10.25 | 18.00 | 12.550 | 4.604 | ||||||||
21 | 11.33 | 13.00 | 16.50 | 19.50 | 19.25 | 15.916 | 3.669 | ||||||||
22 | 10.00 | 13.00 | 19.50 | 19.00 | 18.50 | 16.000 | 4.257 | ||||||||
23 | 12.25 | 19.00 | 14.50 | 15.00 | 18.00 | 15.750 | 2.739 | ||||||||
24 | 19.75 | 16.00 | 12.50 | 16.75 | 17.50 | 16.500 | 2.640 | ||||||||
25 | 15.75 | 17.75 | 19.00 | 19.00 | 10.33 | 16.366 | 3.626 | ||||||||
26 | 19.00 | 24.00 | 25.75 | 17.33 | 19.75 | 21.166 | 3.552 | ||||||||
27 | 19.25 | 22.50 | 16.75 | 11.50 | 18.75 | 17.750 | 4.058 | ||||||||
28 | 10.67 | 16.50 | 17.33 | 16.33 | 13.25 | 14.816 | 2.788 | ||||||||
29 | 19.00 | 20.75 | 13.50 | 18.25 | 16.50 | 17.600 | 2.753 | ||||||||
30 | 5.50 | 17.50 | 16.00 | 9.75 | 11.50 | 12.050 | 4.843 | ||||||||
31 | 20.67 | 8.50 | 14.25 | 14.00 | 18.75 | 15.234 | 4.737 | ||||||||
32 | 18.75 | 16.25 | 14.00 | 21.00 | 22.33 | 18.466 | 3.402 | ||||||||
33 | 13.50 | 17.50 | 18.33 | 17.00 | 15.75 | 16.416 | 1.879 | ||||||||
34 | 20.50 | 16.75 | 17.00 | 21.00 | 19.50 | 18.950 | 1.972 | ||||||||
35 | 18.33 | 15.75 | 15.00 | 20.75 | 15.67 | 17.100 | 2.404 | ||||||||
36 | 16.00 | 20.25 | 24.50 | 15.00 | 16.67 | 18.484 | 3.902 | ||||||||
37 | 18.75 | 19.75 | 12.33 | 15.25 | 21.67 | 17.550 | 3.735 | ||||||||
38 | 22.25 | 13.75 | 20.75 | 21.00 | 16.75 | 18.900 | 3.543 | ||||||||
39 | 18.75 | 16.67 | 13.75 | 12.00 | 18.50 | 15.934 | 2.971 | ||||||||
40 | 12.25 | 14.75 | 11.00 | 17.25 | 8.33 | 12.716 | 3.431 | ||||||||
41 | 21.25 | 20.25 | 14.00 | 17.50 | 17.33 | 18.066 | 2.842 | ||||||||
42 | 21.75 | 15.75 | 15.00 | 19.25 | 16.33 | 17.616 | 2.817 | ||||||||
43 | 19.75 | 16.67 | 17.00 | 16.50 | 15.00 | 16.984 | 1.726 | ||||||||
44 | 18.67 | 21.33 | 18.25 | 16.67 | 17.75 | 18.534 | 1.732 | ||||||||
45 | 15.00 | 16.33 | 19.00 | 17.00 | 17.50 | 16.966 | 1.474 | ||||||||
46 | 14.00 | 19.00 | 17.75 | 26.75 | 15.00 | 18.500 | 5.034 | ||||||||
47 | 17.25 | 20.50 | 18.50 | 19.00 | 20.00 | 19.050 | 1.280 | ||||||||
48 | 18.00 | 10.50 | 11.67 | 16.67 | 18.00 | 14.968 | 3.610 | ||||||||
49 | 8.75 | 20.75 | 21.00 | 8.00 | 21.00 | 15.900 | 6.875 | ||||||||
50 | 16.00 | 12.00 | 18.25 | 12.00 | 14.75 | 14.600 | 2.684 | ||||||||
Phase-II Subgroups of RSS Values Chest-Left Ankle Mote. | |||||||||||||||
No | RSS | ||||||||||||||
1 | 11.50 | 13.50 | 12.00 | 11.50 | 12.00 | 12.100 | 0.000 | −4.096 | 6.41 | −6.41 | |||||
2 | 11.00 | 12.00 | 16.00 | 11.67 | 17.60 | 13.654 | 0.000 | −6.638 | 6.41 | −6.41 | |||||
3 | 12.75 | 15.50 | 6.50 | 10.33 | 18.00 | 12.616 | 0.000 | −10.218 | 6.41 | −6.41 | |||||
4 | 9.50 | 2.00 | 11.60 | 12.00 | 12.00 | 9.420 | 0.000 | −16.994 | 6.41 | −6.41 | |||||
5 | 12.00 | 6.00 | 12.50 | 12.50 | 13.33 | 11.266 | 0.000 | −21.924 | 6.41 | −6.41 | |||||
6 | 12.00 | 13.25 | 11.25 | 12.00 | 13.25 | 12.350 | 0.000 | −25.770 | 6.41 | −6.41 | |||||
7 | 16.33 | 11.75 | 18.00 | 9.75 | 15.00 | 14.166 | 0.000 | −27.800 | 6.41 | −6.41 | |||||
8 | 4.50 | 16.75 | 13.67 | 8.25 | 15.00 | 11.634 | 0.000 | −32.362 | 6.41 | −6.41 | |||||
9 | 12.25 | 4.00 | 11.75 | 6.50 | 12.00 | 9.300 | 0.000 | −39.258 | 6.41 | −6.41 | |||||
10 | 8.50 | 11.00 | 9.50 | 12.25 | 11.75 | 10.600 | 0.000 | −44.854 | 6.41 | −6.41 | |||||
11 | 12.00 | 12.25 | 11.00 | 12.00 | 13.25 | 12.100 | 0.000 | −48.950 | 6.41 | −6.41 | |||||
12 | 18.25 | 12.00 | 17.00 | 11.00 | 12.25 | 14.100 | 0.000 | −51.046 | 6.41 | −6.41 | |||||
13 | 15.50 | 14.00 | 14.00 | 9.00 | 8.75 | 12.250 | 0.000 | −54.992 | 6.41 | −6.41 | |||||
14 | 15.00 | 14.00 | 13.00 | 11.25 | 10.00 | 12.650 | 0.000 | −58.538 | 6.41 | −6.41 | |||||
15 | 4.33 | 12.50 | 14.00 | 12.00 | 8.00 | 10.166 | 0.000 | −64.568 | 6.41 | −6.41 | |||||
16 | 12.75 | 12.00 | 10.75 | 12.00 | 12.00 | 11.900 | 0.000 | −68.864 | 6.41 | −6.41 | |||||
17 | 18.75 | 9.75 | 13.00 | 12.00 | 16.50 | 14.000 | 0.000 | −71.060 | 6.41 | −6.41 | |||||
18 | 16.50 | 12.50 | 12.25 | 14.75 | 10.50 | 13.300 | 0.000 | −73.956 | 6.41 | −6.41 | |||||
19 | 13.75 | 13.67 | 12.00 | 5.00 | 3.50 | 9.584 | 0.000 | −80.568 | 6.41 | −6.41 | |||||
20 | 11.00 | 12.00 | 13.00 | 10.75 | 8.00 | 10.950 | 0.000 | −85.814 | 6.41 | −6.41 | |||||
21 | 13.25 | 12.00 | 11.00 | 12.00 | 11.00 | 11.850 | 0.000 | −90.160 | 6.41 | −6.41 | |||||
22 | 18.00 | 11.00 | 11.67 | 16.00 | 16.25 | 14.584 | 0.000 | −91.772 | 6.41 | −6.41 | |||||
23 | 13.75 | 11.00 | 4.67 | 17.50 | 11.50 | 11.684 | 0.000 | −96.284 | 6.41 | −6.41 | |||||
24 | 6.67 | 11.67 | 12.75 | 11.33 | 7.67 | 10.018 | 0.000 | −102.462 | 6.41 | −6.41 | |||||
25 | 11.50 | 12.33 | 6.67 | 10.00 | 2.67 | 8.634 | 0.000 | −110.024 | 6.41 | −6.41 |
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Abbas, N.; Abujiya, M.R.; Riaz, M.; Mahmood, T. Cumulative Sum Chart Modeled under the Presence of Outliers. Mathematics 2020, 8, 269. https://doi.org/10.3390/math8020269
Abbas N, Abujiya MR, Riaz M, Mahmood T. Cumulative Sum Chart Modeled under the Presence of Outliers. Mathematics. 2020; 8(2):269. https://doi.org/10.3390/math8020269
Chicago/Turabian StyleAbbas, Nasir, Mu’azu Ramat Abujiya, Muhammad Riaz, and Tahir Mahmood. 2020. "Cumulative Sum Chart Modeled under the Presence of Outliers" Mathematics 8, no. 2: 269. https://doi.org/10.3390/math8020269
APA StyleAbbas, N., Abujiya, M. R., Riaz, M., & Mahmood, T. (2020). Cumulative Sum Chart Modeled under the Presence of Outliers. Mathematics, 8(2), 269. https://doi.org/10.3390/math8020269