A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs
Abstract
:1. Introduction
2. The k-of-k DR Nonparametric Control Charts Based on Order Statistics and Runs
- Step 1.
- Draw a reference sample of size m from the in-control process.
- Step 2.
- Determine the control limits of the monitoring scheme by the aid of the corresponding reference ordered observations.
- Step 3.
- Draw independent test samples of size n from the process.
- Step 4.
- Pick up appropriately chosen ordered test observations as monitoring statistics.
- Step 5.
- Activate a k-of-k DR runs-type rule.
- Step 6.
- Declare whether the process is in- or out-of control, by combining the plotting statistics and the DR runs rule defined in the previous steps.
- ●
- Ref. [13] can be viewed as a chart
- ●
- Ref. [19] can be viewed as a chart
- ●
- Ref. [14] can be viewed as a chart
- ●
- Ref. [20] can be viewed as a chart.
3. Main Results for the Proposed k-of-k DR Nonparametric Control Charts
- (i)
- When the chart is applied, success probability p of the aforementioned geometric distribution of order coincides to the probability that the set of conditions stated in (1) is not satisfied. However, Ref. [13] proved that is determined by the aid of the following double sum
- (ii)
- On the other hand, under the chart, the success probability p coincides now to the probability that the set of conditions stated in (2) is not satisfied. Taking into advantage that can be expressed by the aid of the following sum
- (ii)
- The unconditional Average Run Length and the unconditional Variance of thechart is given by
- (ii)
- The unconditional Average Run Length and the unconditional Variance of the chart is given by
- (ii)
- The unconditional Average Run Length and the unconditional Variance of thechart is given by
- ●
- a chart with design parameters . In other words, the practitioner should select the 22nd and the 98th ordered reference observation as the control limits and work with test samples of size . Moreover, if the remaining parameters are determined as , then the resulting chart achieves an in-control ARL equal to 371.26, or
- ●
- a chart with design parameters . In other words, the practitioner should select the 21st and the 73th ordered reference observation as the control limits and work with test samples of size . Moreover, if the remaining parameters are determined as , then the resulting chart achieves an in-control ARL equal to 376.41, or
- ●
- a chart with design parameters . In other words, the practitioner should select the 36th and the 84th ordered reference observation as the control limits and work with test samples of size . Then, if the remaining parameters are determined as , the resulting chart achieves an in-control ARL equal to 370.25.
- ●
- In addition, a similar numerical investigation has been carried out for the class of charts. More precisely, in Table 2, the in-control ARLs of the monitoring schemes are provided for several values of design parameters.
- ●
- a chart with design parameters (with exact in-control ARL equal to 491.42) or
- ●
- a chart with design parameters (with exact in-control ARL equal to 510.87) or
- ●
- a chart with design parameters (with exact in-control ARL equal to 492.12).
4. Numerical Comparisons
5. Discussion and Some Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Reference Sample Size m | |||||||||
---|---|---|---|---|---|---|---|---|---|
50 | 100 | 150 | 200 | ||||||
ARLo | n | (a, b, j, r) | ARLin | (a, b, j, r) | ARLin | (a, b, j, r) | ARLin | (a, b, j, r) | ARLin |
370 | 5 | (6,45,2,2) (9,41,2,2) (11,38,2,2) | 368.64 351.71 364.51 | (10,91,2,2) (13,87,2,3) (22,98,2,3) | 365.67 364.52 371.26 | (12,122,2,2) (13,121,2,3) (20,119,2,3) | 366.70 377.10 365.78 | (18,187,2,2) (27,181,2,3) (34,173,2,3) | 369.21 362.67 360.67 |
11 | (4,37,4,6) (5,33,5,6) (6,35,4,7) | 378.61 372.34 362.10 | (23,85,6,5) (33,87,6,5) (29,86,5,6) | 369.64 366.63 367.58 | (30,129,5,5) (32,127,5,6) (37,124,5,6) | 361.71 375.42 367.91 | (40,174,5,5) (42,170,5,6) (40,176,4,7) | 359.21 374.24 362.29 | |
15 | (18,46,8,7) (17,45,6,8) (12,33,6,8) | 368.10 350.61 370.17 | (14,74,8,7) (21,73,7,7) (33,87,6,7) | 370.86 376.41 374.80 | (26,118,7,7) (22,74,7,7) (34,90,6,7) | 375.83 364.85 366.75 | (28,152,8,7) (43,152,8,7) (54,167,7,8) | 369.78 366.94 370.89 | |
500 | 5 | (6,48,2,2) (9,43,2,2) (12,43,2,2) | 507.10 496.09 472.40 | (8,82,2,2) (13,89,2,3) (21,97,2,3) | 497.87 514.02 497.31 | (11,122,2,2) (13,123,2,3) (21,123,2,3) | 494.04 504.12 504.76 | (16,175,2,2) (27,186,2,3) (35,180,2,3) | 498.45 489.24 503.83 |
11 | (8,43,4,6) (15,41,5,5) (12,42,4,7) | 512.97 473.40 509.58 | (31,79,6,5) (33,88,6,5) (30,88,5,6) | 470.09 493.29 470.89 | (32,138,5,5) (30,126,5,6) (35,123,5,6) | 505.59 504.22 504.88 | (36,169,5,5) (39,167,6,6) (46,165,5,6) | 478.11 491.04 504.88 | |
15 | (16,46,7,7) (11,37,6,8) (9,31,5,8) | 493.73 460.81 533.89 | (23,93,6,7) (25,78,7,7) (34,92,6,7) | 479.20 482.58 485.70 | (26,119,7,7) (34,87,8,7) (22,70,6,7) | 475.13 489.27 478.03 | (37,163,7,7) (38,161,7,8) (47,161,7,8) | 491.73 482.89 502.69 |
Reference Sample Size m | |||||
---|---|---|---|---|---|
100 | 200 | ||||
ARLo | n | (a, b, c, d, i, j, r1, r2) | ARLin | (a, b, c, d, i, j, r1, r2) | ARLin |
370 | 25 | (6,43,55,92,5,21,1,1) (8,43,53,87,6,21,2,1) (13,45,56,85,5,20,2,1) | 356.11 382.20 364.56 | (9,83,103,183,5,21,1,1) (8,71,108,176,5,21,1,1) (9,91,118,170,5,21,2,1) | 375.78 358.49 367.35 |
30 | (7,44,53,90,6,25,2,1) (9,43,52,86,6,25,2,1) (12,42,56,81,5,20,2,1) | 367.36 381.07 362.14 | (9,74,108,178,6,24,1,1) (13,72,117,175,6,25,2,1) (14,70,119,170,6,25,2,1) | 378.37 370.24 388.40 | |
500 | 25 | (6,47,55,92,5,21,1,1) (7,43,53,87,6,21,2,1) (12,42,56,85,5,20,2,1) | 491.42 487.87 492.12 | (8,81,103,184,5,21,1,1) (13,75,117,179,5,21,1,1) (6,97,115,170,5,21,2,1) | 511.35 499.30 501.97 |
30 | (7,44,47,90,6,25,2,1) (8,42,53,86,6,25,2,1) (12,48,56,81,5,20,2,1) | 489.71 510.87 496.10 | (9,76,105,178,6,24,1,1) (13,78,117,175,6,25,2,1) (14,75,114,167,5,20,2,1) | 500.44 485.42 500.47 |
Reference Sample Size m | |||||||||
---|---|---|---|---|---|---|---|---|---|
50 | 100 | 150 | 200 | ||||||
ARLo | n | (a, b, j, r) | ARLout | (a, b, j, r) | ARLout | (a, b, j, r) | ARLin | (a, b, j, r) | ARLout |
370 | 5 | (6,45,2,2) (9,41,2,2) (11,38,2,2) | 55.99 57.79 59.95 | (10,91,2,2) (13,87,2,3) (22,98,2,3) | 59.21 83.59 50.57 | (12,122,2,2) (13,121,2,3) (20,119,2,3) | 81.16 147.59 111.78 | (18,187,2,2) (27,181,2,3) (34,173,2,3) | 61.48 71.68 73.31 |
11 | (4,37,4,6) (5,33,5,6) (6,35,4,7) | 237.35 312.24 133.71 | (23,85,6,5) (33,87,6,5) (29,86,5,6) | 74.64 49.88 41.85 | (30,129,5,5) (32,127,5,6) (37,124,5,6) | 50.99 67.52 60.43 | (40,174,5,5) (42,170,5,6) (40,176,4,7) | 48.76 68.88 46.98 | |
15 | (18,46,8,7) (17,45,6,8) (12,33,6,8) | 23.18 12.47 12.11 | (14,74,8,7) (21,73,7,7) (33,87,6,7) | 189.89 91.17 18.84 | (26,118,7,7) (22,74,7,7) (34,90,6,7) | 96.90 80.01 16.94 | (28,152,8,7) (43,152,8,7) (54,167,7,8) | 175.17 105.05 49.12 | |
500 | 5 | (6,48,2,2) (9,43,2,2) (12,43,2,2) | 59.94 64.57 53.14 | (8,82,2,2) (13,89,2,3) (21,97,2,3) | 103.05 97.53 61.73 | (11,122,2,2) (13,123,2,3) (21,123,2,3) | 105.44 173.89 122.80 | (16,175,2,2) (27,186,2,3) (35,180,2,3) | 83.14 80.85 81.17 |
11 | (8,43,4,6) (15,41,5,5) (12,42,4,7) | 57.85 38.62 29.45 | (31,79,6,5) (33,88,6,5) (30,88,5,6) | 75.43 57.37 41.85 | (32,138,5,5) (30,126,5,6) (35,123,5,6) | 48.01 89.98 80.30 | (36,169,5,5) (39,167,6,6) (46,165,5,6) | 71.79 117.15 83.11 | |
15 | (16,46,7,7) (11,37,6,8) (9,31,5,8) | 22.44 38.90 31.17 | (23,93,6,7) (25,78,7,7) (34,92,6,7) | 27.20 67.15 17.91 | (26,119,7,7) (34,87,8,7) (22,70,6,7) | 111.20 43.76 73.85 | (37,163,7,7) (38,161,7,8) (47,161,7,8) | 93.99 103.54 78.48 |
Reference Sample Size m | |||||
---|---|---|---|---|---|
100 | 200 | ||||
ARLo | n | (a, b, c, d, i, j, r1, r2) | ARLout | (a, b, c, d, i, j, r1, r2) | ARLout |
370 | 25 | (6,43,55,92,5,21,1,1) (8,43,53,87,6,21,2,1) (13,45,56,85,5,20,2,1) | 36.27 118.84 9.91 | (9,83,103,183,5,21,1,1) (8,71,108,176,5,21,1,1) (9,91,118,170,5,21,2,1) | 70.75 307.17 215.94 |
30 | (7,44,53,90,6,25,2,1) (9,43,52,86,6,25,2,1) (12,42,56,81,5,20,2,1) | 32.89 32.80 4.98 | (9,74,108,178,6,24,1,1) (13,72,117,175,6,25,2,1) (14,70,119,170,6,25,2,1) | 67.75 72.17 91.94 | |
500 | 25 | (6,47,55,92,5,21,1,1) (7,43,53,87,6,21,2,1) (12,42,56,85,5,20,2,1) | 36.69 203.41 13.67 | (8,81,103,184,5,21,1,1) (13,75,117,179,5,21,1,1) (6,97,115,170,5,21,2,1) | 112.97 51.25 487.09 |
30 | (7,44,47,90,6,25,2,1) (8,42,53,86,6,25,2,1) (12,48,56,81,5,20,2,1) | 44.30 57.63 4.98 | (9,76,105,178,6,24,1,1) (13,78,117,175,6,25,2,1) (14,75,114,167,5,20,2,1) | 84.55 66.59 59.65 |
Normal Distribution (0 + θ, 1 + δ) | Laplace Distribution (0 + θ, 1 + δ) | ||||
---|---|---|---|---|---|
θ | δ | Chart a = 12, b = 84, j = 3, r = 2 | Competitor 1 | Chart a = 12, b = 84, j = 3, r = 2 | Competitor 1 |
0 | 0 | 475.84 | 458.07 | 475.84 | 458.07 |
0.25 | 0 | 176.43 | 248.92 | 263.59 | 374.63 |
0.5 | 0 | 45.77 | 81.88 | 108.07 | 257.35 |
1 | 0 | 6.30 | 10.00 | 13.82 | 84.08 |
1.5 | 0 | 2.60 | 2.63 | 3.39 | 22.29 |
0.25 | 0.05 | 124.01 | 160.15 | 192.43 | 268.20 |
0.5 | 0.05 | 37.91 | 59.08 | 84.65 | 187.85 |
1 | 0.05 | 6.23 | 8.81 | 12.68 | 65.12 |
1.5 | 0.05 | 2.67 | 2.75 | 3.42 | 18.61 |
0.25 | 0.10 | 91.21 | 109.40 | 145.26 | 198.46 |
0.5 | 0.10 | 32.11 | 44.65 | 68.14 | 141.62 |
1 | 0.10 | 6.17 | 7.90 | 11.76 | 51.92 |
1.5 | 0.10 | 2.51 | 2.52 | 3.44 | 15.89 |
0.25 | 0.15 | 69.60 | 78.44 | 112.82 | 151.09 |
0.5 | 0.15 | 27.67 | 35.01 | 56.10 | 109.75 |
1 | 0.15 | 6.10 | 7.19 | 10.99 | 42.42 |
1.5 | 0.15 | 2.68 | 2.84 | 3.46 | 13.84 |
0.25 | 0.20 | 54.74 | 58.51 | 89.79 | 117.88 |
0.5 | 0.20 | 24.19 | 28.27 | 47.09 | 87.09 |
1 | 0.20 | 6.04 | 6.61 | 10.36 | 35.39 |
1.5 | 0.20 | 2.29 | 2.45 | 3.48 | 12.24 |
Exponential Distribution (λ) | ||
---|---|---|
Shift | Chart | Competitor 2 |
0.000 | 492.12 | 497.21 |
0.025 | 402.57 | 485.98 |
0.050 | 319.50 | 464.96 |
0.075 | 246.08 | 434.80 |
0.100 | 184.13 | 396.84 |
0.125 | 134.15 | 353.09 |
0.150 | 95.51 | 306.01 |
0.175 | 66.80 | 258.20 |
0.200 | 46.23 | 212.12 |
0.225 | 31.95 | 169.80 |
0.250 | 22.27 | 132.63 |
0.275 | 15.84 | 101.32 |
0.300 | 11.60 | 75.90 |
0.325 | 8.81 | 55.97 |
0.350 | 6.96 | 40.77 |
0375 | 5.72 | 29.45 |
0.400 | 4.87 | 21.18 |
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Panayiotou, N.I.; Triantafyllou, I.S. A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs. Stats 2023, 6, 279-292. https://doi.org/10.3390/stats6010017
Panayiotou NI, Triantafyllou IS. A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs. Stats. 2023; 6(1):279-292. https://doi.org/10.3390/stats6010017
Chicago/Turabian StylePanayiotou, Nikolaos I., and Ioannis S. Triantafyllou. 2023. "A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs" Stats 6, no. 1: 279-292. https://doi.org/10.3390/stats6010017
APA StylePanayiotou, N. I., & Triantafyllou, I. S. (2023). A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs. Stats, 6(1), 279-292. https://doi.org/10.3390/stats6010017