A simulation study is constructed to evaluate the performance of the ESNACC. We will compare the type I error of ESNACC with those of CACC, BACC and TSNACC. The
simulated data are generated from the standard skew normal and gamma distributions.
Note that the pdf of the gamma distribution is f (x) = 1
Γ(k)θkxk−1e−x/θ, x > 0,
where θ > 0 and k > 0 are the scale and shape parameters, respectively. The skewness k3 = 2/√
k and the kurtosis k4 = 6/k are independent of the scale parameter θ.
In our simulation study, the observations are generated from the standard skew normal distributions with skewness parameter λ = 0 to 5.5 in increments of 0.5 and 6 to 10 in increments of 1, and gamma distribution with parameters θ = 1, k = 5 to 60 in increments of 5. For the standard skew normal distribution SN (λ), the specification limits will be set as µZ± 6σZ = a1ρ ± 6(1 − a21ρ2). Also for the gamma distribution with parameters θ = 1 and k, we have the mean µG = k and the standard deviation σG =√
k, the specification limits will be set as µG ± 6σG = k ± 6√
k. The steps of the simulations are described below.
Step 1. Generate m=10,000 samples of size 5 from each specified distribution.
Step 2. Assume that the parameters of the underlying distribution are unknown. Given the acceptable process level δ and the type I error α, the control limits of CACC, BACC, TSNACC and ESNACC are established based on the data from Step 1.
Step 3. Calculate the type I errors αL= P (X < LCL|µX = µL) and αU = P (X >
UCL|µX = µU).
Tables 15 to 18 give the simulated type I errors of CACC, BACC, TSNACC and ESNACC under the standard skew normal distributions, where the nominal type I error α = 0.01 or 0.05. Also the acceptable level δ = 0.0027 in Tables 15 and 16, and δ = 0.01 in Tables 17 and 18. The corresponding simulated results for the gamma distributions are given in Tables 19 and 22. It is noted that all the acceptance control charts are designed based on the nominal type I error α = 0.01 or 0.05. Hence a performance is better if its estimated type I error is closer to α. From the simulation results, it can be found that.
1. The CACC performs well only for the normal distributions.
2. The performance of BACC is not good under the skew normal distributions.
3. If the underlying distribution is a skew normal distribution, the performance of ESNACC is better than that of TSNACC, especially for large skewness parameter λ.
4. When the underlying distribution is a gamma distribution, the performances of BACC, TSNACC and ESNACC are close.
6 Conclusions
Non-normal data often occur in industrial processes. Using the conventional control chart for non-normal data will not obtain the correct nominal type I and type II errors.
Thus the development of designing the control charts under non-normality has become one of the important issues. In this paper, based on the exact distribution of sample mean for skew normal distributions, the ESNACC is proposed. The charting constants are also provided in tables for practical implementation. Simulation results show that if the underlying distribution is a skew normal distribution, our ESNACC performs significantly better than CACC, BACC and TSNACC. Also if the underlying distribution is a gamma distribution, the performances of BACC, TSNACC and ESNACC are close, and obviously better than that of CACC.
References
[1] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scan-dinavian Journal of Statistics12, 171-178.
[2] Bai, D.S. and Choi, I.S. (1995). X and R control charts for skewed populations.
Journal of Quality Technology27, 120-131.
[3] Burr, I.W. (1942). Cumulative frequency distribution. Annals of Mathematical Statistics13, 215-232.
[4] Burr, I.W. (1973). Parameters for a general system of distributions to match a grid of α3and α4, Communications in Statistics 2, 1-21.
[5] Chan, L.K. and Cui, H.J. (2003). Skewness correction X and R charts for skewed distributions. Naval Reserach Logistic 50, 555-573.
[6] Chang, Y.S. and Bai, D.S. (2001). Control charts for positively-skewed populations with weighed standard deviation. Quality and Reliability Engineering International 17, 397-406.
[7] Chou, C.Y., Chen C.H. and Liu, H.R. (2005). Acceptance control charts for non-normal data. Journal of Applied Statistics 32, 25-36.
[8] Das, N. and Bhattacharya, A. (2008). A New non-parametric control chart for con-trolling variability. Quality Technology and Quantitative Management 4, 351-361.
[9] Li, C.I., Su, N.C., Su, P.F. and Shyr, Y. (2014). The design of X and R control charts for skew normal distributed data. Accepted by Communication in Statistics-Theory
& methods.
[10] Shewhart, W.A. (1931). Economic control of quality of manufactured product, American Society for Quality Control, Milwaukee, WI.
[11] Su, N.C. and Gupta A.K. (2014). One some sampling distribution for skew normal population. Submitted.
[12] Tadikamalla, P.R. and Popescu, D.G. (2007). Kurtosis correction method for X and R control charts for long-tailed symmetrical distributions. Naval Research Logistics 54, 371-383.
[13] Tsai, T.R. (2007). Skew normal distribution and the design of control charts for averages. International Journal of Reliabilty, Quality and Safety Engineering 14, 49-63.
[14] Tsai, T.R. and Chiang, J.Y. (2008). The design of acceptance control chart for non-normal data. Journal of the Chinese Institute of Industrial Engineers 25, 127-135.
[15] Yourstone, S.A. and Zimmer, W.J. (1992). Non-normality and the design of control charts for averages. Decision Sciences 23, 1099-1113.
Density
0.9 1.0 1.1 1.2 1.3 1.4
012345
skew normal normal
Figure 1: The histogram of the primer thickness data taken from Das and Bhattacharya (2008)
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Figure 2: Normal Q-Q plot of the primer thickness data taken from Das and Bhattacharya (2008)
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0.9 1.0 1.1 1.2 1.3 1.4 1.5
0.91.01.11.21.31.4
Theoretical Quantiles
Sample Quantiles
Figure 3: Skew Normal Q-Q plot of the primer thickness data taken from Das and Bhat-tacharya (2008)
Table1:Theupper100ppercentagepointzp,λoftheSN(λ)distribution p λ0.99730.00270.990.010.950.050.90.1 0-2.782152.78215-2.326342.32634-1.644851.64485-1.281541.28155 0.5-2.219852.98349-1.801132.54762-1.173261.89966-0.837571.55604 1-1.626122.99977-1.281552.57496-0.760061.95451-0.478271.63222 1.5-1.204982.99998-0.924872.57582-0.496071.95974-0.261011.64377 2-0.927792.99998-0.695012.57583-0.334491.95996-0.133811.64480 2.5-0.740982.99998-0.542542.57583-0.231761.95996-0.056101.64485 3-0.609662.99998-0.436802.57583-0.163131.95996-0.006101.64485 3.5-0.513652.99998-0.360352.57583-0.115181.959960.027571.64485 4-0.440982.99998-0.303072.57583-0.080441.959960.051081.64485 4.5-0.384372.99998-0.258912.57583-0.054461.959960.068011.64485 5-0.339232.99998-0.224002.57583-0.034551.959960.080511.64485 5.5-0.302512.99998-0.195852.57583-0.018931.959960.089931.64485 6-0.272112.99998-0.172742.57583-0.006531.959960.097131.64485 7-0.224972.99998-0.137272.575830.011741.959960.107121.64485 8-0.190232.99998-0.111542.575830.024241.959960.113381.64485 9-0.163662.99998-0.092152.575830.033131.959960.117421.64485 10-0.142832.99998-0.077102.575830.039621.959960.120101.64485
Table2:Theupper100ppercentagepointzn,p,λwithn=2 p λ0.99730.00270.990.010.950.050.90.1 0-1.967281.96728-1.644971.64498-1.163091.16309-0.906160.90616 0.5-1.469352.20778-1.172061.90176-0.726471.44640-0.488371.20434 1-1.003182.25528-0.755001.96568-0.379401.54029-0.176751.31761 1.5-0.693222.26241-0.487311.97707-0.171681.565700.001191.35251 2-0.497812.25851-0.323141.98098-0.050991.572870.100671.36467 2.5-0.370792.25416-0.218571.978960.022241.576640.158981.36931 3-0.284352.25400-0.148741.979870.069111.578850.195081.37244 3.5-0.222312.25249-0.099871.984670.100501.579280.218621.37370 4-0.176802.23373-0.064401.983110.122391.579650.234551.37471 4.5-0.142172.24803-0.037781.983110.138121.580170.245811.37535 5-0.115062.23660-0.017451.981770.149761.580190.253911.37581 5.5-0.093432.24110-0.001391.979180.158531.579810.259921.37606 6-0.076062.241100.011171.980690.165341.580090.264461.37674 7-0.049222.245430.030011.984170.174941.580590.270901.37692 8-0.031192.242210.043291.981190.181151.581290.274931.37688 9-0.017242.249640.052281.979700.185301.580300.277741.37719 10-0.006582.239620.058841.983530.188341.581400.279631.37704
Table3:Theupper100ppercentagepointzn,p,λofwithn=3 p λ0.99730.00270.990.010.950.050.90.1 0-1.606271.60627-1.343111.34311-0.949660.94966-0.739900.73990 0.5-1.135871.86597-0.892611.61709-0.528181.24602-0.333511.04879 1-0.723231.93301-0.518401.70226-0.208861.35865-0.042081.17862 1.5-0.459261.94950-0.287181.72342-0.023811.395180.120071.22451 2-0.297281.95421-0.149441.731250.081031.408440.209011.24264 2.5-0.194711.95701-0.063841.732970.143311.415060.260441.25113 3-0.126111.95365-0.007921.736400.182561.418400.292031.25571 3.5-0.078581.961050.030361.739840.208441.419940.312551.25823 4-0.043611.957330.057671.737680.226371.421210.326411.25976 4.5-0.017931.952900.077641.737460.239151.421440.336131.26080 50.001961.951870.092941.737260.248461.422900.343321.26244 5.50.017391.954200.104671.737490.255561.423130.348491.26279 60.030461.956430.113711.742420.260871.424070.352751.26295 70.047941.958340.126091.737480.268731.423310.358551.26355 80.061261.956080.135661.740850.273621.423930.362411.26349 90.070011.959560.140751.737570.277231.423060.365041.26453 100.077541.953440.146601.734680.279591.423640.367041.26425
Table4:Theupper100ppercentagepointzn,p,λwithn=5 p λ0.99730.00270.990.010.950.050.90.1 0-1.244201.24420-1.040261.04026-0.735600.73560-0.573130.57313 0.5-0.800701.52433-0.611781.33199-0.329071.04527-0.178070.89267 1-0.439261.61793-0.278701.44003-0.036541.177780.093691.03955 1.5-0.218591.64564-0.081761.475780.127051.226520.240731.09693 2-0.088161.655900.031841.488410.217391.246900.319881.12140 2.5-0.007751.660440.100441.494900.270191.256910.365191.13379 30.044551.663370.144301.498230.302921.262240.392841.14067 3.50.080151.661690.173551.500620.324281.265170.410851.14473 40.106091.667670.193951.501960.339001.267260.422881.14733 4.50.124231.666330.208601.501670.349481.268390.431531.14950 50.137671.665770.220021.502950.357121.269660.437751.15049 5.50.148411.668110.228411.504200.362861.270450.442241.15143 60.156221.666700.234251.504500.367321.271430.445971.15246 70.169091.665080.243941.504390.373771.272580.451461.15329 80.177351.667270.249941.505770.377731.272630.454361.15451 90.183121.667220.255311.503440.380201.273070.456761.15467 100.187241.670880.257761.504220.382501.273650.458611.15477
Table5:Theupper100ppercentagepointzn,p,λwithn=10 p λ0.99730.00270.990.010.950.050.90.1 0-0.879790.87979-0.735660.73566-0.520140.52014-0.405260.40526 0.5-0.462631.18129-0.328801.04573-0.128430.84335-0.021510.73563 1-0.150041.30371-0.035091.180000.138010.996820.230900.89992 1.50.030411.347010.130011.230690.281451.059180.363430.96921 20.132671.365470.221821.252280.358661.086950.433471.00077 2.50.193751.373780.275651.263020.402891.100990.473031.01701 30.232481.378390.309431.268980.430001.108990.497091.02642 3.50.257911.382350.331541.272350.447611.113960.512621.03210 40.275951.383760.346721.275120.459511.117210.523131.03597 4.50.288261.385000.357601.276170.468021.119450.530531.03845 50.298121.386120.365611.277760.474291.120990.535891.04045 5.50.305531.387340.371701.278780.478951.122320.540031.04181 60.311741.387400.376511.279230.482631.123150.543091.04297 70.319291.387890.382821.280320.487681.124490.547561.04452 80.324231.388500.387561.280730.491171.125320.550631.04539 90.327931.388860.390811.281170.493451.125810.552421.04592 100.330891.388610.392801.281280.495051.126200.553921.04656
Table 6: The values of d∗2 = d∗2(n, λ)
n
λ 2 3 5 10
0 1.12911 1.69367 2.32711 3.08115 0.5 1.05192 1.58093 2.17469 2.87833 1 0.92882 1.39735 1.91545 2.54088 1.5 0.84086 1.25966 1.73410 2.29813 2 0.78527 1.17657 1.62037 2.14905 2.5 0.75200 1.12598 1.54586 2.05087 3 0.73051 1.09016 1.50187 1.98587 3.5 0.71187 1.06904 1.46737 1.94245 4 0.70263 1.05240 1.44628 1.90981 4.5 0.69443 1.04051 1.42725 1.88546 5 0.68948 1.03173 1.41449 1.86638 5.5 0.68484 1.02533 1.40719 1.85437 6 0.68127 1.01888 1.39842 1.84060 7 0.67686 1.01191 1.38780 1.82527 8 0.67252 1.00833 1.38038 1.81071 9 0.67035 1.00566 1.37539 1.80380 10 0.66705 1.00023 1.36985 1.79676
Table7:Thevaluesofzp,λ/d
∗ 2withn=2 p λ0.99730.00270.990.010.950.050.90.1 0-2.464022.46402-2.060332.06033-1.456771.45677-1.135001.13501 0.5-2.110282.83622-1.712232.42187-1.115351.80589-0.796231.47923 1-1.750743.22966-1.379762.77229-0.818312.10429-0.514921.75730 1.5-1.433043.56776-1.099923.06333-0.589952.33064-0.310401.95487 2-1.181493.82032-0.885063.28019-0.425952.49591-0.170402.09457 2.5-0.985343.98931-0.721463.42528-0.308192.60632-0.074602.18729 3-0.834574.10669-0.597953.52607-0.223312.68301-0.008352.25165 3.5-0.721554.21421-0.506203.61838-0.161802.753250.038722.31060 4-0.627614.26967-0.431343.66600-0.114482.789490.072692.34101 4.5-0.553514.32005-0.372833.70926-0.078432.822400.097942.36863 5-0.492024.35109-0.324893.73591-0.050112.842680.116772.38565 5.5-0.441734.38053-0.285973.76119-0.027642.861920.131312.40180 6-0.399414.40349-0.253563.78090-0.009592.876910.142572.41438 7-0.332374.43220-0.202803.805550.017342.895670.158252.43012 8-0.282864.46078-0.165853.830090.036052.914340.168582.44579 9-0.244144.47523-0.137473.842500.049422.923790.175172.45372 10-0.214134.49738-0.115593.861520.059392.938260.180042.46586
Table8:Thevaluesofzp,λ/d
∗ 2withn=3 p λ0.99730.00270.990.010.950.050.90.1 0-1.642671.64267-1.373551.37355-0.971170.97117-0.756660.75667 0.5-1.404141.88717-1.139291.61147-0.742131.20161-0.529790.98425 1-1.163722.14676-0.917131.84275-0.543931.39873-0.342271.16809 1.5-0.956592.38158-0.734222.04486-0.393811.55577-0.207201.30493 2-0.788552.54977-0.590712.18927-0.284291.66583-0.113731.39796 2.5-0.658082.66434-0.481842.28764-0.205831.74068-0.049821.46082 3-0.559242.75186-0.400682.36279-0.149641.79786-0.005601.50881 3.5-0.480482.80623-0.337082.40948-0.107741.833390.025791.53863 4-0.419022.85060-0.287982.44757-0.076431.862370.048531.56295 4.5-0.369412.88317-0.248822.47553-0.052341.883650.065361.58081 5-0.328802.90773-0.217112.49662-0.033491.899690.078041.59427 5.5-0.295042.92588-0.191012.51220-0.018461.911550.087711.60423 6-0.267072.94439-0.169542.52809-0.006411.923640.095331.61437 7-0.222322.96468-0.135652.545520.011601.936900.105851.62550 8-0.188662.97518-0.110612.554540.024041.943770.112441.63126 9-0.162742.98308-0.091632.561320.032941.948920.116761.63559 10-0.142802.99928-0.077082.575230.039611.959510.120071.64447
Table9:Thevaluesofzp,λ/d
∗ 2withn=5 p λ0.99730.00270.990.010.950.050.90.1 0-1.195541.19554-0.999670.99967-0.706820.70682-0.550700.55070 0.5-1.020771.37191-0.828231.17149-0.539510.87353-0.385140.71552 1-0.848951.56609-0.669061.34431-0.396811.02039-0.249690.85213 1.5-0.694871.72999-0.533341.48539-0.286061.13011-0.150510.94790 2-0.572581.85142-0.428921.58966-0.206431.20958-0.082581.01508 2.5-0.479331.94065-0.350971.66627-0.149921.26788-0.036291.06403 3-0.405941.99750-0.290841.71508-0.108621.30502-0.004061.09521 3.5-0.350052.04446-0.245581.75541-0.078501.335700.018791.12096 4-0.304902.07427-0.209551.78100-0.055621.355180.035321.13730 4.5-0.269312.10192-0.181401.80474-0.038161.373240.047651.15246 5-0.239832.12089-0.158361.82103-0.024421.385630.056921.16286 5.5-0.214982.13189-0.139181.83047-0.013451.392820.063911.16889 6-0.194582.14526-0.123531.84195-0.004671.401550.069461.17622 7-0.162102.16167-0.098911.856040.008461.412280.077181.18522 8-0.137812.17329-0.080801.866020.017561.419870.082131.19159 9-0.118992.18118-0.067001.872800.024091.425020.085381.19592 10-0.104272.19001-0.056291.880380.028921.430790.087671.20076
Table10:Thevaluesofzp,λ/d
∗ 2withn=10 p λ0.99730.00270.990.010.950.050.90.1 0-0.902960.90296-0.755030.75502-0.533840.53384-0.415930.41593 0.5-0.771231.03653-0.625760.88511-0.407620.65999-0.290990.54060 1-0.639981.18060-0.504371.01341-0.299130.76923-0.188230.64238 1.5-0.524331.30540-0.402441.12083-0.215860.85275-0.113570.71526 2-0.431721.39595-0.323401.19859-0.155640.91201-0.062260.76536 2.5-0.361301.46278-0.264541.25597-0.113010.95568-0.027350.80203 3-0.307001.51066-0.219961.29708-0.082150.98695-0.003070.82828 3.5-0.264441.54443-0.185511.32607-0.059301.009010.014190.84679 4-0.230901.57082-0.158691.34873-0.042121.026260.026740.86127 4.5-0.203861.59111-0.137321.36616-0.028881.039520.036070.87239 5-0.181761.60738-0.120021.38012-0.018511.050140.043140.88131 5.5-0.163131.61778-0.105611.38905-0.010211.056940.048500.88701 6-0.147841.62989-0.093851.39945-0.003551.064850.052770.89365 7-0.123251.64358-0.075201.411200.006431.073790.058680.90116 8-0.105061.65680-0.061601.422550.013391.082430.062610.90840 9-0.090731.66315-0.051091.428000.018371.086580.065100.91188 10-0.079501.66966-0.042911.433600.022051.090830.066840.91546
Table11:Thevaluesofzn,p,λ/d
∗ 2withn=2 p λ0.99730.00270.990.010.950.050.90.1 0-1.742331.74232-1.456871.45688-1.030091.03009-0.802540.80254 0.5-1.396822.09880-1.114201.80788-0.690611.37501-0.464271.14489 1-1.080062.42811-0.812862.11631-0.408481.65833-0.190301.41859 1.5-0.824422.69060-0.579542.35125-0.204171.862030.001421.60849 2-0.633932.87609-0.411502.52267-0.064932.002980.128191.73784 2.5-0.493072.99754-0.290652.631580.029572.096590.211401.82089 3-0.389243.08553-0.203622.710260.094612.161300.267041.87874 3.5-0.312293.16417-0.140292.787950.141182.218490.307111.92971 4-0.251633.17912-0.091652.822430.174192.248210.333821.95654 4.5-0.204733.23723-0.054402.855730.198902.275490.353971.98054 5-0.166883.24391-0.025302.874300.217202.291870.368271.99544 5.5-0.136433.27243-0.002042.889980.231482.306820.379532.00930 6-0.111653.289580.016392.907330.242692.319310.388192.02083 7-0.072723.317430.044332.931430.258462.335190.400232.03427 8-0.046383.334030.064362.945910.269362.351280.408802.04734 9-0.025723.355920.078002.953230.276432.357420.414322.05443 10-0.009873.357500.088212.973590.282352.370730.419212.06438
Table12:Thevaluesofzn,p,λ/d
∗ 2withn=3 p λ0.99730.00270.990.010.950.050.90.1 0-0.948400.94840-0.793020.79302-0.560710.56071-0.436860.43686 0.5-0.718481.18030-0.564611.02287-0.334090.78816-0.210960.66340 1-0.517581.38334-0.370991.21821-0.149470.97231-0.030110.84347 1.5-0.364591.54764-0.227981.36816-0.018901.107580.095320.97210 2-0.252671.66094-0.127021.471440.068871.197080.177651.05616 2.5-0.172931.73806-0.056701.539080.127271.256740.231301.11115 3-0.115681.79207-0.007261.592780.167461.301090.267881.15185 3.5-0.073501.834400.028401.627480.194981.328240.292371.17697 4-0.041441.859860.054801.651160.215101.350440.310161.19703 4.5-0.017231.876860.074621.669810.229841.366100.323041.21171 50.001901.891850.090081.683840.240821.379150.332761.22362 5.50.016961.905930.102091.694580.249251.387980.339881.23160 60.029891.920170.111601.710130.256031.397680.346211.23955 70.047381.935300.124601.717040.265571.406560.354331.24868 80.060751.939910.134541.726460.271361.412160.359411.25305 90.069621.948520.139951.727780.275671.415040.362991.25740 100.077531.952990.146561.734280.279531.423310.366961.26396
Table13:Thevaluesofzn,p,λ/d
∗ 2withn=5 p λ0.99730.00270.990.010.950.050.90.1 0-0.534660.53466-0.447020.44702-0.316100.31610-0.246280.24628 0.5-0.368190.70094-0.281320.61250-0.151320.48065-0.081880.41048 1-0.229330.84467-0.145500.75180-0.019080.614890.048910.54272 1.5-0.126050.94899-0.047150.851030.073270.707290.138820.63256 2-0.054401.021930.019650.918560.134160.769520.197410.69207 2.5-0.005011.074120.064980.967030.174780.813080.236240.73344 30.029661.107530.096080.997580.201700.840440.261570.75950 3.50.054621.132430.118271.022670.220990.862200.279990.78012 40.073351.153080.134111.038500.234390.876220.292390.79329 4.50.087041.167510.146161.052140.244860.888700.302350.80539 50.097331.177650.155551.062540.252470.897610.309470.81336 5.50.105461.185420.162321.068940.257860.902820.314270.81824 60.111711.191840.167511.075850.262670.909190.318910.82412 70.121841.199800.175781.084010.269320.916970.325310.83102 80.128481.207830.181071.090840.273640.921940.329160.83637 90.133141.212180.185631.093100.276430.925610.332090.83952 100.136681.219760.188171.098100.279230.929770.334790.84299
Table14:Thevaluesofzn,p,λ/d
∗ 2withn=10 p λ0.99730.00270.990.010.950.050.90.1 0-0.285540.28554-0.238760.23876-0.168810.16881-0.131530.13153 0.5-0.160730.41041-0.114230.36331-0.044620.29300-0.007470.25558 1-0.059050.51310-0.013810.464410.054320.392310.090870.35418 1.50.013230.586130.056570.535520.122470.460890.158140.42174 20.061740.635380.103220.582710.166890.505780.201700.46568 2.50.094470.669860.134410.615850.196450.536840.230650.49589 30.117070.694100.155820.639000.216530.558440.250310.51686 3.50.132780.711650.170680.655020.230430.573480.263900.53134 40.144490.724550.181550.667670.240610.584990.273920.54245 4.50.152890.734570.189660.676850.248230.593730.281380.55077 50.159730.742680.195890.684620.254120.600620.287130.55747 5.50.164760.748140.200450.689600.258280.605230.291220.56182 60.169370.753770.204560.695010.262210.610210.295060.56665 70.174930.760380.209740.701440.267180.616070.299990.57226 80.179060.766830.214040.707310.271260.621480.304090.57734 90.181800.769970.216660.710260.273560.624130.306260.57985 100.184160.772840.218610.713110.275520.626800.308290.58247
Table 15: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.0027, α = 0.01
method CACC BACC TSNACC ESNACC
λ αL αU αL αU αL αU αL αU
0 0.0096* 0.0096 0.0065 0.0126 0.0089 0.0104* 0.0089 0.0104 0.5 0.0107* 0.0081 0.0051 0.0147 0.0092 0.0101* 0.0092 0.0101 1 0.0188 0.0041 0.0038 0.0112 0.0098 0.0098 0.0098* 0.0098*
1.5 0.0394 0.0017 0.0040 0.0134 0.0101* 0.0092 0.0103 0.0094*
2 0.0793 0.0009 0.0028 0.0124 0.0096 0.0095 0.0101* 0.0098*
2.5 0.1459 0.0006 0.0026 0.0123 0.0105* 0.0103* 0.0113 0.0107 3 0.2083 0.0004 0.0020 0.0119 0.0099* 0.0095 0.0109 0.0099*
3.5 0.2705 0.0003 0.0017 0.0087 0.0091 0.0091 0.0103* 0.0094*
4 0.3476 0.0003 0.0000 0.0000 0.0079 0.0108* 0.0092* 0.0114 4.5 0.3921 0.0002 0.0000 0.0000 0.0104* 0.0091 0.0121 0.0096*
5 0.4444 0.0002 0.0042 0.0181 0.0070 0.0104* 0.0085* 0.0108 5.5 0.4684 0.0002 0.0046 0.0142 0.0086 0.0088 0.0105* 0.0094*
6 0.5081 0.0002 0.0000 0.0000 0.0107* 0.0087 0.0130 0.0091*
7 0.5527 0.0001 0.0000 0.0000 0.0071 0.0089 0.0092* 0.0093*
8 0.5981 0.0001 0.0000 0.0000 0.0071 0.0092 0.0093* 0.0098*
9 0.6199 0.0001 0.0106* 0.0167 0.0062 0.0087 0.0080 0.0092*
10 0.6430 0.0001 0.0133 0.0154 0.0081 0.0087 0.0105* 0.0094*
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 16: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.0027, α = 0.05
method CACC BACC TSNACC ESNACC
λ αL αU αL αU αL αU αL αU
0 0.0482* 0.0482 0.0340 0.0618 0.0451 0.0514* 0.0451 0.0514 0.5 0.0528* 0.0417 0.0280 0.0688 0.0466 0.0502 0.0466 0.0501*
1 0.0844 0.0228 0.0224 0.0558 0.0493* 0.0491* 0.0493 0.0491 1.5 0.1505 0.0099 0.0233 0.0625 0.0513 0.0472 0.0513* 0.0472*
2 0.2484 0.0051 0.0177 0.0620 0.0503 0.0489 0.0502* 0.0491*
2.5 0.3713 0.0035 0.0169 0.0607 0.0556 0.0530* 0.0554* 0.0533 3 0.4582 0.0024 0.0132 0.0584 0.0538 0.0490 0.0534* 0.0493*
3.5 0.5308 0.0018 0.0113 0.0452 0.0512 0.0470 0.0509* 0.0473*
4 0.6123 0.0017 0.0000 0.0000 0.0478* 0.0555* 0.0474 0.0558 4.5 0.6507 0.0014 0.0000 0.0000 0.0580 0.0475 0.0576* 0.0478*
5 0.6954 0.0014 0.0219 0.0808 0.0452* 0.0530* 0.0446 0.0541 5.5 0.7126 0.0011 0.0265 0.0683 0.0522 0.0465 0.0516* 0.0475*
6 0.7425 0.0010 0.0000 0.0000 0.0608 0.0451 0.0601* 0.0459*
7 0.7736 0.0009 0.0000 0.0000 0.0474* 0.0462 0.0471 0.0465*
8 0.8049 0.0009 0.0000 0.0000 0.0482* 0.0483 0.0476 0.0490*
9 0.8178 0.0008 0.0445* 0.0769 0.0435 0.0456 0.0427 0.0463*
10 0.8324 0.0008 0.0522 0.0723 0.0521 0.0452 0.0513* 0.0462*
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 17: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.01, α = 0.01
method CACC BACC TSNACC ESNACC
λ αL αU αL αU αL αU αL αU
0 0.0097 0.0097 0.0098* 0.0098 0.0092 0.0101* 0.0092 0.0101 0.5 0.0104* 0.0088 0.0084 0.0111 0.0095 0.0100 0.0095 0.0100*
1 0.0151 0.0059 0.0075 0.0095 0.0098 0.0098 0.0099* 0.0099*
1.5 0.0256 0.0034 0.0077 0.0096 0.0100* 0.0094 0.0102 0.0096*
2 0.0436 0.0022 0.0070 0.0089 0.0096 0.0096 0.0100* 0.0098*
2.5 0.0727 0.0017 0.0072 0.0090 0.0102* 0.0102* 0.0109 0.0106 3 0.1004 0.0013 0.0065 0.0090 0.0096 0.0096 0.0106 0.0098*
3.5 0.1293 0.0011 0.0062 0.0078 0.0090 0.0092 0.0102* 0.0097*
4 0.1687 0.0011 0.0000 0.0000 0.0081 0.0104* 0.0095* 0.0109 4.5 0.1928 0.0010 0.0000 0.0000 0.0097* 0.0093 0.0114 0.0097*
5 0.2225 0.0009 0.0106 0.0119 0.0074 0.0101* 0.0090* 0.0105 5.5 0.2371 0.0008 0.0107 0.0100* 0.0085 0.0091 0.0103* 0.0095 6 0.2616 0.0008 0.0000 0.0000 0.0098* 0.0090 0.0119 0.0093*
7 0.2910 0.0007 0.0000 0.0000 0.0073 0.0091 0.0094* 0.0096*
8 0.3226 0.0007 0.0000 0.0000 0.0074 0.0094 0.0094* 0.0097*
9 0.3390 0.0007 0.0171 0.0111 0.0064 0.0089 0.0086* 0.0095*
10 0.3561 0.0008 0.0196 0.0107 0.0078 0.0089 0.0101* 0.0094*
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 18: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.01, α = 0.05
method CACC BACC TSNACC ESNACC
λ αL αU αL αU αL αU αL αU
0 0.0486* 0.0486 0.0467 0.0511 0.0465 0.0505* 0.0465 0.0505 0.5 0.0515* 0.0443 0.0419 0.0554 0.0475 0.0498* 0.0475 0.0498 1 0.0719 0.0307 0.0380 0.0489 0.0494* 0.0493* 0.0494 0.0493 1.5 0.1110 0.0179 0.0390 0.0479 0.0509 0.0479 0.0508* 0.0480*
2 0.1673 0.0115 0.0361 0.0478 0.0502 0.0491 0.0501* 0.0493*
2.5 0.2412 0.0090 0.0363 0.0472 0.0543 0.0521* 0.0541* 0.0525 3 0.2963 0.0068 0.0321 0.0468 0.0527 0.0492 0.0525* 0.0496*
3.5 0.3461 0.0056 0.0294 0.0410 0.0508 0.0477 0.0506* 0.0482*
4 0.4079 0.0053 0.0000 0.0000 0.0488* 0.0539* 0.0485 0.0542 4.5 0.4386 0.0048 0.0000 0.0000 0.0558 0.0478 0.0554* 0.0482*
5 0.4774 0.0045 0.0428 0.0577 0.0470* 0.0519* 0.0464 0.0528 5.5 0.4913 0.0041 0.0470 0.0510* 0.0515 0.0473 0.0510* 0.0480 6 0.5203 0.0040 0.0000 0.0000 0.0574 0.0466 0.0570* 0.0466*
7 0.5504 0.0036 0.0000 0.0000 0.0482* 0.0468 0.0477 0.0477*
8 0.5835 0.0036 0.0000 0.0000 0.0490* 0.0487 0.0480 0.0494*
9 0.5972 0.0032 0.0617 0.0558 0.0452* 0.0463 0.0445 0.0472*
10 0.6133 0.0032 0.0673 0.0527* 0.0513 0.0465 0.0508* 0.0469
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 19: Type I error of acceptance control chart under gamma distribution with param-eters θ = 1 and k, where δ = 0.0027, α = 0.01
method CACC BACC TSNACC ESNACC
k αL αU αL αU αL αU αL αU
5 0.4267 0.0001 0.0050* 0.0119* 0.0029 0.0045 0.0038 0.0047 10 0.2049 0.0003 0.0098* 0.0108* 0.0119 0.0062 0.0130 0.0064 15 0.1417 0.0006 0.0042 0.0140 0.0142* 0.0084 0.0150 0.0087*
20 0.1030 0.0008 0.0044 0.0112* 0.0153* 0.0082 0.0159 0.0085 25 0.0817 0.0010 0.0059 0.0116 0.0131* 0.0093 0.0136 0.0095*
30 0.0717 0.0012 0.0041 0.0155 0.0129* 0.0102* 0.0133 0.0104 35 0.0605 0.0013 0.0049 0.0129 0.0129* 0.0096 0.0133 0.0098*
40 0.0548 0.0015 0.0074* 0.0117 0.0130 0.0097 0.0133 0.0099*
45 0.0541 0.0019 0.0073* 0.0144 0.0137 0.0113* 0.0140 0.0115 50 0.0500 0.0020 0.0063 0.0148 0.0125* 0.0117* 0.0127 0.0119 55 0.0441 0.0020 0.0060 0.0110 0.0140* 0.0095 0.0142 0.0097*
60 0.0423 0.0022 0.0059 0.0123 0.0138* 0.0098 0.0141 0.0100*
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 20: Type I error of acceptance control chart under gamma distribution with param-eters θ = 1 and k, where δ = 0.0027, α = 0.05
method CACC BACC TSNACC ESNACC
k αL αU αL αU αL αU αL αU
5 0.6831 0.0005 0.0282* 0.0597* 0.0238 0.0257 0.0234 0.0261 10 0.4522 0.0016 0.0422* 0.0534* 0.0593 0.0337 0.0590 0.0340 15 0.3620 0.0033 0.0235 0.0682 0.0674 0.0446 0.0672* 0.0448*
20 0.2934 0.0046 0.0254 0.0569 0.0707 0.0432 0.0706* 0.0433*
25 0.2504 0.0057 0.0307 0.0597 0.0626 0.0479 0.0625* 0.0480*
30 0.2295 0.0073 0.0249 0.0741 0.0619 0.0520* 0.0619* 0.0522 35 0.2031 0.0080 0.0261 0.0617 0.0616 0.0492 0.0615* 0.0493*
40 0.1893 0.0091 0.0354 0.0571 0.0620 0.0495 0.0620* 0.0495*
45 0.1887 0.0111 0.0353* 0.0690 0.0650 0.0566* 0.0650 0.0567 50 0.1782 0.0120 0.0309 0.0702 0.0602 0.0583* 0.0601* 0.0584 55 0.1620 0.0120 0.0323 0.0543 0.0657 0.0485 0.0657* 0.0485*
60 0.1572 0.0131 0.0317 0.0602 0.0653 0.0497 0.0653* 0.0497*
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 21: Type I error of acceptance control chart under gamma distribution with param-eters θ = 1 and k, where δ = 0.01, α = 0.01
method CACC BACC TSNACC ESNACC
k αL αU αL αU αL αU αL αU
5 0.1863 0.0008 0.0080* 0.0098* 0.0030 0.0084 0.0038 0.0086 10 0.0859 0.0013 0.0131 0.0083 0.0088 0.0089 0.0096* 0.0092*
15 0.0625 0.0020 0.0078 0.0105 0.0105* 0.0103* 0.0112 0.0107 20 0.0481 0.0023 0.0080 0.0094 0.0114* 0.0098 0.0119 0.0101*
25 0.0402 0.0026 0.0094 0.0089 0.0104* 0.0104* 0.0108 0.0106 30 0.0366 0.0030 0.0072 0.0110 0.0104* 0.0109* 0.0108 0.0111 35 0.0322 0.0031 0.0085 0.0102* 0.0105* 0.0103 0.0108 0.0106 40 0.0300 0.0034 0.0109 0.0091 0.0107* 0.0103* 0.0110 0.0105 45 0.0300 0.0038 0.0110* 0.0105* 0.0112 0.0114 0.0115 0.0116 50 0.0284 0.0040 0.0099* 0.0105* 0.0106 0.0116 0.0108 0.0117 55 0.0259 0.0040 0.0095 0.0092 0.0115* 0.0100* 0.0117 0.0102 60 0.0252 0.0042 0.0095 0.0099* 0.0115* 0.0102 0.0116 0.0103
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods
Table 22: Type I error of acceptance control chart under gamma distribution with param-eters θ = 1 and k, where δ = 0.01, α = 0.05
method CACC BACC TSNACC ESNACC
k αL αU αL αU αL αU αL αU
5 0.4338 0.0035 0.0393* 0.0508* 0.0239 0.0433 0.0235 0.0439 10 0.2660 0.0067 0.0524 0.0434 0.0476* 0.0456 0.0474 0.0459*
15 0.2155 0.0102 0.0374 0.0542 0.0543 0.0527* 0.0542* 0.0529 20 0.1782 0.0121 0.0398 0.0496 0.0571 0.0498 0.0571* 0.0500*
25 0.1557 0.0137 0.0439 0.0484* 0.0527 0.0524 0.0526* 0.0525 30 0.1456 0.0158 0.0383 0.0566 0.0530 0.0549* 0.0529* 0.0550 35 0.1316 0.0165 0.0402 0.0512* 0.0530 0.0521 0.0529* 0.0522 40 0.1248 0.0178 0.0477* 0.0467 0.0536 0.0519* 0.0536 0.0519 45 0.1256 0.0203 0.0483* 0.0535* 0.0562 0.0569 0.0561 0.0570 50 0.1202 0.0212 0.0442 0.0537* 0.0532 0.0577 0.0532* 0.0578 55 0.1112 0.0210 0.0457* 0.0472 0.0567 0.0504* 0.0567 0.0504 60 0.1090 0.0222 0.0455* 0.0508* 0.0567 0.0511 0.0567 0.0511
”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods