A head-to-head comparative study of the conditional performance of control charts based on estimated parameters

Figures & data

Table 1. In-control AARL and SDARL values for the three charts based on estimated parameters when m subgroups of size n are used in Phase I.

m	n	Chart	AARL	SDARL
m = 50	n = 1	Shewhart	421	1362
		CUSUM	287	808
		EWMA	191	238
	n = 5	Shewhart	206	100
		CUSUM	167	83
		EWMA	147	68
m = 100	n = 1	Shewhart	279	318
		CUSUM	228	225
		EWMA	185	125
	n = 5	Shewhart	202	66
		CUSUM	179	59
		EWMA	163	51

Figure 1. Conditional in-control ARL distribution for the three control charts based on estimated parameters when m = 50 subgroups of size n are used. The boxplots show 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles.

Figure 2. Conditional in-control ARL of the Shewhart chart vs. Q based on m = 50 samples of size n. Chart is designed to have ARL₀ = 200 under known parameters.

Figure 3. Conditional in-control ARL of the CUSUM chart vs. Q based on m = 50 samples of size n. Chart is designed to have ARL₀ = 200 under known parameters.

Figure 4. Conditional in-control ARL of the EWMA chart vs. Q based on m = 50 samples of size n. Chart is designed to have ARL₀ = 200 under known parameters.

Figure 5. Conditional in-control ARL of the CUSUM chart vs. the Shewhart chart. Both charts are based on estimated parameters when m = 50 subgroups of size n are used.

Figure 6. Conditional in-control ARL of the EWMA chart vs. the Shewhart chart. Both charts are based on estimated parameters when m = 50 subgroups of size n are used.

Figure 7. Conditional in-control ARL of the EWMA chart vs. the CUSUM chart. Both charts are based on estimated parameters when m = 50 subgroups of size n are used.

Figure 8. Illustration of Definition 1.

Percentage of Shewhart charts according to the conditional ARL performance compared to CUSUM and EWMA chart.

	Compared to CUSUM chart		Compared to EWMA chart
Shewhart	n = 1 m = 50	n = 5 m = 50	n = 1 m = 50	n = 5 m = 50
Worse performance	34.7%	22.5%	49.3%	27.7%
Equivalent	37.6%	30.1%	14.1%	15.3%
Better performance	27.7%	47.4%	36.7%	57.1%
I)	0%	0%	0%	0%
II)	18.6%	32.5%	19.6%	34.3%
III)	9.1%	14.9%	17.1%	22.8%

Table 3. Percentage of CUSUM charts according to the conditional ARL performance compared to the EWMA chart.

	Compared to EWMA chart
CUSUM	n = 1, m = 50	n = 5, m = 50
Worse performance	45.3%	23.7%
Equivalent	28.3%	32.2%
Better performance	26.3%	44.1%
I)	0%	0%
II)	18.6%	36.9%
III)	7.7%	7.2%