Binomial Confidence Intervals for Rare Events: Importance of Defining Margin of Error Relative to Magnitude of Proportion

Figures & data

Table 1 Sample size comparison for ${ϵ}_R=0.4$.

$p^{*}$	$ϵ$	$\mathrm{W}$	$\text{CP}$	$\text{WS}$	$\text{AC}$
${10}^{-1}$	$4.0·{10}^{-2}$	$2.2·{10}^2$	$3.0·{10}^2$	$2.2·{10}^2$	$2.3·{10}^2$
${10}^{-2}$	$4.0·{10}^{-3}$	$2.4·{10}^3$	$3.3·{10}^3$	$2.5·{10}^3$	$2.5·{10}^3$
${10}^{-3}$	$4.0·{10}^{-4}$	$2.4·{10}^4$	$3.4·{10}^4$	$2.5·{10}^4$	$2.6·{10}^4$
${10}^{-4}$	$4.0·{10}^{-5}$	$2.4·{10}^5$	$3.4·{10}^5$	$2.5·{10}^5$	$2.6·{10}^5$
${10}^{-5}$	$4.0·{10}^{-6}$	$2.4·{10}^6$	$3.4·{10}^6$	$2.5·{10}^6$	$2.6·{10}^6$
${10}^{-6}$	$4.0·{10}^{-7}$	$2.4·{10}^7$	$3.4·{10}^7$	$2.5·{10}^7$	$2.6·{10}^7$

Table 2 Wald-based sample size comparison – fixed ϵ.

p	Margin of error scheme
	1	2	3	4
	$ϵ=4·{10}^{-2}$	$ϵ=4·{10}^{-2}$	$ϵ=4·{10}^{-4}$	$ϵ=4·{10}^{-4}$
	$p^{*}=p$	$p^{*}=0.5$	$p^{*}=p$	$p^{*}=0.5$
${10}^{-1}$	$2.2·{10}^2$	$6.0·{10}^2$	$2.2·{10}^6$	$6.0·{10}^6$
	(93.8)	(94.5)	(95.0)	(95.0)
${10}^{-2}$	$2.4·{10}^1$	$6.0·{10}^2$	$2.4·{10}^5$	$6.0·{10}^6$
	(21.4)	(93.1)	(95.0)	(95.0)
${10}^{-3}$	$3.0·{10}^0$	$6.0·{10}^2$	$2.4·{10}^4$	$6.0·{10}^6$
	(0.3)	(45.2)	(93.1)	(95.0)
${10}^{-4}$	$1.0·{10}^0$	$6.0·{10}^2$	$2.4·{10}^3$	$6.0·{10}^6$
	(0)	(5.8)	(21.3)	(94.9)
${10}^{-5}$	$1.0·{10}^0$	$6.0·{10}^2$	$2.4·{10}^2$	$6.0·{10}^6$
	(0)	(0.6)	(0.2)	(94.9)
${10}^{-6}$	$1.0·{10}^0$	$6.0·{10}^2$	$2.5·{10}^1$	$6.0·{10}^6$
	(0)	(0.1)	(0)	(93.4)

Wald 95% CI coverage shown in parentheses.

Coverage computed with $p=p^{*}$

Table 3 Wald-based sample size comparison – variable ϵ.

$p^{*}$	ϵR
	0.05	0.1	0.2	0.3	0.4	0.5	0.75
${10}^{-1}$	$1.4·{10}^4$	$3.5·{10}^3$	$8.6·{10}^2$	$3.8·{10}^2$	$2.2·{10}^2$	$1.4·{10}^2$	$6.2·{10}^1$
	(94.9)	(94.9)	(95.0)	(94.3)	(93.8)	(93.3)	(94.7)
${10}^{-2}$	$1.5·{10}^5$	$3.8·{10}^4$	$9.5·{10}^3$	$4.2·{10}^3$	$2.4·{10}^3$	$1.5·{10}^3$	$6.8·{10}^2$
	(94.9)	(94.9)	(95.2)	(94.0)	(95.2)	(92.5)	(90.2)
${10}^{-3}$	$1.5·{10}^6$	$3.8·{10}^5$	$9.6·{10}^4$	$4.3·{10}^4$	$2.4·{10}^4$	$1.5·{10}^4$	$6.8·{10}^3$
	(94.9)	(95.0)	(95.0)	(94.7)	(93.1)	(93.3)	(90.4)
${10}^{-4}$	$1.5·{10}^7$	$3.8·{10}^6$	$9.6·{10}^5$	$4.3·{10}^5$	$2.4·{10}^5$	$1.5·{10}^5$	$6.8·{10}^4$
	(95.0)	(95.0)	(95.0)	(94.7)	(93.1)	(93.3)	(90.4)
${10}^{-5}$	$1.5·{10}^8$	$3.8·{10}^7$	$9.6·{10}^6$	$4.3·{10}^6$	$2.4·{10}^6$	$1.5·{10}^6$	$6.8·{10}^5$
	(95.0)	(95.0)	(95.0)	(94.7)	(93.1)	(93.3)	(90.4)
${10}^{-6}$	$1.5·{10}^9$	$3.8·{10}^8$	$9.6·{10}^7$	$4.3·{10}^7$	$2.4·{10}^7$	$1.5·{10}^7$	$6.8·{10}^6$
	(95.0)	(95.0)	(95.0)	(94.7)	(93.1)	(93.3)	(90.4)

Wald 95% CI coverage shown in parentheses.

Table 4 ϵ_R thresholds as per (4) for $a\in \{5,10\}$.

$p^{*}$	a = 5			a = 10
	$\alpha =0.1$	$\alpha =0.05$	$\alpha =0.01$	$\alpha =0.1$	$\alpha =0.05$	$\alpha =0.01$
${10}^{-1}$	0.698	0.832	1.093	0.493	0.588	0.773
${10}^{-2}$	0.732	0.872	1.146	0.518	0.617	0.810
${10}^{-3}$	0.735	0.876	1.151	0.520	0.619	0.814
${10}^{-4}$	0.736	0.876	1.152	0.520	0.620	0.815
${10}^{-5}$	0.736	0.877	1.152	0.520	0.620	0.815
${10}^{-6}$	0.736	0.877	1.152	0.520	0.620	0.815

Table 5 Coverage & relative margin of error tolerances.

Tolerance	Coverage	Relative MoE
Target	$\mathit{\text{CPr}}\in (1-\alpha )100\pm 1\%$	${ϵ}_R\le 0.5$
Acceptable	$\mathit{\text{CPr}}\in (1-\alpha )100\pm 2\%$	${ϵ}_R\le {0.75}^{*}$
Minimally acceptable	$\mathit{\text{CPr}}\in (1-\alpha )100\pm 3\%$	${ϵ}_R\le 1$
Unacceptable	$\mathit{\text{CPr}}\notin (1-\alpha )100\pm 3\%$	${ϵ}_R>1$

${}^{*}$ Considered acceptable as a compromise between ${ϵ}_R\le 0.5$ (desired) and ${ϵ}_R\le 1$ (limit).

Fig. 1 CPr versus ${\widetilde{ϵ}}_R$ for $p^{*}=p={10}^{-1},\alpha =0.01$. Dashed (gray) line represents the nominal CPr value. Dot-dashed (red) lines represent the target CPr and ${\widetilde{ϵ}}_R$ tolerances from Table 5. Sample size range is from n = 100, to n = 800, in steps of 20. Labels shown adjacent to data points depict the first n where both CPr and ${\widetilde{ϵ}}_R$ requirements are satisfied. Additional data labels are referred to within the text.

Fig. 2 CPr versus ${\widetilde{ϵ}}_R$ for $p^{*}=p={10}^{-2},\alpha =0.05$. Dashed (gray) line represents the nominal CPr value. Dot-dashed (red) lines represent the target CPr and ${\widetilde{ϵ}}_R$ tolerances from Table 5. Sample size range is from $n=$ 1000, to $n=$ 8000, in steps of 200. Labels shown adjacent to data points depict the first n where both CPr and ${\widetilde{ϵ}}_R$ requirements are satisfied. Additional data labels are referred to within the text.

Table 6 95% CI performance – $p^{*}=p={10}^{-1}$, “small” n.

n	W		CP		WS		AC
	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
10	65.0	1.40	98.7	2.12	93.0	1.85	93.0	2.11
20	87.6	1.17	98.9	1.46	95.7	1.32	95.7	1.46
30	80.9	1.01	99.2	1.18	97.4	1.07	97.4	1.16
40	91.4	0.89	97.0	1.01	94.3	0.93	95.8	0.99
50	87.9	0.81	97.0	0.90	97.0	0.83	97.0	0.88
60	94.1	0.74	97.2	0.82	95.2	0.76	95.2	0.80
70	92.0	0.69	95.5	0.76	93.2	0.70	97.4	0.73
80	90.0	0.65	97.7	0.71	96.3	0.66	96.3	0.68
90	94.6	0.61	96.7	0.66	95.0	0.62	95.0	0.64
100	93.2	0.58	95.6	0.63	93.6	0.59	97.2	0.61
110	91.9	0.55	97.6	0.60	96.3	0.56	96.3	0.58
120	95.4	0.53	96.8	0.57	95.4	0.54	95.4	0.55
130	94.4	0.51	96.0	0.55	94.5	0.52	94.5	0.53
140	93.5	0.49	96.7	0.53	93.5	0.50	96.7	0.51

Table 7 95% CI performance – $p^{*}=p={10}^{-1}$, “large” n.

n	W		CP		WS		AC
	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
150	92.6	0.48	97.2	0.51	96.0	0.48	96.0	0.49
160	95.5	0.46	96.6	0.49	95.4	0.47	95.4	0.47
170	94.8	0.45	96.0	0.48	94.6	0.45	94.6	0.46
180	94.1	0.43	95.4	0.46	93.9	0.44	96.6	0.45
190	93.4	0.42	96.1	0.45	96.1	0.43	96.1	0.43
200	92.7	0.41	96.7	0.44	95.6	0.42	95.6	0.42
210	95.3	0.40	96.2	0.43	95.1	0.41	95.1	0.41
220	94.8	0.39	95.7	0.42	94.5	0.40	94.5	0.40
230	94.3	0.39	96.4	0.41	94.0	0.39	96.4	0.39
240	93.7	0.38	96.0	0.40	96.0	0.38	96.0	0.38
250	93.2	0.37	96.6	0.39	95.6	0.37	95.6	0.38
260	95.5	0.36	96.2	0.38	95.1	0.36	95.1	0.37
270	95.0	0.36	95.8	0.37	94.7	0.36	94.7	0.36
280	94.6	0.35	95.4	0.37	94.3	0.35	94.3	0.36
290	94.2	0.34	96.1	0.36	96.1	0.35	96.1	0.35
300	93.8	0.34	96.6	0.35	95.7	0.34	95.7	0.34
310	93.3	0.33	96.3	0.35	95.4	0.33	95.4	0.34
320	95.4	0.33	96.0	0.34	95.0	0.33	95.0	0.33
330	95.1	0.32	95.7	0.34	94.7	0.32	94.7	0.33
340	94.7	0.32	95.4	0.33	94.3	0.32	94.3	0.32
350	94.4	0.31	96.0	0.33	96.0	0.31	96.0	0.32

Table 8 95% CI performance – $p^{*}=p={10}^{-6}$, “small” n.

n	W		CP		WS		AC
	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
$7.5·{10}^6$	93.6	0.70	95.8	0.79	93.7	0.75	97.4	0.79
$8.0·{10}^6$	89.2	0.68	96.9	0.76	95.2	0.72	95.2	0.76
$8.5·{10}^6$	91.9	0.66	97.7	0.73	96.3	0.70	96.3	0.74
$9.0·{10}^6$	94.0	0.64	95.7	0.71	93.7	0.68	97.2	0.71
$9.5·{10}^6$	90.3	0.63	96.7	0.69	95.2	0.66	95.2	0.69
$10.0·{10}^6$	92.6	0.61	97.5	0.67	96.3	0.64	96.3	0.67
$10.5·{10}^6$	94.4	0.60	95.7	0.65	93.9	0.63	97.1	0.65
$11.0·{10}^6$	91.2	0.58	96.7	0.64	95.3	0.61	95.3	0.64
$11.5·{10}^6$	93.2	0.57	97.5	0.62	96.3	0.60	96.3	0.62
$12.0·{10}^6$	94.3	0.56	95.8	0.61	94.2	0.58	97.1	0.60
$12.5·{10}^6$	92.1	0.55	96.8	0.60	95.5	0.57	95.5	0.59
$13.0·{10}^6$	93.8	0.54	97.5	0.58	96.4	0.56	96.4	0.58
$13.5·{10}^6$	94.7	0.53	96.0	0.57	94.6	0.55	94.6	0.57
$14.0·{10}^6$	92.9	0.52	96.9	0.56	95.7	0.54	95.7	0.55

Table 9 95% CI performance – $p^{*}=p={10}^{-6}$, “large” n.

n	W		CP		WS		AC
	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
$15.0·{10}^6$	91.9	0.50	96.3	0.54	94.9	0.52	94.9	0.53
$15.5·{10}^6$	93.6	0.49	97.1	0.53	96.0	0.51	96.0	0.52
$16.0·{10}^6$	94.4	0.49	95.6	0.52	94.1	0.50	96.8	0.52
$16.5·{10}^6$	92.7	0.48	96.5	0.51	95.3	0.49	95.3	0.51
$17.0·{10}^6$	94.2	0.47	97.2	0.51	96.2	0.49	96.2	0.50
$17.5·{10}^6$	94.9	0.47	95.9	0.50	94.6	0.48	94.6	0.49
$18.0·{10}^6$	93.5	0.46	96.7	0.49	95.6	0.47	95.6	0.48
$18.5·{10}^6$	94.8	0.45	95.3	0.48	93.9	0.46	96.5	0.48
$19.0·{10}^6$	92.8	0.45	96.2	0.48	95.0	0.46	95.0	0.47
$19.5·{10}^6$	94.1	0.44	97.0	0.47	96.0	0.45	96.0	0.46
$20.0·{10}^6$	94.8	0.44	95.7	0.46	94.4	0.45	94.4	0.46
$20.5·{10}^6$	93.5	0.43	96.5	0.46	95.5	0.44	95.5	0.45
$21.0·{10}^6$	94.7	0.43	95.1	0.45	93.8	0.43	96.3	0.44
$21.5·{10}^6$	92.9	0.42	96.0	0.45	94.9	0.43	94.9	0.44
$22.0·{10}^6$	94.2	0.42	96.8	0.44	95.8	0.42	95.8	0.43
$22.5·{10}^6$	94.7	0.41	95.6	0.44	94.4	0.42	94.4	0.43
$23.0·{10}^6$	93.6	0.41	96.4	0.43	95.4	0.41	95.4	0.42
$23.5·{10}^6$	94.8	0.40	95.1	0.43	96.2	0.41	96.2	0.42
$24.0·{10}^6$	93.1	0.40	96.0	0.42	94.9	0.41	94.9	0.41
$24.5·{10}^6$	94.3	0.39	96.7	0.42	95.8	0.40	95.8	0.41
$25.0·{10}^6$	94.8	0.39	95.5	0.41	94.4	0.40	94.4	0.40

Table 10 95% CI performance comparison – n.

$p^{*}$	n	W		CP		WS		AC
		CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
${10}^{-1}$	nA: $1.4·{10}^2$	93.3	0.49	96.8	0.53	95.2	0.50	95.2	0.51
	nB: $1.6·{10}^2$	94.2	0.47	95.6	0.50	94.1	0.47	95.4	0.48
${10}^{-2}$	nA: $1.6·{10}^3$	94.0	0.49	97.1	0.52	95.9	0.50	95.9	0.52
	nB: $1.8·{10}^3$	94.1	0.45	95.4	0.48	95.4	0.46	95.4	0.47
${10}^{-3}$	nA: $1.6·{10}^4$	94.0	0.49	97.0	0.53	95.7	0.50	95.7	0.52
	nB: $1.7·{10}^4$	94.7	0.47	95.9	0.50	94.7	0.48	95.9	0.49
${10}^{-4}$	nA: $1.6·{10}^5$	94.0	0.49	96.9	0.53	95.7	0.51	95.7	0.52
	nB: $1.7·{10}^5$	94.7	0.47	95.9	0.50	94.7	0.48	95.8	0.49
${10}^{-5}$	nA: $1.6·{10}^6$	94.0	0.49	96.9	0.53	95.7	0.51	95.7	0.52
	nB: $1.7·{10}^6$	94.7	0.47	95.9	0.50	94.7	0.48	95.9	0.49
${10}^{-6}$	nA: $1.6·{10}^7$	94.0	0.49	96.9	0.53	95.7	0.51	95.7	0.52
	nB: $1.7·{10}^7$	94.7	0.47	95.9	0.50	94.7	0.48	95.8	0.49

n_A: First n where at least one interval satisfies $\mathit{\text{CPr}}\in 95\pm 1\%$ and ${\widetilde{ϵ}}_R\le 0.5$.

n_B: First n where all intervals satisfy $\mathit{\text{CPr}}\in 95\pm 1\%$ and ${\widetilde{ϵ}}_R\le 0.5$.

Table 11 95% CI sample size comparison.

$p^{*}$	n	W	CP	WS	AC
${10}^{-1}$	nC	$1.5·{10}^2$	$5.4·{10}^1$	$\mathbf{3}.\mathbf{6}·\mathbf{1}{\mathbf{0}}^{\mathbf{1}}$	$4.2·{10}^1$
	nD	$2.9·{10}^2$	$1.9·{10}^2$	$\mathbf{6}.\mathbf{2}·\mathbf{1}{\mathbf{0}}^{\mathbf{1}}$	$1.0·{10}^2$
	nE	$6.5·{10}^2$	$1.2·{10}^3$	$\mathbf{3}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{2}}$
${10}^{-2}$	nC	$1.5·{10}^3$	$7.1·{10}^2$	$\mathbf{4}.\mathbf{4}·\mathbf{1}{\mathbf{0}}^{\mathbf{2}}$	$5.0·{10}^2$
	nD	$2.8·{10}^3$	$2.0·{10}^3$	$\mathbf{7}.\mathbf{4}·\mathbf{1}{\mathbf{0}}^{\mathbf{2}}$	$1.2·{10}^3$
	nE	$8.0·{10}^3$	$1.0·{10}^4$	$\mathbf{2}.\mathbf{6}·\mathbf{1}{\mathbf{0}}^{\mathbf{3}}$	$4.1·{10}^3$
${10}^{-3}$	nC	$1.8·{10}^4$	$7.0·{10}^3$	$\mathbf{4}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{3}}$	$5.1·{10}^3$
	nD	$2.8·{10}^4$	$1.7·{10}^4$	$\mathbf{7}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{3}}$	$1.2·{10}^4$
	nE	$7.4·{10}^4$	$1.0·{10}^5$	$\mathbf{2}.\mathbf{6}·\mathbf{1}{\mathbf{0}}^{\mathbf{4}}$	$3.7·{10}^4$
${10}^{-4}$	nC	$1.8·{10}^5$	$7.0·{10}^4$	$\mathbf{4}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{4}}$	$5.1·{10}^4$
	nD	$2.8·{10}^5$	$1.6·{10}^5$	$\mathbf{7}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{4}}$	$1.2·{10}^5$
	nE	$7.4·{10}^5$	$1.0·{10}^6$	$\mathbf{2}.\mathbf{9}·\mathbf{1}{\mathbf{0}}^{\mathbf{5}}$	$4.7·{10}^5$
${10}^{-5}$	nC	$1.8·{10}^6$	$7.0·{10}^5$	$\mathbf{4}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{5}}$	$5.1·{10}^5$
	nD	$2.8·{10}^6$	$1.7·{10}^6$	$\mathbf{7}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{5}}$	$1.2·{10}^6$
	nE	$7.4·{10}^6$	$1.0·{10}^7$	$\mathbf{2}.\mathbf{9}·\mathbf{1}{\mathbf{0}}^{\mathbf{6}}$	$4.7·{10}^6$
${10}^{-6}$	nC	$1.8·{10}^7$	$7.0·{10}^6$	$\mathbf{4}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{6}}$	$5.1·{10}^6$
	nD	$2.8·{10}^7$	$1.7·{10}^7$	$\mathbf{7}.\mathbf{5}·\mathbf{1}{\mathbf{0}}^{\mathbf{6}}$	$1.2·{10}^7$
	nE	$7.4·{10}^7$	$1.0·{10}^8$	$\mathbf{2}.\mathbf{9}·\mathbf{1}{\mathbf{0}}^{\mathbf{7}}$	$4.7·{10}^7$

n_C: n₁ where $\mathit{\text{CPr}}\in 95\pm 3\%$ and ${\widetilde{ϵ}}_R\le 1\forall n\ge n_1$

n_D: n₁ where $\mathit{\text{CPr}}\in 95\pm 2\%$ and ${\widetilde{ϵ}}_R\le 0.75\forall n\ge n_1$

n_E: n₁ where $\mathit{\text{CPr}}\in 95\pm 1\%$ and ${\widetilde{ϵ}}_R\le 0.5\forall n\ge n_1$

Smallest sample size for each scheme shown in bold.

Table 12 ${\widetilde{ϵ}}_R$ values corresponding to maintained coverage.

CPr	W	CP	WS	AC
$95\pm 3\%$	0.46–0.50	0.80–0.87	0.98–1.00	0.97–1.00
$95\pm 2\%$	0.34–0.37	0.45–0.52	0.75–0.75	0.60–0.61
$95\pm 1\%$	0.22–0.23	0.17–0.20	0.31–0.39	0.29–0.33

${\widetilde{ϵ}}_R$ values observed across the $p^{*}$ range displayed as minimum–maximum.

Table 13 95% CI performance – $p^{*}=p={10}^{-2}$, small n.

n	W		CP		WS		AC
	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
20	18.2	1.80	98.3	9.19	98.3	8.70	98.3	11.36
40	33.1	1.73	99.3	5.22	93.9	5.08	99.3	6.57
60	45.2	1.65	97.8	3.80	97.8	3.72	97.8	4.72
80	55.2	1.57	99.1	3.07	95.3	3.00	99.1	3.74
100	63.3	1.50	98.2	2.62	92.1	2.55	98.2	3.13

Table 14 95% Wald CI – $\widehat{{ϵ}_R}$ comparison.

$\widehat{p}$	n	$\widehat{ϵ}$	$\widehat{{ϵ}_R}$	$\widehat{p}\pm \widehat{ϵ}$
0.137	6310	0.008	0.06	$[0.129,0.145]$
	605	0.027	0.20	$[0.110,0.164]$
	269	0.041	0.30	$[0.096,0.178]$
	152	0.055	0.40	$[0.082,0.192]$
	97	0.068	0.50	$[0.069,0.205]$

Table 15 Coverage comparison – fixed ϵ.

MoE Scheme	p	W	CP	WS	AC
1. $ϵ=4·{10}^{-2},p^{*}=p$	${10}^{-1}$	93.8	96.9	94.7	95.9
	${10}^{-2}$	21.4	97.6	97.6	97.6
	${10}^{-3}$	00.3	99.7	99.7	99.7
	${10}^{-4}$	00.0	100	100	100
	${10}^{-5}$	00.0	100	100	100
	${10}^{-6}$	00.0	100	100	100
2. $ϵ=4·{10}^{-2},p^{*}=0.5$	${10}^{-1}$	94.5	95.9	95.2	95.2
	${10}^{-2}$	93.1	96.3	94.1	97.8
	${10}^{-3}$	45.2	97.7	97.7	99.7
	${10}^{-4}$	05.8	99.8	94.2	100
	${10}^{-5}$	00.6	99.4	99.4	100
	${10}^{-6}$	00.1	99.9	99.4	100
3. $ϵ=4·{10}^{-4},p^{*}=p$	${10}^{-1}$	95.0	95.0	95.0	95.0
	${10}^{-2}$	95.0	95.1	95.0	95.1
	${10}^{-3}$	93.1	96.0	94.9	94.9
	${10}^{-4}$	21.3	97.5	97.5	99.8
	${10}^{-5}$	00.2	99.8	99.8	100
	${10}^{-6}$	00.0	100	100	100
4. $ϵ=4·{10}^{-4},p^{*}=0.5$	${10}^{-1}$	95.0	95.0	95.0	95.0
	${10}^{-2}$	95.0	95.0	95.0	95.0
	${10}^{-3}$	95.0	95.1	95.0	95.0
	${10}^{-4}$	94.9	95.2	95.0	95.0
	${10}^{-5}$	94.9	96.1	95.5	95.5
	${10}^{-6}$	93.4	96.3	94.0	97.7

Table 16 Coverage comparison – ${ϵ}_R=0.75$.

$p^{*}$	W	CP	WS	AC
${10}^{-1}$	94.7	97.0	94.6	94.6
${10}^{-2}$	90.2	97.0	94.9	97.0
${10}^{-3}$	90.4	96.9	94.6	96.9
${10}^{-4}$	90.4	96.9	94.6	96.9
${10}^{-5}$	90.4	96.9	94.6	96.9
${10}^{-6}$	90.4	96.9	94.6	96.9

Table 17 Comparison of classical and modern CI methods.

$p^{*}$	n	W		CP		WS		AC		GFI		Repro
		CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$	CPr	${\widetilde{ϵ}}_R$
${10}^{-1}$	20	87.6	1.17	98.9	1.46	95.7	1.32	95.7	1.46	97.9	1.31	98.2	1.42
	60	94.1	0.74	97.2	0.82	95.2	0.76	95.2	0.80	95.7	0.76	97.1	0.80
	10^2	93.2	0.58	95.6	0.63	93.6	0.59	97.2	0.61	96.1	0.59	95.8	0.61
	10^3	95.3	0.19	95.5	0.19	94.9	0.19	94.9	0.19	94.8	0.19	94.6	0.18
	10^4	95.1	0.06	95.1	0.06	94.9	0.06	94.9	0.06	94.9	0.06	94.9	0.06
${10}^{-2}$	20	18.2	1.80	98.3	9.19	98.3	8.70	98.3	11.36	98.6	7.84	98.5	9.02
	60	45.2	1.65	97.8	3.80	97.8	3.72	97.8	4.72	97.5	3.26	98.3	3.73
	10^2	63.3	1.50	98.2	2.62	92.1	2.55	98.2	3.13	98.3	2.27	98.2	2.57
	10^3	92.7	0.61	97.6	0.67	96.4	0.64	96.4	0.66	94.4	0.62	96.2	0.65
	10^4	94.6	0.19	95.6	0.20	95.0	0.20	95.0	0.20	94.6	0.19	94.9	0.19
${10}^{-3}$	20	02.0	1.90	98.0	85.01	98.0	81.22	100	> 100	98.2	69.93	98.2	83.91
	60	05.8	1.91	99.8	30.69	94.2	30.84	99.8	42.46	98.8	25.05	99.7	30.26
	10^2	09.5	1.89	99.5	19.00	90.5	19.27	99.5	26.50	99.4	15.58	99.5	18.82
	10^3	63.2	1.51	98.1	2.67	91.9	2.65	98.1	3.29	97.9	2.35	97.9	2.63
	10^4	92.6	0.61	97.5	0.67	96.3	0.64	96.3	0.67	94.4	0.63	96.5	0.66
${10}^{-4}$	20	00.2	1.91	99.8	> 100	99.8	> 100	100	> 100	99.8	> 100	99.8	> 100
	60	00.6	1.94	99.4	> 100	99.4	> 100	100	> 100	99.4	> 100	99.4	> 100
	10^2	01.0	1.94	99.0	> 100	99.0	> 100	100	> 100	99.3	> 100	98.9	> 100
	10^3	09.5	1.90	99.5	19.33	90.5	19.94	100	27.66	99.6	15.83	99.4	19.11
	10^4	63.2	1.52	98.1	2.68	91.9	2.66	98.1	3.29	98.2	2.34	97.4	2.65