Table 2.

Comparing the performance measures of the FLGI rule when the variance is incorrectly assumed to be known (and equal to $0.64^2$), in at least one of the two arms, with those obtained when the variance is assumed unknown (but with an initial estimate, ${\tilde{s}}_{k,0}^2$, of $0.64^2$)

	${\mathit{\mu }}_0={\mathit{\mu }}_1=0.155$					${\mathit{\mu }}_0=0.155,{\mathit{\mu }}_1=0.529$
$\mathit{b}$	${\mathit{t}}_{1-\mathit{\alpha }}$	$\mathit{\alpha }$	$\mathbb{E}\left({\mathit{p}}^{*}\right)$ (SD)	Rel $\mathit{E}\mathit{T}\mathit{O}\%$ (SD)	Bias (SD)	$1-\beta$	$\mathbb{E}\left({\mathit{p}}^{*}\right)$ (SD)	Rel $\mathit{E}\mathit{T}\mathit{O}\%$ (SD)	Bias (SD)
(i) FLGI‐known with ${\sigma }_0^2={\sigma }_1^2=\frac{1}{2}\times 0.64^2$
1	2.001	0.0509	0.4983 (0.27)	–0.04 (3.84)	0.0013 (0.26)	0.4722	0.9215 (0.07)	46.01 (4.34)	0.1430 (0.29)
2	1.977	0.0505	0.5010 (0.26)	0.17 (3.83)	–0.0007 (0.24)	0.5422	0.9142 (0.07)	45.22 (4.32)	0.1209 (0.26)
6	1.925	0.0525	0.4986 (0.24)	0.12 (3.85)	0.0009 (0.20)	0.6642	0.8952 (0.07)	43.29 (4.33)	0.0771 (0.21)
9	1.887	0.0518	0.5002 (0.23)	0.07 (3.82)	–0.0004 (0.18)	0.7241	0.8814 (0.07)	41.77 (4.30)	0.0545 (0.18)
18	1.798	0.0509	0.4999 (0.21)	0.14 (3.83)	0.0010 (0.15)	0.8406	0.8370 (0.07)	36.90 (4.29)	0.0224 (0.14)
36	1.698	0.0500	0.5006 (0.16)	–0.03 (3.83)	–0.0005 (0.12)	0.9341	0.7331 (0.06)	25.53 (4.19)	0.0050 (0.12)
(ii) FLGI with ${\sigma }_0^2={\sigma }_1^2=\frac{1}{2}\times 0.64^2$
1	2.280	0.0502	0.5014 (0.30)	–0.04 (3.84)	0.0017 (0.21)	0.4301	0.9177 (0.07)	45.60 (4.33)	0.0641 (0.20)
2	2.209	0.0484	0.4987 (0.29)	0.04 (3.83)	–0.0013 (0.20)	0.4713	0.9131 (0.07)	45.15 (4.32)	0.0602 (0.20)
6	2.132	0.0491	0.4987 (0.28)	0.01 (3.81)	–0.0017 (0.19)	0.5520	0.9008 (0.07)	43.88 (4.35)	0.0537 (0.19)
9	2.080	0.0505	0.4991 (0.27)	0.04 (3.84)	–0.0007 (0.18)	0.5958	0.8894 (0.07)	42.55 (4.33)	0.0427 (0.18)
18	1.897	0.0516	0.5004 (0.24)	0.04 (3.84)	–0.0004 (0.16)	0.7604	0.8466 (0.07)	37.95 (4.29)	0.0186 (0.15)
36	1.751	0.0501	0.5000 (0.19)	–0.22 (3.83)	–0.0002 (0.12)	0.9110	0.7401 (0.06)	26.33 (4.18)	0.0035 (0.12)
(iii) FLGI‐known with ${\sigma }_0^2={\sigma }_1^2=2\times 0.64^2$
1	1.940	0.0505	0.4998 (0.37)	–0.04 (7.69)	0.0024 (0.81)	0.1517	0.8106 (0.27)	34.25 (10.54)	0.2656 (0.74)
2	1.924	0.0481	0.4997 (0.36)	0.13 (7.64)	–0.0015 (0.72)	0.1719	0.8116 (0.26)	34.29 (10.29)	0.2279 (0.66)
6	1.865	0.0484	0.4999 (0.33)	–0.28 (7.70)	–0.0008 (0.54)	0.2233	0.8018 (0.23)	33.06 (9.85)	0.1441 (0.48)
9	1.790	0.0529	0.4998 (0.31)	–0.21 (7.69)	–0.0009 (0.45)	0.2730	0.7935 (0.21)	32.21 (9.55)	0.1068 (0.40)
18	1.747	0.0514	0.4998 (0.27)	0.49 (7.70)	0.0006 (0.33)	0.3454	0.7593 (0.18)	28.34 (9.10)	0.0487 (0.30)
36	1.677	0.0497	0.5003 (0.20)	–0.03 (7.66)	0.0050 (0.25)	0.4532	0.6837 (0.14)	20.27 (8.53)	0.0108 (0.24)
(iv) FLGI with ${\sigma }_0^2={\sigma }_1^2=2\times 0.64^2$
1	2.386	0.0518	0.5013 (0.32)	–0.04 (7.69)	0.0009 (0.45)	0.1944	0.8117 (0.21)	34.37 (9.55)	0.1326 (0.41)
2	2.323	0.0501	0.4975 (0.31)	–0.21 (7.64)	–0.0037 (0.43)	0.2047	0.8088 (0.21)	33.57 (9.50)	0.1292 (0.4)
6	2.255	0.0502	0.5008 (0.29)	0.05 (7.68)	0.0000 (0.40)	0.2184	0.7945 (0.20)	32.22 (9.37)	0.1117 (0.38)
9	2.138	0.0525	0.5004 (0.28)	–0.07 (7.69)	0.0016 (0.38)	0.2413	0.7827 (0.19)	31.01 (9.27)	0.0936 (0.35)
18	1.888	0.0514	0.4974 (0.25)	–0.02 (7.63)	–0.0028 (0.32)	0.3346	0.7511 (0.17)	27.18 (8.97)	0.0517 (0.29)
36	1.753	0.0480	0.5002 (0.19)	0.13 (7.69)	–0.0012 (0.25)	0.4470	0.6748 (0.13)	19.11 (8.48)	0.0145 (0.24)
Heterogeneous variances
(v) FLGI‐known with ${\sigma }_0^2=0.64^2,{\sigma }_1^2=\frac{1}{2}\times 0.64^2$
1	1.9900	0.0502	0.4284 (0.30)	0.03 (4.61)	0.1030 (0.38)	0.1828	0.9203 (0.09)	45.84 (4.32)	0.2522 (0.41)
2	1.9710	0.0487	0.4358 (0.29)	–0.35 (4.60)	0.0902 (0.35)	0.2200	0.9102 (0.09)	44.85 (4.32)	0.2038 (0.37)
6	1.9170	0.0487	0.4587 (0.27)	0.26 (4.63)	0.0583 (0.28)	0.3399	0.8857 (0.09)	42.06 (4.35)	0.1271 (0.28)
9	1.8730	0.0508	0.4666 (0.26)	–0.08 (4.65)	0.0477 (0.24)	0.4222	0.8703 (0.09)	40.49 (4.36)	0.0952 (0.24)
18	1.7890	0.0532	0.4836 (0.23)	0.09 (4.68)	0.0280 (0.19)	0.5893	0.8240 (0.09)	35.37 (4.43)	0.0427 (0.18)
36	1.7060	0.0533	0.4992 (0.18)	0.07 (4.70)	0.0131 (0.15)	0.7697	0.7237 (0.08)	24.42 (4.56)	0.0117 (0.15)
(vi) FLGI with ${\sigma }_0^2=0.64^2,{\sigma }_1^2=\frac{1}{2}\times 0.64^2$
1	2.425	0.0489	0.4822 (0.31)	–0.07 (4.72)	0.0341 (0.26)	0.2655	0.8897 (0.11)	42.51 (4.58)	0.1104 (0.26)
2	2.331	0.0524	0.4789 (0.30)	–0.07 (4.69)	0.0323 (0.25)	0.2969	0.8833 (0.11)	41.94 (4.59)	0.1044 (0.25)
6	2.266	0.0485	0.4752 (0.28)	0.05 (4.74)	0.0282 (0.24)	0.3431	0.8674 (0.11)	40.08 (4.62)	0.0927 (0.24)
9	2.185	0.0510	0.4722 (0.28)	–0.37 (4.72)	0.0263 (0.23)	0.3861	0.8562 (0.11)	38.81 (4.62)	0.0813 (0.23)
18	1.977	0.0489	0.4721 (0.25)	0.06 (4.75)	0.0197 (0.19)	0.5261	0.8158 (0.10)	34.69 (4.60)	0.0418 (0.19)
36	1.778	0.0511	0.4786 (0.19)	–0.04 (4.72)	0.0120 (0.15)	0.7429	0.7226 (0.09)	24.28 (4.60)	0.0111 (0.15)
(vii) FLGI‐known with ${\sigma }_0^2=0.64^2,{\sigma }_1^2=2\times 0.64^2$
1	1.981	0.0489	0.5892 (0.34)	0.30 (6.45)	–0.1878 (0.65)	0.2669	0.7882 (0.29)	31.44 (11.25)	0.0885 (0.64)
2	1.941	0.0524	0.5747 (0.34)	0.33 (6.51)	–0.1584 (0.59)	0.3005	0.7958 (0.27)	32.39 (10.90)	0.0819 (0.55)
6	1.855	0.0519	0.5517 (0.31)	0.01 (6.53)	–0.1035 (0.45)	0.3741	0.8034 (0.23)	33.30 (10.18)	0.0586 (0.40)
9	1.821	0.0482	0.5422 (0.30)	–0.03 (6.56)	–0.0806 (0.38)	0.4119	0.7995 (0.21)	32.71 (9.84)	0.0454 (0.33)
18	1.735	0.0489	0.5194 (0.26)	–0.34 (6.59)	–0.0469 (0.28)	0.5105	0.7758 (0.17)	30.16 (9.15)	0.0197 (0.24)
36	1.665	0.0475	0.5007 (0.20)	–0.04 (6.63)	–0.0225 (0.22)	0.6098	0.6982 (0.12)	21.74 (8.20)	0.0004 (0.20)
(viii) FLGI with ${\sigma }_0^2=0.64^2,{\sigma }_1^2=2\times 0.64^2$
1	2.238	0.0517	0.5204 (0.32)	–0.05 (6.70)	–0.0480 (0.38)	0.2806	0.8522 (0.19)	38.78 (9.50)	0.0774 (0.34)
2	2.207	0.0484	0.5249 (0.31)	0.24 (6.65)	–0.0433 (0.37)	0.2885	0.8507 (0.18)	38.63 (9.40)	0.0710 (0.32)
6	2.095	0.0509	0.5290 (0.29)	–0.06 (6.69)	–0.0377 (0.34)	0.3353	0.8405 (0.17)	37.31 (9.28)	0.0637 (0.30)
9	2.039	0.0491	0.5297 (0.28)	–0.07 (6.71)	–0.0367 (0.33)	0.3554	0.8299 (0.16)	36.31 (9.14)	0.0501 (0.28)
18	1.839	0.0497	0.5284 (0.25)	0.05 (6.72)	–0.0296 (0.27)	0.4708	0.7943 (0.14)	32.01 (8.71)	0.0220 (0.23)
36	1.700	0.0489	0.5222 (0.19)	0.15 (6.70)	–0.0182 (0.21)	0.6050	0.7017 (0.11)	21.95 (8.05)	0.0030 (0.20)

Note: In the upper half of the table, the true variance of the response is actually half or double 0.64² (as shown) and in the lower half, the true variances are heterogeneous. These results are averaged over 50 000 replications for a trial of size T = 72.

Abbreviation: FLGI, Forward‐Looking Gittins Index.