Table 1.

Performance of our proposed estimation for ∆_EB(t; t₀) and the perturbation-resampling approach to obtain ${\widehat{\sigma }}_{EB}(t,t_0)$, in terms of bias, empirical standard error (ESE), average standard error (ASE) using ${\widehat{\sigma }}_{EB}(t,t_0)$, coverage of the 95% confidence intervals, and Type 1 error/power under three settings, setting (i) (null setting), setting (ii), and setting (iii); performance of ${\widehat{\sigma }}_{EB0}$ from Remark 1, where the standard error (SE) is derived under the null, and corresponding Type 1 error/power when this estimate is used for testing is also shown.

	Setting (i)
	t0 = 0.25	t0 = 0.50	t0 = 0.75
Bias	−0.0001	−0.0002	−0.0001
ESE	0.0072	0.0108	0.0132
ASE	0.0076	0.0107	0.0134
Coverage	0.9570	0.9460	0.9540
Type 1 error	0.0360	0.0480	0.0410
SE derived under the null
${\widehat{\sigma }}_{EB0}/\sqrt{n_B}$	0.0073	0.0106	0.0134
Type 1 error	0.0420	0.0520	0.0440
	Setting (ii)
	t0 = 0.25	t0 = 0.50	t0 = 0.75
Bias	0.0000	0.0002	0.0000
ESE	0.0065	0.0091	0.0114
ASE	0.0068	0.0095	0.0118
Coverage	0.9580	0.9600	0.9560
Power	0.3310	0.6230	0.7650
SE derived under the null
${\widehat{\sigma }}_{EB0}/\sqrt{n_B}$	0.0064	0.0092	0.0117
Power	0.3850	0.6520	0.7760
	Setting (iii)
	t0 = 0.25	t0 = 0.50	t0 = 0.75
Bias	0.0001	0.0001	0.0000
ESE	0.0061	0.0086	0.0108
ASE	0.0066	0.0091	0.0112
Coverage	0.9550	0.9570	0.9540
Power	0.5170	0.8410	0.9420
SE derived under the null
${\widehat{\sigma }}_{EB0}/\sqrt{n_B}$	0.0061	0.0088	0.0111
Power	0.5880	0.8590	0.9440