Table 4.

Simulation results. Proportions of rejection of three test statistics for left-truncated data without satisfying stationarity assumption based on sample size 100 per group with 5000 simulations

X1 ~ Weibull(2,1), $X_2\sim \text{Weibull}(2,1/\sqrt{\beta })$
β	Cen1%	Cen2%	Prop. of rejection	Prop. of rejection	Prop. of rejection
			Proposed test	Truncation logrank	Standard Logrank
			$T_1^{\ast }$	TLT	TLR
A1 ~ Exponential(1/10), A2 ~ Exponential(1/10)
No censoring
1	0%	0%	0.048	0.048	0.049
1.2	0%	0%	0.356	0.233	0.353
1.4	0%	0%	0.820	0.648	0.829
0.8	0%	0%	0.461	0.318	0.452
0.6	0%	0%	0.990	0.941	0.989
Same censoring distributions SC1 = SC2
1	20%	20%	0.050	0.049	0.052
1.2	20%	19%	0.295	0.194	0.291
1.4	20%	17%	0.735	0.521	0.732
0.8	20%	23%	0.419	0.263	0.410
0.6	20%	26%	0.982	0.865	0.983
1	40%	40%	0.051	0.052	0.053
1.2	40%	37%	0.252	0.146	0.237
1.4	40%	34%	0.698	0.454	0.685
0.8	40%	44%	0.366	0.223	0.366
0.6	40%	49%	0.946	0.721	0.937
A1 ~ Uniform(0, 20), A2 ~ Exponential(1/10)
No censoring
1	0%	0%	0.054	0.048	0.056
1.2	0%	0%	0.267	0.242	0.265
1.4	0%	0%	0.763	0.649	0.775
0.8	0%	0%	0.572	0.316	0.567
0.6	0%	0%	0.999	0.928	0.998
Same censoring distributions SC1 = SC2
1	20%	20%	0.055	0.052	0.058
1.2	20%	19%	0.244	0.199	0.242
1.4	20%	17%	0.702	0.568	0.700
0.8	20%	23%	0.504	0.278	0.495
0.6	20%	26%	0.984	0.862	0.979
1	40%	40%	0.056	0.053	0.055
1.2	40%	37%	0.212	0.176	0.220
1.4	40%	34%	0.631	0.460	0.624
0.8	40%	44%	0.422	0.199	0.400
0.6	40%	49%	0.969	0.739	0.959