Table 4.

Simulation results under violation of the independence assumption: Survival and truncation times simulated from a normal copula with correlations ρ_LT , ρ_LR, and ρ_TR, where ρ_XY denotes the correlation between random variables X and Y. The marginal distributions for the survival, left, and right truncation times are set to gamma(10, 1), gamma(3, 1), and gamma(5, 2) distributions, respectively. Here q_L,q_R, q are the proportion of observations missing due to left, right, and double (left and right) truncation, respectively. The average observed sample size is n = 250 for all models. ${\widehat{F}}_{SP}$ denotes the SPMLE, ${\widehat{F}}_{NP}$ denotes the NPMLE, and ${\widehat{F}}_{\mathit{emp}}$ denotes the naïve empirical CDF which ignores double truncation. These estimators were all computed at t_0.1, …, t_0.9, the 1st through 9th deciles of the true survival distribution F₀. For a given estimator $\widehat{F}$, Bias($\widehat{F}$) is the (absolute) difference between $\widehat{F}$ and F₀, averaged across the 9 deciles. Here SD($\widehat{F}$) is standard deviation of $\widehat{F}$ across simulations, $\widehat{SE}(\widehat{F})$ is estimated standard error of $\widehat{F}$, $\mathit{MSE}(\widehat{F})$ is mean squared error of $\widehat{F}$, and Cov($\widehat{F}$) is 95% coverage, all averaged across the 9 deciles. Here ${\widehat{t}}_{0.5}$ is the estimated median value based on $\widehat{F}$. The true median value based on F₀ is t_0.5 = 9:7. Here Bias(${\widehat{t}}_{0.5}$) = ${\widehat{t}}_{0.5}$ - t_0.5 and SD(${\widehat{t}}_{0.5}$) is the standard deviation of ${\widehat{t}}_{0.5}$ across simulations.

Model	ρLT, ρLR, ρRT	qL, qR, q	Estimator	Bias($\widehat{F}$)	SD($\widehat{F}$)	$\widehat{SE}(\widehat{F})$	MSE($\widehat{F}$)	Cov($\widehat{F}$)	Bias(${\widehat{t}}_{0.5}$)	SD(${\widehat{t}}_{0.5}$)
			${\widehat{F}}_{SP}$	0.049	0.038	0.046	0.005	0.744	−0.458	0.343
7	0.5,0.1,0.1	0.09,0.35,0.43	${\widehat{F}}_{NP}$	0.055	0.038	0.024	0.005	0.380	−0.507	0.332
			${\widehat{F}}_{\mathit{emp}}$	0.102	0.025	0.020	0.012	0.315	−0.896	0.202
			${\widehat{F}}_{SP}$	0.042	0.033	0.046	0.004	0.701	0.336	0.268
8	−0.5,0.1,−0.1	0.22,0.37,0.56	${\widehat{F}}_{NP}$	0.044	0.034	0.022	0.004	0.447	0.374	0.276
			${\widehat{F}}_{\mathit{emp}}$	0.068	0.023	0.018	0.006	0.293	−0.285	0.171
			${\widehat{F}}_{SP}$	0.119	0.029	0.058	0.018	0.478	−0.968	0.218
9	−0.1,0.1,−0.5	0.18,0.39,0.55	${\widehat{F}}_{NP}$	0.121	0.029	0.026	0.018	0.101	−0.982	0.223
			${\widehat{F}}_{\mathit{emp}}$	0.124	0.022	0.017	0.021	0.262	−0.942	0.156
			${\widehat{F}}_{SP}$	0.083	0.027	0.055	0.010	0.579	−0.503	0.189
10	−0.5,0.1,−0.5	0.22,0.39,0.58	${\widehat{F}}_{NP}$	0.079	0.028	0.025	0.009	0.279	−0.488	0.195
			${\widehat{F}}_{\mathit{emp}}$	0.106	0.021	0.017	0.016	0.223	−0.638	0.146