Table 3.

Simulation results under misspecification of the truncation distribution: Survival times simulated from a gamma(10, 1) distribution. Left and right truncation times assumed to come from a gamma(α₁, β₁) and gamma(α₂, β₂) distribution, respectively. Model 4 corresponds to misspecification of the right truncation time by simulating it as Unif(0, 20), and the left truncation time as gamma(3,1). Model 5 corresponds to misspecification of the left truncation time by simulating it as Weibull(1, 3), and the right truncation time as gamma(5, 2). Model 6 corresponds to misspecification both truncation times by simulating the left truncation time as Weibull(1, 3) and the right truncation time as Unif (0, 20). Here q_L, q_R, q are the proportion of observations missing due to left, right, and double (left and right) truncation, respectively, and n is the size of the observed sample. ${\widehat{F}}_{SP}$ 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 $\widehat{F}$, MSE($\widehat{F},\mathit{MSE}(\widehat{F})$) 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	qL, qR, q	n	Estimator	Bias($\widehat{F}$)	SD($\widehat{F}$)	$\widehat{\mathrm{SE}}(\widehat{F})$	MSE($\widehat{F}$)	Cov($\widehat{F}$)	Bias(${\widehat{t}}_{0.5}$)	SD(${\widehat{t}}_{0.5}$)
			${\widehat{F}}_{SP}$	0.008	0.074	0.160	0.006	0.968	0.091	0.725
4	0.02,0.50,0.51	50	${\widehat{F}}_{NP}$	0.023	0.075	0.068	0.006	0.878	0.017	0.734
			${\widehat{F}}_{\mathit{emp}}$	0.092	0.056	0.044	0.013	0.455	−0.848	0.461
			${\widehat{F}}_{SP}$	0.009	0.033	0.062	0.001	0.995	0.062	0.310
4	0.02,0.50,0.51	250	${\widehat{F}}_{NP}$	0.007	0.033	0.033	0.001	0.932	−0.014	0.299
			${\widehat{F}}_{\mathit{emp}}$	0.091	0.025	0.020	0.010	0.301	−0.816	0.207
			${\widehat{F}}_{SP}$	0.011	0.084	0.098	0.008	0.886	0.032	0.841
5	0.06,0.52,0.56	50	${\widehat{F}}_{NP}$	0.029	0.087	0.077	0.009	0.834	0.073	0.954
			${\widehat{F}}_{\mathit{emp}}$	0.144	0.054	0.042	0.026	0.441	−1.297	0.423
			${\widehat{F}}_{SP}$	0.007	0.038	0.037	0.002	0.921	−0.032	0.339
5	0.06,0.52,0.56	250	${\widehat{F}}_{NP}$	0.008	0.040	0.039	0.002	0.921	−0.021	0.353
			${\widehat{F}}_{\mathit{emp}}$	0.144	0.024	0.019	0.024	0.331	−1.288	0.201
			${\widehat{F}}_{SP}$	0.009	0.073	0.168	0.006	0.961	0.047	0.698
6	0.06,0.50,0.53	50	${\widehat{F}}_{NP}$	0.027	0.073	0.068	0.006	0.873	−0.030	0.659
			${\widehat{F}}_{\mathit{emp}}$	0.088	0.056	0.044	0.012	0.455	−0.826	0.476
			${\widehat{F}}_{SP}$	0.010	0.034	0.062	0.001	0.992	0.035	0.320
6	0.06,0.50,0.53	250	${\widehat{F}}_{NP}$	0.009	0.035	0.032	0.001	0.924	−0.039	0.365
			${\widehat{F}}_{\mathit{emp}}$	0.086	0.025	0.020	0.009	0.301	−0.771	0.210