Table 1. Simulation results.

q is the proportion of observations missing due to truncation and n is the size of the observed sample. ${\widehat{\beta }}_{uw}$ denotes the naïve unweighted estimator, ${\widehat{\beta }}_{u_{\widehat{\theta }}}$ denotes the proposed parametric weighted estimator, ${\widehat{\beta }}_{w_{np}}$ denotes the proposed nonparametric weighted estimator, and ${\widehat{\beta }}_{cc}$ denotes the unattainable complete case estimator based on both truncated and non-truncated observations. SD is the empirical standard deviation of estimates across simulations, $\widehat{SE}$ is the average of the estimated standard errors, Cov is the coverage of 95% confidence intervals. The true value of β is 1.

q	n	Estimator	Bias	SD	$\widehat{\text{SE}}$	Cov
	50	${\widehat{\beta }}_{uw}$	−0.081	0.574	0.545	0.943
0.20	50	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	−0.011	0.616	0.552	0.927
	50	${\widehat{\beta }}_{w_{np}}$	−0.015	0.616	0.620	0.937
	63	${\widehat{\beta }}_{cc}$	0.003	0.504	0.475	0.943
	100	${\widehat{\beta }}_{uw}$	−0.071	0.375	0.371	0.945
	100	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.003	0.405	0.374	0.940
	100	${\widehat{\beta }}_{w_{np}}$	0.000	0.406	0.408	0.943
	125	${\widehat{\beta }}_{cc}$	−0.005	0.340	0.328	0.941
	250	${\widehat{\beta }}_{uw}$	−0.066	0.235	0.231	0.938
	250	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.007	0.254	0.232	0.925
	250	${\widehat{\beta }}_{w_{np}}$	0.004	0.254	0.250	0.945
	313	${\widehat{\beta }}_{cc}$	0.011	0.205	0.205	0.951
	50	${\widehat{\beta }}_{uw}$	−0.031	0.548	0.536	0.957
0.40	50	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.053	0.593	0.551	0.935
	50	${\widehat{\beta }}_{w_{np}}$	0.045	0.605	0.626	0.934
	83	${\widehat{\beta }}_{cc}$	0.047	0.423	0.404	0.949
	100	${\widehat{\beta }}_{uw}$	−0.092	0.381	0.370	0.939
	100	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	−0.006	0.424	0.381	0.936
	100	${\widehat{\beta }}_{w_{np}}$	−0.009	0.426	0.419	0.938
	167	${\widehat{\beta }}_{cc}$	0.008	0.274	0.282	0.958
	250	${\widehat{\beta }}_{uw}$	−0.084	0.235	0.231	0.927
	250	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.005	0.263	0.235	0.922
	250	${\widehat{\beta }}_{w_{np}}$	0.004	0.266	0.258	0.944
	417	${\widehat{\beta }}_{cc}$	0.008	0.180	0.177	0.948
	50	${\widehat{\beta }}_{uw}$	0.139	0.562	0.542	0.937
0.60	50	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.041	0.547	0.561	0.950
	50	${\widehat{\beta }}_{w_{np}}$	0.034	0.555	0.580	0.939
	125	${\widehat{\beta }}_{cc}$	0.005	0.338	0.326	0.947
	100	${\widehat{\beta }}_{uw}$	0.122	0.374	0.372	0.949
	100	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.014	0.361	0.392	0.970
	100	${\widehat{\beta }}_{w_{np}}$	0.011	0.363	0.382	0.955
	250	${\widehat{\beta }}_{cc}$	−0.004	0.234	0.228	0.936
	250	${\widehat{\beta }}_{uw}$	0.111	0.244	0.232	0.911
	250	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.013	0.234	0.249	0.964
	250	${\widehat{\beta }}_{w_{np}}$	0.005	0.237	0.234	0.937
	625	${\widehat{\beta }}_{cc}$	0.006	0.150	0.144	0.947
	50	${\widehat{\beta }}_{uw}$	−0.127	0.560	0.538	0.937
0.80	50	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	−0.015	0.666	0.633	0.940
	50	${\widehat{\beta }}_{w_{np}}$	−0.004	0.724	0.701	0.947
	250	${\widehat{\beta }}_{cc}$	0.008	0.226	0.233	0.961
	100	${\widehat{\beta }}_{uw}$	−0.122	0.373	0.367	0.940
	100	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	0.013	0.472	0.456	0.924
	100	${\widehat{\beta }}_{w_{np}}$	0.016	0.493	0.472	0.949
	500	${\widehat{\beta }}_{cc}$	0.006	0.162	0.164	0.955
	250	${\widehat{\beta }}_{uw}$	−0.163	0.236	0.228	0.878
	250	${\widehat{\beta }}_{w_{\widehat{\theta }}}$	−0.021	0.316	0.328	0.913
	250	${\widehat{\beta }}_{w_{np}}$	−0.019	0.315	0.294	0.927
	1250	${\widehat{\beta }}_{cc}$	0.000	0.104	0.103	0.949