Table II.

Simulation results for ${\beta }_1$ under (M2) and (M3).

$n$	${\eta }_{i1}$ a	$Q_i$	${\eta }_{i2}$ b	Lin			Bůžková			Liang (extension)			Weighted-Liang (extension)			Sun			Weighted-Sun (extension)
				Bias	ESE	MSE	Bias	ESE	MSE	Bias	ESE	MSE	Bias	ESE	MSE	Bias	ESE	MSE	Bias	ESE	MSE
100	${\eta }_{i1}^{(1)}$	1	${\eta }_{i2}^{(1)}$	−0.155	0.430	0.209	−0.030	0.426	0.182	−0.603	0.496	0.611	0.007	0.421	0.177	−0.595	0.494	0.598	0.016	0.407	0.166
			${\eta }_{i2}^{(2)}$	0.335	0.426	0.294	0.469	0.418	0.394	−0.281	0.464	0.294	0.116	0.405	0.178	−0.305	0.462	0.306	0.008	0.398	0.158
	${\eta }_{i1}^{(2)}$	1	${\eta }_{i2}^{(1)}$	−0.171	0.488	0.267	−0.041	0.484	0.236	−0.592	0.539	0.641	0.002	0.481	0.231	−0.585	0.536	0.630	0.014	0.472	0.223
			${\eta }_{i2}^{(2)}$	0.356	0.528	0.406	0.490	0.528	0.519	−0.223	0.541	0.342	0.134	0.470	0.239	−0.247	0.540	0.353	0.025	0.465	0.217
		$X_{i1}$	${\eta }_{i2}^{(1)}$	0.104	0.415	0.183	0.232	0.411	0.223	−0.358	0.481	0.359	0.004	0.445	0.198	−0.284	0.493	0.324	0.250	0.442	0.258
			${\eta }_{i2}^{(2)}$	0.348	0.483	0.354	0.486	0.485	0.472	−0.233	0.504	0.309	0.045	0.443	0.198	−0.163	0.506	0.283	0.127	0.438	0.208
200	${\eta }_{i1}^{(1)}$	1	${\eta }_{i2}^{(1)}$	−0.166	0.307	0.122	−0.025	0.310	0.097	−0.621	0.351	0.509	0.012	0.302	0.092	−0.615	0.349	0.500	0.018	0.292	0.085
			${\eta }_{i2}^{(2)}$	0.336	0.311	0.209	0.484	0.301	0.325	−0.285	0.342	0.198	0.113	0.294	0.100	−0.304	0.341	0.209	0.007	0.288	0.083
	${\eta }_{i1}^{(2)}$	1	${\eta }_{i2}^{(1)}$	−0.179	0.342	0.149	−0.043	0.350	0.124	−0.607	0.387	0.518	−0.004	0.353	0.124	−0.601	0.385	0.510	0.004	0.346	0.120
			${\eta }_{i2}^{(2)}$	0.338	0.372	0.253	0.492	0.377	0.384	−0.240	0.390	0.210	0.104	0.346	0.131	−0.258	0.391	0.220	−0.002	0.343	0.118
		$X_{i1}$	${\eta }_{i2}^{(1)}$	0.100	0.297	0.098	0.235	0.303	0.147	−0.361	0.352	0.254	0.003	0.324	0.105	−0.289	0.358	0.211	0.257	0.329	0.174
			${\eta }_{i2}^{(2)}$	0.335	0.339	0.227	0.490	0.344	0.358	−0.237	0.367	0.190	0.024	0.323	0.105	−0.168	0.365	0.162	0.110	0.320	0.114

Bias, ${\hat{\beta }}_1-{\beta }_1$, ${\beta }_1=1$; ESE, empirical sample error; MSE, mean-squared error.

^aTwo possible links: ${\eta }_{i1}^{(1)}:{\eta }_{i1}={\eta }_{i2}$; ${\eta }_{i1}^{(2)}:\mathrm{E}\left[{\eta }_{i1}\mid {\eta }_{i2}\right]=\theta \left({\eta }_{i2}-1\right)$, $\theta =1$.

^bLatent vaiiable distributions: ${\eta }_{i2}^{(1)}:{\eta }_{i2}\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left.=1,{\sigma }^2=0.5\right)$; ${\eta }_{i2}^{(2)}:{\eta }_{i2}\sim I\left(X_{i1}⩽0.5\right)\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}[\mathrm{0.5,1.5}]+I\left(X_{i1}>0.5\right)\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\mathrm{1,0.7})$.