Table 2 Simulation results with interval-censored data, including the estimated bias (Bias), the sample standard error (SSE) of the estimates, the average of the standard error estimates (SEE), and the 95% empirical coverage probability (CP)
Proposed method | CL method | Ignoring truncation | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
n | Par | True | Bias | SSE | SEE | CP | Bias | SSE | SEE | CP | Bias | SSE | SEE | CP | ||
\(A^{*}\) follows the uniform distribution | ||||||||||||||||
100 | \(\beta _1\) | 1 | 0.057 | 0.253 | 0.258 | 96.1 | 0.100 | 0.302 | 0.262 | 89.5 | 0.212 | 0.290 | 0.323 | 93.3 | ||
\(\beta _2\) | 1 | 0.079 | 0.408 | 0.411 | 95.2 | 0.114 | 0.510 | 0.427 | 87.4 | 0.214 | 0.471 | 0.451 | 88.2 | |||
\(\Lambda (0.4)\) | 0.16 | 0.013 | 0.103 | 0.104 | 94.0 | 0.009 | 0.103 | – | – | -0.076 | 0.052 | – | – | |||
\(\Lambda (0.8)\) | 0.64 | -0.053 | 0.186 | 0.184 | 93.9 | -0.039 | 0.185 | – | – | -0.236 | 0.113 | – | – | |||
\(\Lambda (1.2)\) | 1.44 | -0.033 | 0.306 | 0.301 | 92.1 | -0.065 | 0.308 | – | – | -0.431 | 0.230 | – | – | |||
300 | \(\beta _1\) | 1 | 0.025 | 0.134 | 0.132 | 93.9 | 0.042 | 0.155 | 0.150 | 93.1 | 0.154 | 0.149 | 0.209 | 90.3 | ||
\(\beta _2\) | 1 | 0.016 | 0.212 | 0.214 | 95.3 | 0.032 | 0.249 | 0.244 | 93.2 | 0.169 | 0.250 | 0.240 | 83.4 | |||
\(\Lambda (0.4)\) | 0.16 | 0.023 | 0.075 | 0.073 | 96.5 | 0.022 | 0.076 | – | – | -0.068 | 0.034 | – | – | |||
\(\Lambda (0.8)\) | 0.64 | -0.006 | 0.131 | 0.132 | 94.2 | 0.002 | 0.127 | – | – | -0.224 | 0.069 | – | – | |||
\(\Lambda (1.2)\) | 1.44 | -0.021 | 0.229 | 0.222 | 95.2 | -0.017 | 0.225 | – | – | -0.427 | 0.139 | – | – | |||
500 | \(\beta _1\) | 1 | 0.005 | 0.100 | 0.100 | 96.1 | 0.017 | 0.121 | 0.115 | 94.3 | 0.132 | 0.107 | 0.189 | 86.6 | ||
\(\beta _2\) | 1 | 0.022 | 0.165 | 0.163 | 93.6 | 0.036 | 0.202 | 0.189 | 91.8 | 0.148 | 0.188 | 0.179 | 75.4 | |||
\(\Lambda (0.4)\) | 0.16 | 0.019 | 0.059 | 0.055 | 93.9 | 0.018 | 0.063 | – | – | -0.066 | 0.028 | – | – | |||
\(\Lambda (0.8)\) | 0.64 | -0.014 | 0.098 | 0.101 | 95.1 | -0.008 | 0.097 | – | – | -0.222 | 0.057 | – | – | |||
\(\Lambda (1.2)\) | 1.44 | -0.022 | 0.186 | 0.182 | 94.6 | -0.019 | 0.189 | – | – | -0.429 | 0.107 | – | – | |||
\(A^{*}\) follows the exponential distribution | ||||||||||||||||
100 | \(\beta _1\) | 1 | 0.084 | 0.250 | 0.266 | 96.8 | 0.115 | 0.292 | 0.259 | 88.9 | 0.164 | 0.273 | 0.309 | 94.2 | ||
\(\beta _2\) | 1 | 0.084 | 0.411 | 0.428 | 96.3 | 0.125 | 0.484 | 0.424 | 88.9 | 0.162 | 0.449 | 0.437 | 90.9 | |||
\(\Lambda (0.4)\) | 0.16 | 0.008 | 0.096 | 0.101 | 96.2 | 0.007 | 0.097 | – | – | -0.062 | 0.056 | – | – | |||
\(\Lambda (0.8)\) | 0.64 | -0.046 | 0.178 | 0.174 | 93.6 | -0.05 | 0.179 | – | – | -0.169 | 0.134 | – | – | |||
\(\Lambda (1.2)\) | 1.44 | -0.043 | 0.289 | 0.303 | 93.5 | -0.037 | 0.290 | – | – | -0.267 | 0.272 | – | – | |||
300 | \(\beta _1\) | 1 | 0.030 | 0.136 | 0.135 | 94.3 | 0.046 | 0.152 | 0.145 | 92.5 | 0.126 | 0.145 | 0.179 | 91.4 | ||
\(\beta _2\) | 1 | 0.019 | 0.223 | 0.221 | 94.9 | 0.034 | 0.254 | 0.239 | 93.0 | 0.136 | 0.248 | 0.218 | 82.1 | |||
\(\Lambda (0.4)\) | 0.16 | 0.021 | 0.066 | 0.068 | 95.9 | 0.023 | 0.067 | – | – | -0.056 | 0.035 | – | – | |||
\(\Lambda (0.8)\) | 0.64 | -0.012 | 0.117 | 0.116 | 94.4 | -0.005 | 0.119 | – | – | -0.169 | 0.079 | – | – | |||
\(\Lambda (1.2)\) | 1.44 | -0.027 | 0.206 | 0.211 | 95.1 | -0.235 | 0.205 | – | – | -0.288 | 0.161 | – | – | |||
500 | \(\beta _1\) | 1 | 0.014 | 0.105 | 0.102 | 94.5 | 0.024 | 0.116 | 0.111 | 93.5 | 0.100 | 0.110 | 0.187 | 90.4 | ||
\(\beta _2\) | 1 | 0.017 | 0.168 | 0.168 | 95.0 | 0.027 | 0.191 | 0.185 | 94.2 | 0.103 | 0.190 | 0.185 | 82.7 | |||
\(\Lambda (0.4)\) | 0.16 | 0.019 | 0.054 | 0.052 | 97.7 | 0.017 | 0.054 | – | – | -0.055 | 0.027 | – | – | |||
\(\Lambda (0.8)\) | 0.64 | -0.006 | 0.096 | 0.099 | 96.5 | -0.003 | 0.094 | – | – | -0.169 | 0.063 | – | – | |||
\(\Lambda (1.2)\) | 1.44 | -0.002 | 0.184 | 0.184 | 95.1 | -0.200 | 0.186 | – | – | -0.288 | 0.131 | – | – | |||