In this paper, we consider statistical inferences on the estimation of the parameters of a Weibull distribution when data are randomly censored. Maximum likelihood estimators (MLEs) and approximate MLEs are derived to estimate the parameters. We consider two cases for the censoring model: the assumption that the censoring distribution does not involve any parameters of interest and a censoring distribution that follows a Weibull distribution. A simulation study is conducted to compare the performances of the estimators. The result shows that the MLEs and the approximate MLEs are similar in terms of biases and mean square errors; in addition, the assumption of the censoring model has a strong influence on the estimation of scale parameter.
In survival studies, data may be subject to random censoring. Let
For a model of random censorship, Koziol and Green (1976) introduced a special model with
Under this model, we have
Hence,
Suppose the lifetime distribution follows a Weibull distribution Weibull(
and the pdf
where
This paper derives the maximum likelihood estimators (MLEs) and the approximate MLEs of the parameters of a Weibull distribution under random censoring. The analysis of censored data usually assumes that the censoring distribution does not involve any parameters of interest. We find the MLEs and the approximate MLEs in this case and those under the Koziol-Green model in (
For the observed random pairs (
where
If the lifetime distribution follows Weibull(
The MLEs of
where
Substituting back
Balakrishnan and Kateri (2008) discussed the existence and uniqueness of the MLEs of a Weibull distribution. The
If the random variable
where
Pareek
Letting
Remind that
when
Since
by taking derivatives with respect to
Clearly, (
for some
with
From (
with
where
Since
is the only positive root of the quadratic
Let us think about the plotting position
which has been studied in Kaplan and Meier (1958), Efron (1967), Breslow and Crowley (1974), and Meier (1975). The modified one is
We put the part (
If we assume the Koziol-Green model (
which is also the likelihood equation of
As for the approximate MLE of
and the quadratic equation in
where
and
and
In this section, a simulation study is conducted to compare the performance of the MLEs with the approximate MLEs in terms of biases and mean squared errors (MSEs) under the different assumptions of the censoring model. Random samples are generated when the true model is the Koziol-Green model in (
From Tables 1?4, where the estimation process is conducted under the assumption that the censoring distribution does not involve any parameters, we observe the following. First, the approximate MLEs
From Tables 5?8, where the estimation is conducted under the true model (the Koziol-Green model in (
In this paper, we study randomly censored data from a Weibull distribution. We derive the approximate MLEs in explicit forms using the type I extreme value distribution of the minimum since the MLEs of the parameters do not have explicit solutions. In the simulation study, we generate random samples under the Koziol-Green model, compute the estimators ignoring the censoring model, and compare them with the estimators from the true model. Consequently, we find that the censoring model has a strong influence on the scale parameter
Averages of the MLE
MSE( | MSE( | MSE( | mean( | mean( | RE( | RE( | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.4 | 20 | 2.285 | 2.129 | 2.034 | 0.666 | 0.375 | 0.238 | 0.155 | 0.251 | 56.243 | 35.738 |
30 | 2.153 | 2.108 | 2.023 | 0.316 | 0.242 | 0.151 | 0.045 | 0.130 | 76.709 | 47.707 | |
40 | 2.110 | 2.062 | 1.997 | 0.225 | 0.169 | 0.105 | 0.047 | 0.112 | 75.040 | 46.785 | |
50 | 2.092 | 2.074 | 2.012 | 0.152 | 0.145 | 0.086 | 0.018 | 0.079 | 95.276 | 56.290 | |
0.6 | 20 | 2.193 | 2.136 | 2.079 | 0.345 | 0.287 | 0.224 | 0.056 | 0.114 | 83.332 | 64.858 |
30 | 2.141 | 2.110 | 2.066 | 0.208 | 0.173 | 0.139 | 0.031 | 0.075 | 83.006 | 66.802 | |
40 | 2.095 | 2.078 | 2.041 | 0.133 | 0.123 | 0.099 | 0.017 | 0.054 | 92.247 | 74.165 | |
50 | 2.088 | 2.079 | 2.047 | 0.106 | 0.098 | 0.078 | 0.009 | 0.041 | 92.162 | 73.708 | |
0.8 | 20 | 2.168 | 2.146 | 2.121 | 0.234 | 0.214 | 0.196 | 0.023 | 0.047 | 91.435 | 83.964 |
30 | 2.102 | 2.091 | 2.073 | 0.136 | 0.130 | 0.121 | 0.010 | 0.029 | 95.579 | 88.635 | |
40 | 2.080 | 2.076 | 2.061 | 0.100 | 0.097 | 0.091 | 0.004 | 0.019 | 97.166 | 90.474 | |
50 | 2.071 | 2.062 | 2.049 | 0.077 | 0.077 | 0.072 | 0.009 | 0.022 | 99.128 | 93.077 |
MLE = maximum likelihood estimator, RE = relative efficiency, MSE = mean squared error.
Averages of the MLE
MSE( | MSE( | MSE( | mean( | mean( | RE( | RE( | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.4 | 20 | 0.407 | 0.416 | 0.424 | 0.026 | 0.025 | 0.024 | ?0.010 | ?0.017 | 95.312 | 92.540 |
30 | 0.403 | 0.406 | 0.414 | 0.017 | 0.017 | 0.016 | ?0.003 | ?0.011 | 97.566 | 94.282 | |
40 | 0.404 | 0.407 | 0.413 | 0.012 | 0.011 | 0.011 | ?0.003 | ?0.009 | 97.588 | 94.681 | |
50 | 0.399 | 0.400 | 0.406 | 0.010 | 0.009 | 0.009 | ?0.001 | ?0.007 | 99.598 | 95.153 | |
0.6 | 20 | 0.616 | 0.621 | 0.628 | 0.042 | 0.041 | 0.040 | ?0.006 | ?0.012 | 97.853 | 95.309 |
30 | 0.609 | 0.612 | 0.618 | 0.025 | 0.024 | 0.024 | ?0.003 | ?0.009 | 97.631 | 95.252 | |
40 | 0.611 | 0.613 | 0.618 | 0.019 | 0.019 | 0.018 | ?0.002 | ?0.007 | 98.324 | 96.440 | |
50 | 0.607 | 0.608 | 0.612 | 0.015 | 0.015 | 0.015 | ?0.001 | ?0.005 | 99.375 | 97.387 | |
0.8 | 20 | 0.827 | 0.830 | 0.833 | 0.108 | 0.108 | 0.108 | ?0.002 | ?0.006 | 99.854 | 100.396 |
30 | 0.814 | 0.815 | 0.818 | 0.081 | 0.081 | 0.082 | ?0.001 | ?0.004 | 99.926 | 100.799 | |
40 | 0.813 | 0.813 | 0.816 | 0.069 | 0.070 | 0.070 | 0.000 | ?0.003 | 100.201 | 101.267 | |
50 | 0.808 | 0.809 | 0.811 | 0.062 | 0.062 | 0.063 | ?0.001 | ?0.003 | 100.837 | 101.928 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.
Averages of the MLE
MSE( | MSE( | MSE( | mean( | mean( | RE( | RE( | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.4 | 20 | 0.560 | 0.528 | 0.503 | 0.040 | 0.023 | 0.014 | 0.032 | 0.057 | 56.973 | 34.912 |
30 | 0.545 | 0.526 | 0.506 | 0.021 | 0.015 | 0.010 | 0.019 | 0.038 | 72.217 | 45.068 | |
40 | 0.527 | 0.520 | 0.502 | 0.013 | 0.012 | 0.007 | 0.007 | 0.024 | 86.643 | 52.821 | |
50 | 0.522 | 0.515 | 0.500 | 0.010 | 0.009 | 0.005 | 0.008 | 0.023 | 85.355 | 52.461 | |
0.6 | 20 | 0.549 | 0.535 | 0.521 | 0.023 | 0.017 | 0.014 | 0.014 | 0.028 | 77.586 | 60.953 |
30 | 0.529 | 0.521 | 0.510 | 0.012 | 0.010 | 0.008 | 0.008 | 0.019 | 84.453 | 67.114 | |
40 | 0.521 | 0.519 | 0.509 | 0.008 | 0.008 | 0.006 | 0.002 | 0.011 | 90.331 | 72.481 | |
50 | 0.514 | 0.513 | 0.505 | 0.006 | 0.006 | 0.005 | 0.001 | 0.010 | 99.539 | 80.320 | |
0.8 | 20 | 0.539 | 0.535 | 0.529 | 0.015 | 0.015 | 0.013 | 0.004 | 0.010 | 98.553 | 90.588 |
30 | 0.528 | 0.526 | 0.521 | 0.008 | 0.008 | 0.008 | 0.002 | 0.007 | 101.966 | 94.120 | |
40 | 0.519 | 0.516 | 0.513 | 0.006 | 0.006 | 0.005 | 0.002 | 0.006 | 94.039 | 87.951 | |
50 | 0.513 | 0.512 | 0.509 | 0.005 | 0.004 | 0.004 | 0.001 | 0.004 | 97.638 | 91.554 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.
Averages of the MLE
MSE( | MSE( | MSE( | mean( | mean( | RE( | RE( | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
0.4 | 20 | 0.404 | 0.411 | 0.419 | 0.027 | 0.026 | 0.025 | ?0.007 | ?0.015 | 96.387 | 93.325 |
30 | 0.402 | 0.407 | 0.414 | 0.017 | 0.016 | 0.016 | ?0.006 | ?0.012 | 93.121 | 90.685 | |
40 | 0.404 | 0.405 | 0.412 | 0.013 | 0.012 | 0.012 | ?0.002 | ?0.008 | 97.581 | 93.924 | |
50 | 0.405 | 0.408 | 0.414 | 0.010 | 0.009 | 0.009 | ?0.003 | ?0.008 | 95.671 | 93.205 | |
0.6 | 20 | 0.616 | 0.622 | 0.628 | 0.089 | 0.089 | 0.091 | ?0.005 | ?0.012 | 100.767 | 102.439 |
30 | 0.610 | 0.614 | 0.620 | 0.070 | 0.071 | 0.073 | ?0.004 | ?0.010 | 101.851 | 104.216 | |
40 | 0.608 | 0.609 | 0.614 | 0.062 | 0.062 | 0.064 | 0.000 | ?0.005 | 99.876 | 102.249 | |
50 | 0.605 | 0.606 | 0.610 | 0.056 | 0.056 | 0.058 | ?0.001 | ?0.005 | 100.045 | 102.423 | |
0.8 | 20 | 0.838 | 0.840 | 0.843 | 0.249 | 0.251 | 0.253 | ?0.002 | ?0.005 | 100.629 | 101.450 |
30 | 0.816 | 0.817 | 0.820 | 0.208 | 0.209 | 0.211 | ?0.001 | ?0.004 | 100.506 | 101.394 | |
40 | 0.813 | 0.814 | 0.816 | 0.195 | 0.196 | 0.198 | ?0.001 | ?0.004 | 100.410 | 101.269 | |
50 | 0.814 | 0.814 | 0.816 | 0.190 | 0.191 | 0.192 | 0.000 | ?0.003 | 100.053 | 100.850 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.
Averages of the MLE
MSE( | MSE( | mean( | RE( | ||||
---|---|---|---|---|---|---|---|
0.4 | 20 | 2.152 | 2.143 | 0.196 | 0.192 | 0.008 | 97.737 |
30 | 2.103 | 2.100 | 0.114 | 0.113 | 0.003 | 98.783 | |
40 | 2.065 | 2.063 | 0.076 | 0.076 | 0.002 | 99.039 | |
50 | 2.064 | 2.063 | 0.059 | 0.059 | 0.001 | 100.314 | |
0.6 | 20 | 2.143 | 2.135 | 0.178 | 0.175 | 0.008 | 98.278 |
30 | 2.112 | 2.107 | 0.113 | 0.111 | 0.004 | 98.351 | |
40 | 2.075 | 2.073 | 0.076 | 0.076 | 0.002 | 99.309 | |
50 | 2.075 | 2.074 | 0.062 | 0.062 | 0.001 | 100.179 | |
0.8 | 20 | 2.154 | 2.146 | 0.181 | 0.177 | 0.008 | 97.821 |
30 | 2.096 | 2.092 | 0.108 | 0.107 | 0.004 | 99.238 | |
40 | 2.078 | 2.076 | 0.079 | 0.079 | 0.002 | 99.509 | |
50 | 2.063 | 2.062 | 0.063 | 0.063 | 0.001 | 100.080 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.
Averages of the MLE
MSE( | MSE( | mean( | RE( | ||||
---|---|---|---|---|---|---|---|
0.4 | 20 | 0.418 | 0.419 | 0.024 | 0.024 | ?0.0003 | 99.791 |
30 | 0.410 | 0.410 | 0.016 | 0.016 | ?0.0001 | 99.830 | |
40 | 0.408 | 0.409 | 0.011 | 0.011 | ?0.0001 | 99.971 | |
50 | 0.402 | 0.402 | 0.009 | 0.009 | 0.0000 | 100.001 | |
0.6 | 20 | 0.623 | 0.624 | 0.041 | 0.041 | ?0.0005 | 99.776 |
30 | 0.614 | 0.615 | 0.024 | 0.024 | ?0.0003 | 99.850 | |
40 | 0.615 | 0.615 | 0.018 | 0.018 | ?0.0001 | 99.881 | |
50 | 0.609 | 0.609 | 0.015 | 0.015 | 0.0000 | 99.970 | |
0.8 | 20 | 0.831 | 0.831 | 0.108 | 0.108 | ?0.0006 | 100.030 |
30 | 0.816 | 0.816 | 0.081 | 0.081 | ?0.0004 | 100.123 | |
40 | 0.814 | 0.814 | 0.070 | 0.070 | ?0.0002 | 100.077 | |
50 | 0.810 | 0.810 | 0.062 | 0.062 | 0.0000 | 100.020 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.
Averages of the MLE
MSE( | MSE( | mean( | RE( | ||||
---|---|---|---|---|---|---|---|
0.4 | 20 | 0.533 | 0.531 | 0.011 | 0.011 | 0.002 | 99.037 |
30 | 0.525 | 0.524 | 0.007 | 0.007 | 0.001 | 98.778 | |
40 | 0.518 | 0.518 | 0.005 | 0.005 | 0.000 | 99.569 | |
50 | 0.513 | 0.513 | 0.003 | 0.003 | 0.000 | 99.698 | |
0.6 | 20 | 0.538 | 0.536 | 0.011 | 0.011 | 0.002 | 98.230 |
30 | 0.521 | 0.520 | 0.006 | 0.006 | 0.001 | 99.288 | |
40 | 0.517 | 0.516 | 0.005 | 0.005 | 0.000 | 99.520 | |
50 | 0.511 | 0.511 | 0.004 | 0.004 | 0.000 | 100.309 | |
0.8 | 20 | 0.536 | 0.534 | 0.012 | 0.012 | 0.002 | 98.801 |
30 | 0.526 | 0.525 | 0.007 | 0.007 | 0.001 | 98.819 | |
40 | 0.517 | 0.516 | 0.005 | 0.005 | 0.001 | 99.561 | |
50 | 0.512 | 0.512 | 0.004 | 0.004 | 0.000 | 99.591 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.
Averages of the MLE
MSE( | MSE( | mean( | RE( | ||||
---|---|---|---|---|---|---|---|
0.4 | 20 | 0.413 | 0.414 | 0.025 | 0.025 | ?0.0004 | 99.877 |
30 | 0.410 | 0.410 | 0.016 | 0.016 | ?0.0002 | 99.951 | |
40 | 0.408 | 0.408 | 0.012 | 0.012 | ?0.0001 | 100.029 | |
50 | 0.410 | 0.410 | 0.009 | 0.009 | 0.0000 | 99.977 | |
0.6 | 20 | 0.623 | 0.624 | 0.090 | 0.090 | ?0.0004 | 100.069 |
30 | 0.616 | 0.616 | 0.072 | 0.072 | ?0.0003 | 100.077 | |
40 | 0.611 | 0.611 | 0.063 | 0.063 | ?0.0001 | 100.071 | |
50 | 0.608 | 0.608 | 0.057 | 0.057 | 0.0000 | 100.006 | |
0.8 | 20 | 0.841 | 0.842 | 0.251 | 0.252 | ?0.0007 | 100.168 |
30 | 0.818 | 0.819 | 0.210 | 0.210 | ?0.0004 | 100.114 | |
40 | 0.814 | 0.815 | 0.196 | 0.196 | ?0.0002 | 100.078 | |
50 | 0.815 | 0.815 | 0.191 | 0.191 | ?0.0001 | 100.028 |
MLE = maximum likelihood estimator, MSE = mean squared error, RE = relative efficiency.