Helsel & Cohn (1988)
|
ZDL |
25 |
500 |
Log-normal |
60 |
RMSE |
Mean |
MLE: Significant bias in the estimates of means and standard deviations |
HDL |
Mixture of two log-normals |
Bias |
Median |
DL |
Delta |
|
Standard deviation |
MLE |
|
|
Interquartile ranges |
ROS |
|
|
|
Kroll & Stedinger (1996)
|
MLE |
10 |
5000 |
Log-normal |
20 |
RMSE |
Percentile 10,90 |
MLE: Suitable for estimating quantiles and interquartile ranges in highly censored data; |
ROS |
25 |
Mixture of two log-normals |
60 |
Mean |
ROS: Suitable for estimating means and standard deviations in medium to long time series with short to medium censoring |
|
50 |
Gamma |
80 |
Standard Deviation |
|
|
|
Delta |
|
Interquartile |
|
|
|
|
|
Ranges |
|
She (1997)
|
HDL |
21 |
1000 |
Log-normal |
Three randomly between |
Bias |
Mean |
HDL: Best for CV = 1.00 and 2.00 |
KM |
Gamma |
10 and 80 |
Standard error |
Standard Deviation |
KM: Second-best technique, similar to MLE |
MLE |
|
|
|
|
MLE: Best for CV = 0.25, 0.50. |
ROS |
|
|
|
|
Means: Worse estimates for higher CV values |
Shunway et al. (2002)
|
MLE |
20 |
500 |
Log-normal |
50 |
Bias |
Mean |
ROS: No bias for the log-normal distribution, but larger standard error for highly asymmetrical series |
ROS |
50 |
Gamma |
80 |
Confidence interval |
Variance |
MLE: Recommended to use a bias corrector |
Hewett & Ganser (2007)
|
HDL |
mai/19 |
100 |
Log-normal |
jan/50 |
Bias |
Mean |
MLE: Recommended for all scenarios |
LR2 |
20-100 |
Contaminated log-normal |
50-80 |
RMSE |
95th quantile |
ROS: Recommended for estimating averages |
DL |
|
|
|
|
|
KM: Presented poor estimates |
KM |
|
|
|
|
|
LD: Overestimated the mean and underestimated the 95th percentile |
MLE |
|
|
|
|
|
|
ROS |
|
|
|
|
|
|
Authors
|
Methods
|
Elements
|
Random Samples
|
Distribution
|
Censoring Percentage
|
Accuracy Measure |
Evaluated Stats
|
Conclusions Related to the Log-normal Distribution
|
Antweiller & Taylor (2008)
|
ZDL |
34-841 |
44 |
No specific distributions |
Randomly between |
Bias |
Mean |
KM: Achieved the best results for censoring up to 70%, except when estimating the median |
|
HDL |
14 and 95 |
Percentile |
ROS and HDL: Yielded reasonable results |
|
DL |
|
25, 50 and 75 |
No method yielded suitable results for censoring greater than 70% |
|
KM |
|
Standard deviation |
|
|
MLE |
|
Interquartile range |
|
|
ROS |
|
|
|
Niemann (2016)
|
ZDL |
50 |
10000 |
Log-normal |
5 to 60 |
Bias |
Mean |
HDL, LR2: Good for ratings up to 30% |
|
HDL |
RMSE |
MLE: Exhibited significant bias and high RMSE |
|
LR2 |
Confidence interval |
HDL: Stood out for censorship rates exceeding 50%, providing unbiased estimates and low RMSE |
|
DL |
|
|
|
KM |
|
|
|
MLE |
|
|
Tekindal et al. (2017)
|
LR2 |
20 |
10000 |
Log-normal |
5 |
Bias |
Mean |
ROS: Recommended for estimating mean values; |
|
DL |
80 |
Exponential |
25 |
Median |
LR2: Exhibited less bias when estimating medians |
|
KM |
140 |
Weibull |
45 |
Standard deviation |
KM, DL: Demonstrated similar performance, with the overestimation of means and the underestimation of standard deviations |
|
MLE |
200 |
|
65 |
|
MLE: Worst scenario |
|
ROS |
260 |
|
|
|
|
Canales et al. (2018)
|
LR2 |
100 |
10000 |
Log-normal |
< 10 |
Bias |
Mean |
ROS: Performed better in series with a high percentage of censored data |
|
DL |
35 |
RMSE |
MLE: Showed poor performance, with a high RMSE, especially in series with pronounced asymmetry |
|
KM |
65 |
|
|
|
MLE |
90 |
|
|
|
ROS |
97 |
|
|
George et al. (2021)
|
HDL |
20 |
1000 |
Log-normal |
30 |
|
Mean |
KM: Overestimated means and underestimated standard deviations, performing less poorly in highly skewed distributions |
|
MLE |
50 |
Moderately and highly Asymmetrical |
50 |
Standard deviation |
ROS: Demonstrated the best performance |
|
ROS |
|
|
80 |
|
HDL: Provided reasonable estimates for means but performed poorly for standard deviations |
|
KM |
|
|
|
|
MLE: Performed poorly in asymmetrical series |