We consider goodness-of-fit test statistics for Weibull distributions when data are randomly censored and the parameters are unknown. Koziol and Green (
In the statistical analysis of life time data, the goodness-of-fit test procedure is important to choose the distribution that adequately fits the data. Classical goodness-of-fit tests are usually based on graphical analysis, moments such as skewness or kurtosis, chi-squared type, the empirical distribution function (EDF), or regression and correlations. Many studies are conducted so that these analyses are generalized to censored data.
Akritas (1988), Hollander and Peña (1992) generalized chi-squared test statistics to censored cases. Koziol and Green (1976), Koziol (1980), and Nair (1981) adapted test statistics based on EDF or weighted empirical process to randomly censored data. The Koziol-Green statistic is a generalized version of the Cramér-von Mises statistic for randomly censored data based on the Kaplan-Meier product limit estimator of the distribution function. Chen (1984) considered a correlation statistic for randomly censored data.
Kim (2012) studied the Koziol-Green and Kolmogorov-Smirnov statistics for a randomly censored exponential distribution with an unknown scale parameter. In this paper, we apply these statistics for Weibull distributions when data are randomly censored and the parameters are unknown. Liao and Shimokawa (1999) proposed a new statistic based on the Kolmogorov-Smirnov, Cramér-von Mises and Anderson-Darling statistics and applied it to test Weibull distributions with estimated parameters for a complete sample. Liao and Shimokawa (1999) considered two procedures for the estimation: the first is widely used maximum likelihood estimators (MLEs) and the second is estimators based on graphical plotting method with the least squares on Weibull probability paper. We will generalize their new statistic as well as classical ones to randomly censored Weibull distributions with estimated parameters.
Section 2 presents the test statistics for randomly censored Weibull distributions. Section 3 describes the parameter estimation procedures. Section 4 contains a power comparison of the proposed statistics. Section 4 ends the paper with some conclusion remarks.
Let
We assume that
Suppose we wish to test the null hypothesis
with
The Kaplan-Meier estimator in (
By Michael and Schucany (1986), the Kaplan-Meier estimator in (
or
and it reduces to (
Using (
with
Another popular EDF based test statistic is the Cramér-von Mises statistic. Koziol and Green (1976) generalized it to
for randomly censored data using the Kaplan-Meier estimator. It measures the discrepancy between
with
The statistics
In this paper, we want to test if the lifetimes
and the probability density function (pdf)
Note that
where
If we let
and use the test statistic (
i.e., it is composite and includes some unknown parameters. We therefore need to estimate the unknown parameters
If we estimate
and consider the similar test statistic
with
Liao and Shimokawa (1999) proposed a new test statistic for a Weibull distribution with estimated parameters for a complete data set. Their new statistic is to combine the idea of the Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling statistics. If we generalize their new statistic to randomly censored data, it becomes
This can also be written as
by (
For the observed random pairs (
the log-likelihood function
when
The MLEs of
where
Substituting back
Balakrishnan and Kateri (2008) discussed the existence and uniqueness of the MLEs
Another estimation method is to use graphical plotting method. The general concept of probability plotting is described in D’Agostino (1986). Graphical methods for survival distribution fitting are in Lee and Wang (2003). For Weibull plotting, we use type I extreme value distributions. By taking the logarithm of the distribution function of
The plot of the ordered value
We plot only uncensored data for randomly censored data. The plot therefore should be
for the
where
according to (
A simulation study is conducted to give the null distributions and compare the power of the test statistics. The statistics
Similar to Liao and Shimokawa (1999), we can show that the test statistics are not dependent on the true values of
We use the random censorship model proposed in Koziol and Green (1976) to control the censored ratio. It is
where
In Tables 1
The model in (
The values of
The asymptotic null distributions of the statistics remain unknown. The asymptotic distribution of
Next, we investigate the power of the test statistics. Table 4 and Table 5 present the power of the statistics at the significance level
Gamma distribution, Gamma(
Lognormal distribution, lognorm(
Log-logistic distribution, log-logistic(
Log double exponential, log-DE, DE with pdf
Half-logistic with
The power results indicate the following. The meaningfully good power is written in bold in each alternative. First,
Second, the power in decreasing order seem to be judged as
As an illustrative example for goodness-of-fit testing, we take the remission times of 21 patients with acute leukemia who received 6-mercaptopurine (6-MP). The study was to assess the ability of the 6-MP treatment to maintain remission. We randomized 42 patients to receive 6-MP or a placebo. The study was ended in one year. The data were originally reported by Freireich
Lee and Wang (2003) checked the Weibull hazard plot for the given data and showed the Weibull distribution provided a good fit.
When we compute the statistics proposed in this paper for the remission data above, we have
The
In this paper, the Kolmogorov-Smirnov, Koziol-Green, and Liao-Shimokawa statistics are applied to test randomly censored Weibull distributions with estimated parameters. Statistics are generalized to randomly censored cases using the Kaplan-Meier product limit of the distribution function just like Koziol and Green (1976).
The upper percentage points of the statistics are provided by simulations. The parameters are estimated by the MLEs and the graphical plotting method. The null distributions depend on the estimation method since the test statistics are not distribution free when the parameters are estimated. Through the simulation study, the Liao-Shimokawa statistic showed relatively good power among competitors. However the null distribution of the statistic changes too much upon the parameter estimation. Hence we do not recommend the use of the statistic simply because of the good power. We instead recommend the Koziol and Green statistic since it shows slightly better power than the Kolmogorov-Smirnov; in addition, the null distribution does not heavily depend on the estimation procedure.
Upper tail percentage points for the test statistics
Statistic | |||||||||
---|---|---|---|---|---|---|---|---|---|
0.01 | 0.025 | 0.05 | 0.10 | 0.15 | 0.25 | 0.5 | |||
20 | 0.6 | 3.33 | 3.01 | 2.73 | 2.38 | 2.14 | 1.81 | 1.34 | |
0.5 | 2.67 | 2.34 | 2.08 | 1.79 | 1.61 | 1.38 | 1.08 | ||
0.4 | 2.05 | 1.76 | 1.56 | 1.34 | 1.23 | 1.09 | 0.89 | ||
0.2 | 1.20 | 1.10 | 1.02 | 0.94 | 0.89 | 0.81 | 0.70 | ||
30 | 0.6 | 3.48 | 3.12 | 2.81 | 2.47 | 2.23 | 1.88 | 1.40 | |
0.5 | 2.71 | 2.30 | 2.05 | 1.77 | 1.61 | 1.38 | 1.08 | ||
0.4 | 1.96 | 1.70 | 1.51 | 1.31 | 1.21 | 1.07 | 0.88 | ||
0.2 | 1.17 | 1.09 | 1.01 | 0.93 | 0.87 | 0.80 | 0.69 | ||
40 | 0.6 | 3.60 | 3.20 | 2.92 | 2.53 | 2.28 | 1.94 | 1.43 | |
0.5 | 2.76 | 2.41 | 2.13 | 1.82 | 1.63 | 1.40 | 1.09 | ||
0.4 | 1.94 | 1.68 | 1.49 | 1.31 | 1.20 | 1.07 | 0.89 | ||
0.2 | 1.14 | 1.06 | 0.98 | 0.91 | 0.86 | 0.79 | 0.68 | ||
50 | 0.6 | 3.75 | 3.36 | 3.00 | 2.61 | 2.35 | 2.00 | 1.48 | |
0.5 | 2.71 | 2.40 | 2.11 | 1.82 | 1.63 | 1.41 | 1.11 | ||
0.4 | 1.89 | 1.61 | 1.44 | 1.26 | 1.17 | 1.05 | 0.88 | ||
0.2 | 1.14 | 1.06 | 0.98 | 0.90 | 0.85 | 0.78 | 0.68 | ||
100 | 0.6 | 3.98 | 3.60 | 3.21 | 2.80 | 2.52 | 2.16 | 1.60 | |
0.5 | 2.70 | 2.40 | 2.11 | 1.84 | 1.65 | 1.43 | 1.13 | ||
0.4 | 1.81 | 1.57 | 1.40 | 1.24 | 1.15 | 1.04 | 0.88 | ||
0.2 | 1.12 | 1.03 | 0.97 | 0.89 | 0.84 | 0.78 | 0.67 | ||
20 | 0.6 | 3.33 | 3.01 | 2.76 | 2.43 | 2.23 | 1.93 | 1.48 | |
0.5 | 2.70 | 2.39 | 2.15 | 1.90 | 1.73 | 1.52 | 1.20 | ||
0.4 | 2.13 | 1.90 | 1.70 | 1.50 | 1.37 | 1.21 | 0.98 | ||
0.2 | 1.33 | 1.22 | 1.13 | 1.03 | 0.97 | 0.89 | 0.76 | ||
30 | 0.6 | 3.48 | 3.14 | 2.86 | 2.54 | 2.33 | 2.03 | 1.55 | |
0.5 | 2.75 | 2.41 | 2.16 | 1.92 | 1.75 | 1.54 | 1.21 | ||
0.4 | 2.10 | 1.87 | 1.68 | 1.48 | 1.37 | 1.22 | 0.98 | ||
0.2 | 1.33 | 1.21 | 1.13 | 1.04 | 0.98 | 0.89 | 0.76 | ||
40 | 0.6 | 3.63 | 3.26 | 2.98 | 2.63 | 2.41 | 2.09 | 1.60 | |
0.5 | 2.82 | 2.55 | 2.29 | 1.98 | 1.81 | 1.57 | 1.24 | ||
0.4 | 2.09 | 1.87 | 1.68 | 1.50 | 1.38 | 1.23 | 1.00 | ||
0.2 | 1.31 | 1.20 | 1.11 | 1.01 | 0.96 | 0.87 | 0.75 | ||
50 | 0.6 | 3.76 | 3.41 | 3.08 | 2.71 | 2.49 | 2.15 | 1.65 | |
0.5 | 2.79 | 2.50 | 2.25 | 1.99 | 1.83 | 1.60 | 1.26 | ||
0.4 | 2.05 | 1.84 | 1.65 | 1.47 | 1.36 | 1.21 | 0.99 | ||
0.2 | 1.31 | 1.19 | 1.11 | 1.02 | 0.96 | 0.87 | 0.75 | ||
100 | 0.6 | 4.03 | 3.65 | 3.30 | 2.91 | 2.67 | 2.34 | 1.80 | |
0.5 | 2.80 | 2.54 | 2.30 | 2.04 | 1.88 | 1.66 | 1.31 | ||
0.4 | 2.06 | 1.83 | 1.68 | 1.50 | 1.39 | 1.24 | 1.00 | ||
0.2 | 1.29 | 1.18 | 1.10 | 1.01 | 0.95 | 0.87 | 0.73 |
Upper tail percentage points for the test statistics
Statistic | |||||||||
---|---|---|---|---|---|---|---|---|---|
0.01 | 0.025 | 0.05 | 0.10 | 0.15 | 0.25 | 0.5 | |||
20 | 0.6 | 2.77 | 2.03 | 1.54 | 1.05 | 0.78 | 0.52 | 0.27 | |
0.5 | 1.43 | 1.00 | 0.71 | 0.48 | 0.38 | 0.28 | 0.18 | ||
0.4 | 0.69 | 0.47 | 0.36 | 0.26 | 0.22 | 0.18 | 0.12 | ||
0.2 | 0.23 | 0.19 | 0.17 | 0.14 | 0.12 | 0.11 | 0.08 | ||
30 | 0.6 | 2.64 | 1.89 | 1.42 | 0.98 | 0.75 | 0.49 | 0.27 | |
0.5 | 1.24 | 0.84 | 0.61 | 0.44 | 0.34 | 0.26 | 0.17 | ||
0.4 | 0.51 | 0.37 | 0.29 | 0.23 | 0.20 | 0.16 | 0.11 | ||
0.2 | 0.22 | 0.18 | 0.16 | 0.13 | 0.12 | 0.10 | 0.07 | ||
40 | 0.6 | 2.57 | 1.87 | 1.37 | 0.92 | 0.71 | 0.49 | 0.27 | |
0.5 | 1.15 | 0.81 | 0.58 | 0.39 | 0.32 | 0.24 | 0.16 | ||
0.4 | 0.49 | 0.35 | 0.28 | 0.22 | 0.19 | 0.15 | 0.11 | ||
0.2 | 0.21 | 0.18 | 0.15 | 0.13 | 0.11 | 0.09 | 0.07 | ||
50 | 0.6 | 2.58 | 1.83 | 1.36 | 0.93 | 0.71 | 0.49 | 0.27 | |
0.5 | 0.98 | 0.70 | 0.53 | 0.37 | 0.30 | 0.23 | 0.15 | ||
0.4 | 0.43 | 0.32 | 0.26 | 0.21 | 0.18 | 0.14 | 0.11 | ||
0.2 | 0.21 | 0.18 | 0.15 | 0.13 | 0.11 | 0.09 | 0.07 | ||
100 | 0.6 | 2.22 | 1.61 | 1.21 | 0.84 | 0.67 | 0.47 | 0.28 | |
0.5 | 0.76 | 0.56 | 0.44 | 0.33 | 0.27 | 0.22 | 0.15 | ||
0.4 | 0.33 | 0.27 | 0.22 | 0.18 | 0.16 | 0.14 | 0.10 | ||
0.2 | 0.20 | 0.17 | 0.14 | 0.12 | 0.10 | 0.09 | 0.06 | ||
20 | 0.6 | 2.79 | 2.08 | 1.64 | 1.21 | 0.95 | 0.67 | 0.37 | |
0.5 | 1.60 | 1.13 | 0.87 | 0.64 | 0.52 | 0.39 | 0.24 | ||
0.4 | 0.84 | 0.62 | 0.49 | 0.37 | 0.31 | 0.24 | 0.16 | ||
0.2 | 0.31 | 0.26 | 0.22 | 0.18 | 0.16 | 0.13 | 0.09 | ||
30 | 0.6 | 2.75 | 2.08 | 1.64 | 1.20 | 0.94 | 0.68 | 0.38 | |
0.5 | 1.39 | 1.04 | 0.82 | 0.60 | 0.50 | 0.37 | 0.23 | ||
0.4 | 0.76 | 0.57 | 0.44 | 0.34 | 0.29 | 0.23 | 0.15 | ||
0.2 | 0.30 | 0.25 | 0.21 | 0.17 | 0.15 | 0.13 | 0.09 | ||
40 | 0.6 | 2.72 | 2.06 | 1.59 | 1.18 | 0.94 | 0.69 | 0.39 | |
0.5 | 1.40 | 1.03 | 0.80 | 0.59 | 0.48 | 0.36 | 0.22 | ||
0.4 | 0.71 | 0.56 | 0.44 | 0.34 | 0.28 | 0.22 | 0.15 | ||
0.2 | 0.31 | 0.25 | 0.21 | 0.17 | 0.15 | 0.12 | 0.08 | ||
50 | 0.6 | 2.75 | 2.07 | 1.62 | 1.20 | 0.96 | 0.69 | 0.39 | |
0.5 | 1.27 | 0.95 | 0.75 | 0.57 | 0.47 | 0.36 | 0.22 | ||
0.4 | 0.68 | 0.52 | 0.41 | 0.32 | 0.27 | 0.22 | 0.14 | ||
0.2 | 0.30 | 0.25 | 0.21 | 0.17 | 0.15 | 0.12 | 0.08 | ||
100 | 0.6 | 2.52 | 1.93 | 1.55 | 1.18 | 0.96 | 0.70 | 0.41 | |
0.5 | 1.08 | 0.85 | 0.69 | 0.53 | 0.44 | 0.34 | 0.21 | ||
0.4 | 0.61 | 0.49 | 0.40 | 0.31 | 0.26 | 0.21 | 0.14 | ||
0.2 | 0.30 | 0.25 | 0.21 | 0.17 | 0.14 | 0.12 | 0.08 |
Upper tail percentage points for the test statistics
Statistic | |||||||||
---|---|---|---|---|---|---|---|---|---|
0.01 | 0.025 | 0.05 | 0.10 | 0.15 | 0.25 | 0.5 | |||
20 | 0.6 | 0.80 | 0.73 | 0.68 | 0.62 | 0.58 | 0.53 | 0.42 | |
0.5 | 0.90 | 0.82 | 0.76 | 0.70 | 0.67 | 0.61 | 0.51 | ||
0.4 | 0.97 | 0.89 | 0.83 | 0.77 | 0.73 | 0.68 | 0.59 | ||
0.2 | 1.14 | 1.05 | 0.98 | 0.91 | 0.87 | 0.82 | 0.73 | ||
30 | 0.6 | 0.72 | 0.66 | 0.61 | 0.56 | 0.52 | 0.48 | 0.40 | |
0.5 | 0.82 | 0.75 | 0.70 | 0.64 | 0.61 | 0.56 | 0.48 | ||
0.4 | 0.91 | 0.83 | 0.77 | 0.72 | 0.68 | 0.63 | 0.55 | ||
0.2 | 1.03 | 0.96 | 0.91 | 0.84 | 0.80 | 0.75 | 0.67 | ||
40 | 0.6 | 0.67 | 0.62 | 0.57 | 0.53 | 0.50 | 0.45 | 0.39 | |
0.5 | 0.76 | 0.71 | 0.66 | 0.61 | 0.58 | 0.53 | 0.46 | ||
0.4 | 0.86 | 0.79 | 0.73 | 0.68 | 0.65 | 0.60 | 0.53 | ||
0.2 | 1.02 | 0.94 | 0.87 | 0.81 | 0.77 | 0.72 | 0.64 | ||
50 | 0.6 | 0.66 | 0.59 | 0.55 | 0.51 | 0.48 | 0.44 | 0.38 | |
0.5 | 0.74 | 0.68 | 0.63 | 0.58 | 0.55 | 0.51 | 0.44 | ||
0.4 | 0.82 | 0.75 | 0.70 | 0.65 | 0.62 | 0.58 | 0.51 | ||
0.2 | 0.99 | 0.91 | 0.85 | 0.79 | 0.75 | 0.70 | 0.62 | ||
100 | 0.6 | 0.56 | 0.53 | 0.49 | 0.45 | 0.43 | 0.39 | 0.34 | |
0.5 | 0.66 | 0.61 | 0.57 | 0.53 | 0.50 | 0.46 | 0.40 | ||
0.4 | 0.75 | 0.69 | 0.65 | 0.60 | 0.57 | 0.53 | 0.46 | ||
0.2 | 0.92 | 0.84 | 0.79 | 0.73 | 0.69 | 0.64 | 0.56 | ||
20 | 0.6 | 1.22 | 0.96 | 0.82 | 0.71 | 0.66 | 0.59 | 0.49 | |
0.5 | 1.62 | 1.16 | 0.96 | 0.82 | 0.76 | 0.68 | 0.57 | ||
0.4 | 2.05 | 1.35 | 1.09 | 0.93 | 0.84 | 0.76 | 0.64 | ||
0.2 | 2.84 | 1.73 | 1.36 | 1.14 | 1.03 | 0.91 | 0.77 | ||
30 | 0.6 | 1.16 | 0.88 | 0.75 | 0.65 | 0.60 | 0.55 | 0.46 | |
0.5 | 1.45 | 1.10 | 0.91 | 0.77 | 0.70 | 0.63 | 0.54 | ||
0.4 | 1.63 | 1.20 | 1.02 | 0.87 | 0.80 | 0.71 | 0.60 | ||
0.2 | 2.30 | 1.50 | 1.23 | 1.04 | 0.95 | 0.84 | 0.72 | ||
40 | 0.6 | 1.01 | 0.82 | 0.71 | 0.63 | 0.58 | 0.52 | 0.44 | |
0.5 | 1.26 | 0.97 | 0.85 | 0.74 | 0.67 | 0.61 | 0.52 | ||
0.4 | 1.58 | 1.15 | 0.95 | 0.82 | 0.76 | 0.68 | 0.58 | ||
0.2 | 1.98 | 1.40 | 1.15 | 0.99 | 0.90 | 0.81 | 0.68 | ||
50 | 0.6 | 0.97 | 0.78 | 0.69 | 0.60 | 0.56 | 0.50 | 0.43 | |
0.5 | 1.12 | 0.93 | 0.81 | 0.70 | 0.65 | 0.58 | 0.50 | ||
0.4 | 1.34 | 1.06 | 0.91 | 0.79 | 0.73 | 0.65 | 0.56 | ||
0.2 | 1.63 | 1.30 | 1.11 | 0.96 | 0.88 | 0.79 | 0.67 | ||
100 | 0.6 | 0.79 | 0.69 | 0.61 | 0.54 | 0.51 | 0.46 | 0.38 | |
0.5 | 0.92 | 0.79 | 0.71 | 0.63 | 0.59 | 0.53 | 0.45 | ||
0.4 | 1.08 | 0.93 | 0.82 | 0.72 | 0.67 | 0.60 | 0.51 | ||
0.2 | 1.30 | 1.11 | 0.98 | 0.87 | 0.80 | 0.72 | 0.61 |
Power comparison of
Distribution | Censoring ratio | ||||||
---|---|---|---|---|---|---|---|
Weibull | 0.10 | 0.09 | 0.10 | 0.10 | 0.10 | 0.10 | |
0.10 | 0.10 | 0.09 | 0.09 | 0.10 | 0.10 | ||
0.11 | 0.10 | 0.10 | 0.11 | 0.10 | 0.10 | ||
0.11 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | ||
Gamma(0.5) | 0.10 | 0.10 | 0.10 | 0.10 | 0.08 | 0.08 | |
0.10 | 0.09 | 0.11 | 0.08 | 0.08 | 0.06 | ||
0.10 | 0.12 | 0.13 | 0.09 | 0.09 | 0.07 | ||
0.17 | 0.18 | 0.16 | 0.13 | 0.15 | 0.07 | ||
Gamma(2) | 0.10 | 0.10 | 0.11 | 0.12 | 0.12 | ||
0.10 | 0.12 | 0.12 | 0.13 | 0.13 | |||
0.12 | 0.12 | 0.13 | 0.14 | 0.15 | |||
0.09 | 0.11 | 0.14 | 0.13 | 0.13 | |||
Lognorm(0.5) | 0.11 | 0.11 | 0.25 | 0.16 | 0.20 | ||
0.13 | 0.20 | 0.32 | 0.22 | 0.28 | |||
0.18 | 0.28 | 0.41 | 0.32 | 0.36 | |||
0.26 | 0.41 | 0.53 | 0.35 | 0.43 | |||
Lognorm(1) | 0.10 | 0.11 | 0.25 | 0.15 | 0.19 | ||
0.12 | 0.17 | 0.33 | 0.24 | 0.30 | |||
0.18 | 0.27 | 0.40 | 0.33 | 0.36 | |||
0.27 | 0.41 | 0.54 | 0.34 | 0.41 | |||
Lognorm(2) | 0.11 | 0.11 | 0.25 | 0.15 | 0.19 | ||
0.13 | 0.18 | 0.32 | 0.22 | 0.29 | |||
0.19 | 0.27 | 0.40 | 0.32 | 0.35 | |||
0.26 | 0.41 | 0.54 | 0.34 | 0.41 | |||
Log-logistic(0.5) | 0.11 | 0.11 | 0.20 | 0.14 | 0.14 | ||
0.13 | 0.18 | 0.27 | 0.17 | 0.20 | |||
0.20 | 0.29 | 0.39 | 0.24 | 0.25 | |||
0.36 | 0.48 | 0.61 | 0.26 | 0.31 | |||
Log-logistic(1) | 0.11 | 0.11 | 0.18 | 0.13 | 0.14 | ||
0.12 | 0.17 | 0.28 | 0.17 | 0.20 | |||
0.19 | 0.28 | 0.39 | 0.24 | 0.24 | |||
0.35 | 0.48 | 0.60 | 0.28 | 0.31 | |||
Log-logistic(2) | 0.11 | 0.11 | 0.20 | 0.14 | 0.15 | ||
0.13 | 0.19 | 0.28 | 0.17 | 0.19 | |||
0.18 | 0.28 | 0.38 | 0.24 | 0.25 | |||
0.36 | 0.48 | 0.60 | 0.26 | 0.30 | |||
Log-double exponential | 0.10 | 0.11 | 0.11 | 0.09 | 0.14 | ||
0.13 | 0.20 | 0.12 | 0.12 | 0.24 | |||
0.26 | 0.37 | 0.17 | 0.16 | 0.38 | |||
0.58 | 0.71 | 0.31 | 0.39 | 0.71 | |||
Half-logistic | 0.10 | 0.09 | 0.10 | 0.09 | 0.08 | 0.07 | |
0.09 | 0.10 | 0.11 | 0.09 | 0.09 | 0.07 | ||
0.11 | 0.10 | 0.12 | 0.09 | 0.08 | 0.07 | ||
0.13 | 0.12 | 0.12 | 0.10 | 0.12 | 0.07 |
Power comparison of
Distribution | Censoring ratio | ||||||
---|---|---|---|---|---|---|---|
Weibull | 0.10 | 0.11 | 0.10 | 0.11 | 0.10 | 0.10 | |
0.09 | 0.09 | 0.10 | 0.10 | 0.11 | 0.10 | ||
0.11 | 0.10 | 0.10 | 0.10 | 0.11 | 0.10 | ||
0.09 | 0.09 | 0.09 | 0.10 | 0.11 | 0.09 | ||
Gamma(0.5) | 0.10 | 0.11 | 0.13 | 0.10 | 0.07 | 0.09 | |
0.10 | 0.10 | 0.14 | 0.10 | 0.10 | 0.10 | ||
0.11 | 0.15 | 0.16 | 0.10 | 0.12 | 0.10 | ||
0.22 | 0.25 | 0.22 | 0.19 | 0.24 | 0.13 | ||
Gamma(2) | 0.11 | 0.11 | 0.13 | 0.12 | 0.14 | ||
0.11 | 0.14 | 0.15 | 0.15 | 0.18 | |||
0.13 | 0.14 | 0.16 | 0.17 | 0.19 | |||
0.11 | 0.16 | 0.19 | 0.17 | 0.18 | |||
Lognorm(0.5) | 0.12 | 0.15 | 0.44 | 0.26 | 0.37 | ||
0.15 | 0.33 | 0.55 | 0.40 | 0.56 | |||
0.28 | 0.51 | 0.66 | 0.54 | 0.65 | |||
0.53 | 0.72 | 0.82 | 0.65 | 0.75 | |||
Lognorm(1) | 0.12 | 0.16 | 0.43 | 0.27 | 0.36 | ||
0.16 | 0.31 | 0.57 | 0.40 | 0.57 | |||
0.28 | 0.52 | 0.66 | 0.55 | 0.65 | |||
Lognorm(2) | 0.12 | 0.16 | 0.44 | 0.24 | 0.37 | ||
0.16 | 0.31 | 0.55 | 0.41 | 0.56 | |||
0.28 | 0.52 | 0.67 | 0.55 | 0.65 | |||
0.55 | 0.72 | 0.82 | 0.64 | 0.75 | |||
Log-logistic(0.5) | 0.12 | 0.14 | 0.31 | 0.17 | 0.21 | ||
0.15 | 0.30 | 0.46 | 0.27 | 0.34 | |||
0.31 | 0.52 | 0.62 | 0.36 | 0.40 | |||
0.66 | 0.79 | 0.86 | 0.50 | 0.58 | |||
Log-logistic(1) | 0.12 | 0.16 | 0.31 | 0.17 | 0.21 | ||
0.16 | 0.29 | 0.46 | 0.27 | 0.33 | |||
0.31 | 0.52 | 0.62 | 0.36 | 0.41 | |||
0.67 | 0.80 | 0.85 | 0.50 | 0.57 | |||
Log-logistic(2) | 0.11 | 0.14 | 0.30 | 0.17 | 0.22 | ||
0.16 | 0.29 | 0.46 | 0.28 | 0.34 | |||
0.32 | 0.53 | 0.62 | 0.37 | 0.40 | |||
0.66 | 0.80 | 0.85 | 0.51 | 0.58 | |||
Log-double exponential | 0.10 | 0.16 | 0.12 | 0.11 | 0.19 | ||
0.20 | 0.37 | 0.17 | 0.15 | 0.35 | |||
0.48 | 0.68 | 0.25 | 0.28 | 0.55 | |||
0.89 | 0.94 | 0.61 | 0.68 | 0.90 | |||
Half-logistic | 0.11 | 0.10 | 0.12 | 0.09 | 0.08 | 0.09 | |
0.10 | 0.10 | 0.12 | 0.09 | 0.08 | 0.09 | ||
0.10 | 0.11 | 0.12 | 0.09 | 0.10 | 0.09 | ||
0.14 | 0.15 | 0.14 | 0.13 | 0.15 | 0.10 |