The use of direct MLE and PWM matching for censored data has been applied using well known distributions such as log Normal and Weibull distributions (Wang et al. 2010a). The theoretical justification for using MLE and PWM to fit any continuous distribution to data can be found elsewhere (Hosking 1990, Fisher 1922, Aldrich 1997). For the SR method, as long as the simulated data is sufficiently close to the true underlying distribution and the GLD can model the simulated uncensored data accurately, it will yield an accurate model.

Unlike many standard statistical distributions, GLDs are characterised by inverse quantile functions. This means to use ML under SR or MLE directly, it is necessary to get the probability density functions using numerical methods such as the Newton–Raphson procedure. Other methods such as PWM do not require this numerical step in the optimisation algorithm. However, all fitting methods are affected by the sudden shape change problem in GLD parameter estimation. For relatively small change in one of the four parameters, GLD can exhibit a dramatic change in shape. For example, RS GLD (0,1,0.5,1) is an increasing function from -1 to 1 but RS GLD (0,1,0.5,0.75) is a parabola shaped function from -1 to 1, even though the change in the fourth parameter is only 0.25. In other situations, RS GLD (0,1,1.5,2) and RS GLD (0,1,1.5,2.5) (with a 0.5 change in the fourth parameter) do exhibit similar shapes and there is a smoother transition in shapes as the fourth parameter changes. This property means the standard theory examining the lower bound for the variability of parameters is not particularly useful for GLDs. The theoretical examinations of fitting methods for GLD are drawn out by numerical computations, potential abrupt changes in shape of distributions from small changes in parameters and perhaps the most elegant strategy at the present time is to use simulations to compare between the methods. Instead of comparing whether the parameters of fitted GLD are close to the true GLD, the emphasis is on whether the fitted GLD quantiles are sufficiently close to the true quantiles from some other known statistical distribution. This is illustrated below.

To assess the performance of various estimation methods, survival curves were generated from 2000 observations from symmetric and skewed Normal distribution with parameters: location = 20, scale = 2, shape = -5 or 5 or 0 for left skewed, right skewed and symmetric shape respectively. The motivation for using skewed Normal rather than Weibull distribution is to facilitate a better comparison across different scenarios using the same distribution with different parameters. Also, it would be rather unfair comparison if one were to use extended Weibull distributions (exponentiated Weibull and four parameter Weibull) to fit Weibull distributed data. Instead, the primary focus here is to examine how well the GLDs and extended Weibull distributions fit data from other distributions, since the true distribution is never known in practice. Additionally, Gumbel distribution (an extreme value distribution) with location parameter 15 and scale parameter 5 is also used in this comparison. These distributions are primarily chosen to examine the behaviour of these fitting algorithms over a range of different shapes.

This entire process is repeated for 200 observations to allow assessment of effect of sample size on the performance of proposed fitting methods. To create right censored data, observations greater than quantile ranging from 0.5 (median) to 0.9 are censored. Similarly, to create left censored data, observations less than quantile ranging from 0.1 to 0.5 (median) are censored. Five estimation methods: ML (maximum likelihood)/LM (L moments)/QS (quantile matching) under SR and MLE and PPWM matching were applied over 100 simulation runs. When fitting RS GLD using MLE with half of the data being censored (i.e., at 0.5), sometimes it is desirable to use the SR-ML method to generate the initial values to start the optimisation process, rather than using randomised search as it would lead to a better performance. This strategy is used in this article.

To give reader an idea as to the degree of accuracy attained by GLDs in comparison to other distributions, this article also assesses the performance of the exponentiated Weibull distribution (Mudholkar and Srivastava 1993, Mudholkar et al. 1995, Singh et al. 2005) and the four parameter Weibull distribution (Wahed et al. 2009, Jeong 2006) under the same simulation scenarios. Maximum likelihood estimation via Newton–Raphson algorithm is used to fit both distributions to survival data. The set of initial values used to begin the optimisation process is obtained as follows: 1000 initial values from 0 to 100 are randomly using Sobol sequence generator for parameters of both distributions. From these 1000 set of values, the set of initial values that maximises the likelihood is used in the optimisation process.

The sample size 200 and 2000 were chosen to reflect that the number of patients in many Phase III and IV trials are in the vicinity of 200 and some large meta-analysis may combine several studies and reach around 2000 patients. The primary intention of 200 and 2000 sample size is to allow comparison as to the accuracy of the estimates as sample size increases. The general pattern of improved accuracy and numerical precision for larger sample sizes is seen from Figs. 4, 5, 6 and 7 and this is an expected result.

To compare the performance between methods, the relative error was computed. The relative error is defined as the absolute difference between fitted and true quantile divided by the true quantile. This is computed using 100 equally spaced quantiles from 1 % quantile up to the censored quantile for right censored data. For left censored data, this is computed using 100 equally spaced quantiles from the censored quantile up to the 99 % quantile. The log mean and log variance of the relative error among five estimation methods for different types of censoring and different statistical distributions are shown in Figs. 4, 5, 6 and 7. The log transformation is designed to solve the problem of extreme results, to ensure a fairer and clearer comparison across different methods.

Within Figs. 4, 5, 6 and 7, the emphasis is on the performance of different methods. In Fig. 4, it is clear that the exponentiated Weibull is among the worst performing distribution except with respect to fitting of Gumbel distribution data and the four parameter Weibull has fairly comparable performance with GLD but performs less well for Gumbel data. The precision comparison in Fig. 5 indicates similar conclusion as Fig. 4 and the general pattern in these two figures is reflected in Figs. 6 and 7 but with added variability.

Within GLD methods, Figs. 4 and 5 show that the most accurate methods appear to be MLE and PWM matching while quantile matching, ML and LM under SR method appear to be more variable in a number of cases. This is expected as ML/LM/QS under SR introduce extra variability through simulation, due to the nature of the SR algorithm in this article**.** However, this is not always true. The advantage of using SR is seen in Fig. 6 for Gumbel Distribution with 10 % left censored data (GB-LC0.1) for FMKL GLD PWM and FMKL GLD under SR-LM. It is clear that in this example, the direct use of PWM does not result in a fitting result as good as using SR-LM. This is likely due to the difficulty in ascertaining a suitable set of initial values to ensure proper convergence to find the best possible GLD fit, an area where SR can provide valuable guidance and input.

The log variability of relative error plot shows that the methods provided give quite precise results with direct MLE tends to outperforms PWM. ML/LM/QS under SR all have similar performance and often perform slightly worse than MLE or PWM (Figs. 5 and 7).

The perceived, generic pattern of superior performances of MLE or PWM over ML/LM SR should be interpreted with caution. There are also cases, as shown in the example above, where LM SR can in fact outperform from direct use of PWM. Also, note that the optimality of the simulation results comes from the fact that the true distributions are known and a single GLD is an adequate approximation to the true underlying distribution. In practical situations, the true underlying distribution is unknown and may require a mixture of GLDs. When dealing with mixture of GLDs, it is harder to fit a distribution to censored data using direct MLE or PWMs, since this requires maximising or minimising a much more complex objective function which can be difficult in practice. The ML/LM/QS under SR, on the other hand, can be adapted more easily fit mixture of GLDs and the success of these methods in fitting mixtures has already been documented elsewhere (Su 2007b, 2010a, 2010b). SR also tends to give better model for empirical data, as illustrated in Application in empirical data modelling. The main message is that the theoretical loss of efficiency and accuracy using SR is likely to be minimal as evident in these simulation studies but as illustrated below, SR can provide additional information to aid the fitting of a suitable distribution which is not attainable by using an estimation method such as PWM or MLE directly.