The Weibull distribution, with shape parameter $k>0$ and scale parameter $\lambda>0$, is one of the most popular parametric distributions in survival analysis with complete or censored data. Although inference of the parameters of the Weibull distribution is commonly done through maximum likelihood, it is well established that the maximum likelihood estimate of the shape parameter is inadequate due to the associated large bias when the sample size is small or the proportion of censored data is large. This manuscript demonstrates how the Bayesian information-theoretic minimum message length principle coupled with a suitable choice of weakly informative prior distributions, can be used to infer Weibull distribution parameters given complete data or data with type I censoring. Empirical experiments show that the proposed minimum message length estimate of the shape parameter is superior to the maximum likelihood estimate and appears superior to other recently proposed modified maximum likelihood estimates in terms of Kullback-Leibler risk. Lastly, we derive an extension of the proposed method to data with type II censoring.

翻译：威布尔分布（形状参数 $k>0$，尺度参数 $\lambda>0$）是生存分析中处理完整或删失数据时最常用的参数分布之一。尽管威布尔分布的参数推断通常通过极大似然方法进行，但现有研究表明，当样本量较小或删失数据比例较大时，形状参数的极大似然估计会因显著偏差而表现不佳。本文展示了如何结合贝叶斯信息论中的最小信息长度原则与恰当的弱信息先验分布，在完整数据或第一类删失数据条件下推断威布尔分布参数。实证实验表明，本文提出的形状参数最小信息长度估计在Kullback-Leibler风险指标上优于极大似然估计，且优于近期提出的其他改进型极大似然估计。最后，我们推导了该方法在第二类删失数据情形下的扩展形式。