  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.
