  We develop two new classes of tests for the Weibull distribution based on
Stein's method. The proposed tests are applied in the full sample case as well
as in the framework of random right censoring. We investigate the finite sample
performance of the new tests using a comprehensive Monte Carlo study. In both
the absence and presence of censoring, it is found that the newly proposed
classes of tests outperform competing tests against the majority of the
distributions considered. In the cases where censoring is present we consider
various censoring distributions. Some remarks on the asymptotic properties of
the proposed tests are included. The paper presents another result of
independent interest; the test initially proposed in Krit (2014) for use with
full samples is amended to allow for testing for the Weibull distribution in
the presence of censoring. The techniques developed in the paper are
illustrated using two practical examples. In the first, we consider the
survival times of patients with a certain type of leukemia. The second example
is concerned with the initial remission times of leukemia patients, where the
observed remission times are subject to random right censoring. We further
include some concluding remarks along with avenues for future research.
