Here we will consider a function of κ-statistics, the κ-Weibull distribution, and compare it to the well-known Weibull distribution. The κ-Weibull will be also compared to the 3-parameter extended Weibull function, obtained according to the Marshall–Olkin extended distributions. The log-logistic distribution will be considered for comparison too, such as the exponentiated Weibull, the Burr and the q-Weibull distributions. The most important observation, coming from the proposed calculations, is that the κ-Weibull hazard function is strongly depending on the values of parameter κ, a parameter which is deeply influencing the behaviour of the tail of the probability distribution. As a consequence, the κ-Weibull function turns out to be quite relevant for generalizations of the Weibull approach to modeling failure times. Discussions about the Maximum Likelihood approach for Weibull, κ-Weibull and Burr distributions will be also given.