Summary: The Weibull distribution has been observed as one of the most useful distribution for modelling and analysing life time data in engineering, biology, and others. Studies have been done vigorously in the literature to determine the best method for estimating its parameters. Recently, much attention has been given to the Bayesian estimation approach for parameter estimation which is in contention with other estimation methods. We examine the performance of the maximum likelihood estimator and Bayesian estimator using extensions of the Jeffreys prior information with three loss functions, namely, the linear exponential loss, general entropy loss, and the square error loss function for estimating the two-parameter Weibull failure time distribution. These methods are compared using the mean squared error through simulation studies with varying sample sizes. The results show that the Bayesian estimator using extensions of Jeffreys’ prior under linear exponential loss functions in most cases gives the smallest mean squared error and the absolute bias for both the scale parameter \(\alpha\) and the shape parameter \(\beta\) for the given values of the extension of Jeffreys’ prior.