Usually, we estimate the unknown parameter by observing a random sample and using the usual methods of estimation such as maximum likelihood method. In some situations, we have information about the real parameter in the form of a guess. In these cases, one may shrink the maximum likelihood or other estimators towards a guess value and construct a shrinkage estimator. In this paper, we study the behavior of a Bayes shrinkage estimator for the scale parameter of exponential distribution based on censored samples under an asymmetric and scale invariant loss function. To do this, we propose a Bayes shrinkage estimator and compute the relative efficiency between this estimator and the best linear estimator within a subclass with respect to sample size, hyperparameters of the prior distribution and the vicinity of the guess and real parameter. Also, the obtained results are extended to Weibull and Rayleigh lifetime distributions.