In most cases, we use a symmetric loss such as the quadratic loss in a usual estimation problem. But, in the non-regular case when the regularity conditions do not necessarily hold, it seems to be more reasonable to choose an asymmetric loss than the symmetric one. In this paper, we consider the Bayes estimation under the linear exponential (LINEX) loss which is regarded as a typical example of asymmetric loss. We also compare the Bayes risks of estimators under the LINEX loss for a family of truncated distributions and a location parameter family of truncated distributions.