First, from the Bayesian viewpoint, the information inequality applicable to the non-regular case is discussed. It is shown to be able to construct an estimator which minimizes locally the variance of any estimator satisfying weaker conditions than the unbiasedness condition, from which an information inequality is derived. The Hammersley-Chapman-Robbins inequality is obtained as a special case of the inequality. Next, the Kiefer-type information inequality is extended to the asymptotic situation and is applied to the case of a family of truncated distributions. Further, for a family of non-regular distributions with a location parameter including the uniform and truncated distributions, the stochastic expansion of the Bayes estimator is given and the asymptotic lower bound for the Bayes risk is obtained and shown to be sharp.