The concept of asymptotic sufficiency was extended to the non-regular case in Sect. 1.4 . In this chapter, for a one-directional family of distributions, the second order asymptotic sufficiency is discussed, and a set of the extremal statistics and an ancillary statistic is shown to be second order asymptotically sufficient. The two-sided asymptotic efficiency is also discussed in the sense that the estimator has asymptotically the most concentration probability (CP) in any symmetric interval around a true parameter in the class of all asymptotically median unbiased (AMU) estimators. Indeed, the bound for the CP of AMU estimators is obtained in non-regular cases, and some examples are also given.