In Chap. 4 , the phenomenon “first order efficiency implies second order efficiency” was derived in the regular cases, which yields the problem how to discriminate between second order asymptotically efficient estimators. In order to do so, the concept of asymptotic deficiency introduced by Hodges and Lehmann (1970) is useful. In this chapter, under suitable regularity conditions, the asymptotic deficiency is characterized by the terms of order \(1/n\) in the asymptotic variances of asymptotically efficient estimators. But the asymptotic deficiency is unavailable in non-regular cases. As an intermediate concept between first order asymptotic efficiency and asymptotic deficiency, the normalized deficiency is proposed. The concept is shown to be very useful in discriminating first order but not the next order asymptotically efficient estimators which often appear in some kind of non-regular cases like the two-sided exponential case.