The asymptotic loss of the amount of information (extended to as Rényi measure) associated with a statistic is discussed in non-regular cases. It is shown that the second order asymptotic loss of information in reducing to a statistic consisting of extremes and asymptotically ancillary statistic vanishes. The result corresponds to the fact that the statistic is second order asymptotically sufficient in the sense of Theorem 2.2 . Some examples on truncated distributions are given.