Loss of information of a statistic for a family of non-regular distributions

In the non-regular case, the asymptotic loss of amount of information (extended to as Rényi measure) associated with a statistic is discussed. It is shown that the second order asymptotic loss of information in reducing to a statistic consisting of extreme values and an asymptotically ancillary statistic vanishes. This result corresponds to the fact that the statistic is second order asymptotically sufficient in the sense of Akahira (1991, Metron, 49, 133–143). Some examples on truncated distributions are also given.