(Received January 28, 1994; revised April 10, 1995)
Abstract. 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.
Key words and phrases: Amount of Rényi type information, loss of information, non-regular case, extreme statistics, asymptotically ancillary statistic, truncated distributions.