Loss of information of a statistic for a family of non-regular distributions, II: more general case

In the paper of Akahira (Ann Inst Statist Math 48:349–364, <CitationRef CitationID="CR9">1996</CitationRef>), it was shown that the second order asymptotic loss of information in reducing to a statistic consisting of extreme values and an asymptotically ancillary statistic vanished for a family of non-regular distributions whose densities have the same values and the sum of differential coefficients at the endpoints of the bounded support is equal to zero. In this paper, the result can be shown to be extended to the case of a family of non-regular distributions without the above restriction. Copyright The Institute of Statistical Mathematics, Tokyo 2012