Limiting behavior of relative Rényi entropy in a non-regular location shift family

In a regular distribution family, Cramér-Rao inequality holds, and the maximum likelihood estimator (MLE) converges to a normal distribution whose variance is the inverse of the Fisher information because the Fisher information converges and is well-defined in this family. However, in a non-regular location shift family which is generated by a distribution of R whose support is not R (e.g., a Weibull distribution, gamma distribution, or beta distribution), the Fisher information diverges and cannot be defined. Thus, one might think that a substitute information quantity is necessary for a discussion of the asymptotic theory. Akahira and Takeuchi [1] proposed the limit of the Hellinger affinity − log ∫