Two Non-Regular Extensions of Large Deviation Bound

Abstract

We formulate two types of extensions of the large deviation theory initiated by Bahadur in a non-regular setting. One can be regarded as a bound of the point estimation; the other can be regarded as the limit of a bound of the interval estimation. Both coincide in the regular case but do not necessarily coincide in a non-regular case. Using the limits of relative Rényi entropies, we derive their upper bounds and give a necessary and sufficient condition for the coincidence of the two upper bounds. We also discuss the attainability of these two bounds in several non-regular location shift families.

Keywords:

• Interval estimation
• Large deviation
• Non-regular family
• Point estimation
• Relative Rényi entropy

Mathematics Subject Classification:

• 62F12
• 62F25

Acknowledgment

The present study was supported in part by the Laboratory for Mathematical Neuroscience, Brain Science Institute, RIKEN and MEXT through a Grant-in-Aid for Scientific Research on Priority Area “Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI),” No. 18079014. The author is grateful for Professor Shun-ichi Amari's helpful discussions on this topics.