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Two Non-Regular Extensions of Large Deviation Bound

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.