  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 R\'{e}nyi
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.
