Bahadur asymptotic efficiency in the zone of moderate deviation probabilities
Mikhail Ermakov
TL;DR
This paper addresses pointwise Bahadur-type lower bounds for estimator efficiency in the zone of moderate deviations for IID models with parameter $\theta \in \Theta \subset \mathbf{R}^d$. Building on the same assumptions that yield the Hajek–Le Cam locally asymptotically minimax bound, it derives a Bahadur analogue in the moderate-deviation regime, showing the local bound is a special case and extending to multidimensional, one-sided, and cone-restricted bounds. The main condition ensures absolute continuity with finite Fisher information, enabling a series of bounds: (i) a LAN-like lower bound in moderate deviations, (ii) a one-dimensional Bahadur efficiency bound with two-sided and one-sided tails, and (iii) a general multidimensional lower bound employing KL-type divergences and cone geometry. The results clarify the tail behavior of estimator risks beyond consistency and provide tools for precision assessment of confidence regions in moderate-deviation scales, with direct implications for statistical inference in vector-parameter models.
Abstract
For a sequence of independent identically distributed random variables having a distribution function with an unknown parameter from a set $Θ\subset \mathbf{R}^d$, we prove an analogue of the lower bound of Bahadur asymptotic efficiency for the zone of moderate deviation probabilities. The assumptions coincide with assumptions conditions under which the locally asymptotically minimax lower bound of Hajek-Le Cam was proved. The lower bound for local Bahadur asymptotic efficiency is a special case of this lower bound.