Rare Event Analysis for Minimum Hellinger Distance Estimators via Large Deviation Theory.

Area of Science:

• Statistics
• Probability Theory
• Econometrics

Background:

• Hellinger distance is a common alternative to maximum likelihood estimation.
• Asymptotic distributions of Hellinger distance estimators are known, but rare event probabilities are not.
• Model misspecification is a practical concern in statistical modeling.

Purpose of the Study:

• To analyze rare event probabilities induced by Hellinger distance estimators under model misspecification.
• To characterize the exponential decay of these probabilities using large deviation theory.
• To provide an explicit representation of the rate function, even for non-differentiable cases.

Main Methods:

• Application of large deviation theory to Hellinger distance estimators.
• Analysis under potential model misspecification in one and higher dimensions.
• Characterization of decay rates using a rate function (convex conjugate of a limiting cumulant generating function).
• Investigation of geometric considerations for explicit representation of the lower bound.

Main Results:

• Rare event probabilities decay exponentially.
• A rate function characterizes this exponential decay.
• Explicit representation of the rate function is achieved, including non-differentiable cases.
• Modulus of continuity properties of the affinity are analyzed.

Conclusions:

• Large deviation theory effectively analyzes rare event probabilities for Hellinger distance estimators under misspecification.
• The derived rate function provides a precise characterization of these probabilities.
• The findings offer new insights into the behavior of Hellinger distance estimators in the presence of model uncertainty.