Certain distributions do not have a closed-form density, but it is simple
to draw samples from them. For such distributions, simulated minimum Hellinger
distance (SMHD) estimation appears to be useful. Since the method is
distance-based, it happens to be naturally robust. This paper is a follow-up to
a previous paper where the SMHD estimators were only shown to be consistent;
this paper establishes their asymptotic normality. For any parametric family of
distributions for which all positive integer moments exist, asymptotic
properties for the SMHD method indicate that the variance of the SMHD
estimators attains the lower bound for simulation-based estimators, which is
based on the inverse of the Fisher information matrix, adjusted by a constant
that reflects the loss of efficiency due to simulations. All these features
suggest that the SMHD method is applicable in many fields such as finance or
actuarial science where we often encounter distributions without closed-form
density.
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