Abstract
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.
Highlights
In actuarial science and finance, we often have to fit data with a distribution that is continuous
For any parametric family of distributions for which all positive integer moments exist, asymptotic properties for the simulated minimum Hellinger distance (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
Tamura and Boos worked with kernel density estimates with nonrandom bandwidths and the assumption that the parametric family has a closed-form density. They can relax the requirement that the parametric family needs a compact support, as given by Beran [3] in his seminal paper. They have obtained the result that, in general, if parametric families have positive integer moments of all orders, the minimum Hellinger distance (MHD) estimators in the univariate case will have the same efficiency as the maximum likelihood (ML) estimators and the asymptotic covariance matrix can be based on the Fisher information matrix just as for the ML estimators
Summary
In actuarial science and finance, we often have to fit data with a distribution that is continuous. They can relax the requirement that the parametric family needs a compact support, as given by Beran [3] in his seminal paper Rather, they have obtained the result that, in general, if parametric families have positive integer moments of all orders, the minimum Hellinger distance (MHD) estimators in the univariate case will have the same efficiency as the ML estimators and the asymptotic covariance matrix can be based on the Fisher information matrix just as for the ML estimators. It only requires an adjustment by a constant which depends on τ, the ratio between the sample size U drawn from the parametric family and the original sample size n of the data. Though the objective function to be minimized is nonsmooth, we can establish asymptotic normality for the SMHD estimators and show that the SMHD estimators attain the lower bound within the class of simulated estimators just as, for the parametric model, the MHD or ML estimators attain the lower bound based on the Fisher information matrix when simulations are not needed because the parametric model density has a closed-form expression
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