Robust Estimation With Exponentially Tilted Hellinger Distance

This paper is concerned with estimation of parameters defined by general estimating equations in the form of a moment condition model. In this context, Kitamura, Otsu and Evdokimov (2013a) have introduced the minimum Hellinger distance (HD) estimator which is asymptotically semiparametrically efficient when the model assumption holds (correct specification) and achieves optimal minimax robust properties under small deviations from the model (local misspecification). In this paper, we evaluate the performance of inference procedures of interest under two complementary types of misspecification, local and global. First, we show that HD is not robust to global misspecification in the sense that HD may cease to be root n convergent when the functions defining the moment conditions are unbounded. Second, in the spirit of Schennach (2007), we introduce the exponentially tilted Hellinger distance (ETHD) estimator by combining the Hellinger distance and the Kullback-Leibler information criterion. Our estimator shares the same desirable asymptotic properties as HD under correct specification and local misspecification, and remains well-behaved under global misspecification. ETHD is therefore the first estimator that is efficient under correct specification, and robust to both global and local misspecification.