Second Order Efficiency of Minimum Contrast Estimators in a Curved Exponential Family

This paper presents a sufficient condition for second order efficiency of an estimator. The condition is easily checked in the case of minimum contrast estimators. The $\alpha^\ast$-minimum contrast estimator is defined and proved to be second order efficient for every $\alpha, 0 < \alpha < 1$. The Fisher scoring method is also considered in the light of second order efficiency. It is shown that a contrast function is associated with the second order tensor and the affine connection. This fact leads us to prove the above assertions in the differential geometric framework due to Amari.