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.
"Second Order Efficiency of Minimum Contrast Estimators in a Curved Exponential Family." Ann. Statist. 11 (3) 793 - 803, September, 1983. https://doi.org/10.1214/aos/1176346246