Optimal Berry-Esseen bound for statistical estimations and its application to SPDE

We consider asymptotically normal statistics of the form F-n/G(n), where F-n and G(n) are functionals of Gaussian fields. For these statistics, we establish an optimal Berry-Esseen bound for the Central Limit Theorem (CLT) of the sequence F-n/G(n), is phi (n) in the following sense: there exist constants 0 < c < C < infinity such that c <= d(Kol) (F-n/G(n), Z)/phi(n) <= C, where d(Kol) (F-n, Z) = sup(z is an element of R). vertical bar Pr(F-n, <= z) - Pr(Z <= z)vertical bar. As an example, we find an optimal Berry-Esseen bound for the CLT of the maximum likelihood estimators for parameters occurring in parabolic stochastic partial differential equations. (C) 2017 Elsevier Inc. All rights reserved.