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

Abstract

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 φ ( n ) in the following sense: there exist constants 0 < c < C < ∞ such that c ≤ d Kol ( F n / G n , Z ) / φ ( n ) ≤ C , where d Kol ( F n , Z ) = sup z ∈ R ∣ Pr ( F n ≤ z ) − Pr ( Z ≤ z ) ∣ . 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.