Before revision:  Correspondingly, we give conditions on the parameter process and the block length under which we can establish the local central limit theorem, and the boundedness of moments of order 2\kappa of the local estimators, where \kappa > 1. Under those assumptions, the global estimator asymptotic bias explodes asymptotically . As a consequence, we provide a non-na\"{i}ve estimator , which is constructed as the na\"{i}ve one when applying a first-order bias reduction to the local MLE.
After revision:  Correspondingly, we give conditions on the parameter process and the block length under which we can establish the local central limit theorem, and the boundedness of moments of order 2\kappa of the local estimators, where \kappa > 1. Under those assumptions, the global estimator asymptotic bias explodes asymptotically , thus we provide a non-na\"{i}ve estimator , which is constructed as the na\"{i}ve one when applying a first-order bias reduction to the local MLE.