Asymptotic minimax results for stochastic process families with critical points

We give two local asymptotic minimax bounds for models which admit a local quadratic approximation at every parameter point, but are not necessarily locally asymptotically normal or mixed normal. Such parameter points appear as critical points for stochastic process models exhibiting both stationary and explosive behavior. The first result shows that, for estimators normalized with the random Fisher information, the classical bound for the mixed normal case remains valid. However, the bound is not attained by asymptotically centering estimators. The second result refers to filtered models. It gives a sharp bound for estimators based on observing the path of a process until the random Fisher information exceeds a given constant.