We examine the asymptotic properties of local M-estimators under three sets of high-level conditions. These conditions are sufficiently general to cover the minimum volume predictive region, the conditional maximum score estimator for a panel data discrete choice model and many other widely used estimators in statistics and econometrics. Specifically, they allow for discontinuous criterion functions of weakly dependent observations which may be localized by kernel smoothing and contain nuisance parameters with growing dimension. Furthermore, the localization can occur around parameter values rather than around a fixed point and the observations may take limited values which lead to set estimators. Our theory produces three different nonparametric cube root rates for local M-estimators and enables valid inference building on novel maximal inequalities for weakly dependent observations. The standard cube root asymptotics is included as a special case. The results are illustrated by various examples such as the Hough transform estimator with diminishing bandwidth, the maximum score-type set estimator and many others.
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