  This paper examines 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, 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 whose dimension may grow to
infinity. Furthermore, the localization can occur around parameter values
rather than around a fixed point and the observation may take limited values,
which leads to set estimators. Our theory produces three different
nonparametric cube root rates and enables valid inference for the local
M-estimators, building on novel maximal inequalities for weakly dependent data.
Our results include the standard cube root asymptotics as a special case. To
illustrate the usefulness of our results, we verify our conditions for various
examples such as the Hough transform estimator with diminishing bandwidth,
maximum score-type set estimator, and many others.
