Local M-estimation with Discontinuous Criterion for Dependent and Limited Observations

Abstract

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.

References in corpus (4)

• Inconsistency of bootstrap: The Grenander estimator
• Empirical processes indexed by estimated functions
• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• Confidence sets for split points in decision trees

Cited by in corpus (7)

• Robust inference for threshold regression models
• Factor-Driven Two-Regime Regression
• Quantile regression approach to conditional mode estimation
• Two-Stage Maximum Score Estimator
• Semiparametric Estimation of Dynamic Binary Choice Panel Data Models
• On Gaussian Approximation for M-Estimator
• Semiparametric Discrete Choice Models for Bundles