## The Annals of Statistics

- Ann. Statist.
- Volume 46, Number 1 (2018), 344-369.

### Local M-estimation with discontinuous criterion for dependent and limited observations

Myung Hwan Seo and Taisuke Otsu

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 46, Number 1 (2018), 344-369.

**Dates**

Received: March 2016

Revised: October 2016

First available in Project Euclid: 22 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1519268433

**Digital Object Identifier**

doi:10.1214/17-AOS1552

**Mathematical Reviews number (MathSciNet)**

MR3766955

**Zentralblatt MATH identifier**

06865114

**Subjects**

Primary: 62F12: Asymptotic properties of estimators

Secondary: 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 62G20: Asymptotic properties

**Keywords**

Cube root asymptotics maximal inequality mixing process partial identification parameter-dependent localization

#### Citation

Seo, Myung Hwan; Otsu, Taisuke. Local M-estimation with discontinuous criterion for dependent and limited observations. Ann. Statist. 46 (2018), no. 1, 344--369. doi:10.1214/17-AOS1552. https://projecteuclid.org/euclid.aos/1519268433

#### Supplemental materials

- Supplement to “Local M-estimation with discontinuous criterion for dependent and incomplete observation”. The supplement contains all the proofs of the theorems and lemmas, details for illustrations and additional examples.Digital Object Identifier: doi:10.1214/17-AOS1552SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.