The Annals of Statistics

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

Myung Hwan Seo and Taisuke Otsu

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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

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

Received: March 2016
Revised: October 2016
First available in Project Euclid: 22 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators
Secondary: 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 62G20: Asymptotic properties

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


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.

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  • Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator. Econometrica 73 1175–1204.
  • Adamczak, R. (2008). A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 1000–1034.
  • Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H. and Tukey, J. W. (1972). Robust Estimates of Location: Survey and Advances. Princeton Univ. Press, Princeton, NJ.
  • Anevski, D. and Hössjer, O. (2006). A general asymptotic scheme for inference under order restrictions. Ann. Statist. 34 1874–1930.
  • Banerjee, M. and McKeague, I. W. (2007). Confidence sets for split points in decision trees. Ann. Statist. 35 543–574.
  • Baraud, Y. (2010). A Bernstein-type inequality for suprema of random processes with applications to model selection in non-Gaussian regression. Bernoulli 16 1064–1085.
  • Belloni, A., Chen, D., Chernozhukov, V. and Hansen, C. (2012). Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica 80 2369–2429.
  • Bühlmann, P. and Yu, B. (2002). Analyzing bagging. Ann. Statist. 30 927–961.
  • Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18 17–39.
  • Chan, K. S. (1993). Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Ann. Statist. 21 520–533.
  • Chen, X. (2007). Chapter 76. Large sample sieve estimation of semi-nonparametric models. In Handbook of Econometrics 6B 5549–5632. Elsevier, Amsterdam.
  • Chen, X., Hansen, L. P. and Carrasco, M. (2010). Nonlinearity and temporal dependence. J. Econometrics 155 155–169.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31–41.
  • Chernozhukov, V., Hong, H. and Tamer, E. (2007). Estimation and confidence regions for parameter sets in econometric models. Econometrica 75 1243–1284.
  • de Jong, R. M. and Woutersen, T. (2011). Dynamic time series binary choice. Econometric Theory 27 673–702.
  • Doukhan, P., Massart, P. and Rio, E. (1995). Invariance principles for absolutely regular empirical processes. Ann. Inst. Henri Poincaré Probab. Stat. 31 393–427.
  • Gautier, E. and Kitamura, Y. (2013). Nonparametric estimation in random coefficients binary choice models. Econometrica 81 581–607.
  • Goldenshluger, A. and Zeevi, A. (2004). The Hough transform estimator. Ann. Statist. 32 1908–1932.
  • Hirano, K. and Porter, J. R. (2003). Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica 71 1307–1338.
  • Honoré, B. E. and Kyriazidou, E. (2000). Panel data discrete choice models with lagged dependent variables. Econometrica 68 839–874.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
  • Koo, B. and Seo, M. H. (2015). Structural-break models under mis-specification: Implications for forecasting. J. Econometrics 188 166–181.
  • Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer, New York.
  • Lee, M.-J. (1989). Mode regression. J. Econometrics 42 337–349.
  • Manski, C. F. (1975). Maximum score estimation of the stochastic utility model of choice. J. Econometrics 3 205–228.
  • Manski, C. F. and Tamer, E. (2002). Inference on regressions with interval data on a regressor or outcome. Econometrica 70 519–546.
  • Merlevède, F., Peligrad, M. and Rio, E. (2009). Bernstein inequality and moderate deviations under strong mixing conditions. In High Dimensional Probability V: The Luminy Volume. Inst. Math. Stat. (IMS) Collect. 5 273–292. IMS, Beachwood, OH.
  • Merlevède, F., Peligrad, M. and Rio, E. (2011). A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probab. Theory Related Fields 151 435–474.
  • Nickl, R. and Söhl, J. (2016). Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions. Working paper, arXiv:1510.05526v2.
  • Paulin, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. Electron. J. Probab. 20 1–32.
  • Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
  • Pollard, D. (1989). Asymptotics via empirical processes. Statist. Sci. 4 341–366.
  • Polonik, W. and Yao, Q. (2000). Conditional minimum volume predictive regions for stochastic processes. J. Amer. Statist. Assoc. 95 509–519.
  • Prakasa Rao, B. L. S. (1969). Estkmation of a unimodal density. Sankhy$\bar{a}$ Ser. A 31 23–36.
  • Rio, E. (1997). About the Lindeberg method for strongly mixing sequences. ESAIM Probab. Stat. 1 35–61.
  • Romano, J. P. and Shaikh, A. M. (2008). Inference for identifiable parameters in partially identified econometric models. J. Statist. Plann. Inference 138 2786–2807.
  • Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871–880.
  • Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 1953–1977.
  • Seo, M. H and Otsu, T. (2018). Supplement to “Local M-estimation with discontinuous criterion for dependent and limited observations.” DOI:10.1214/17-AOS1552SUPP.
  • Talagrand, M. (2005). The Generic Chaining. Springer, Berlin.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • van der Vaart, A. W. and Wellner, J. A. (2007). Empirical processes indexed by estimated functions. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 234–252. IMS, Beachwood, OH.
  • Yao, W., Lindsay, B. G. and Li, R. (2012). Local modal regression. J. Nonparametr. Stat. 24 647–663.
  • Zinde-Walsh, V. (2002). Asymptotic theory for some high breakdown point estimators. Econometric Theory 18 1172–1196.

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.