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Optimal Rates of Convergence of Parameter Estimators in the Binary Response Model with Weak Distributional Assumptions

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  • Horowitz, Joel L.

Abstract

The smoothed maximum score estimator of the coefficient vector of a binary response model is consistent and, after centering and suitable normalization, asymptotically normally distributed under weak assumptions [5]. Its rate of convergence in probability is N − , where h ≥ 2 is an integer whose value depends on the strength of certain smoothness assumptions. This rate of convergence is faster than that of the maximum score estimator of Manski [11,12], which converges at the rate N −1/3 under assumptions that are somewhat weaker than those of the smoothed estimator. In this paper I prove that under the assumptions of smoothed maximum score estimation, N − is the fastest achievable rate of convergence of an estimator of the coefficient vector of a binary response model. Thus, the smoothed maximum score estimator has the fastest possible rate of convergence. The rate of convergence is defined in a minimax sense so as to exclude superefficient estimators.

Suggested Citation

  • Horowitz, Joel L., 1993. "Optimal Rates of Convergence of Parameter Estimators in the Binary Response Model with Weak Distributional Assumptions," Econometric Theory, Cambridge University Press, vol. 9(01), pages 1-18, January.
  • Handle: RePEc:cup:etheor:v:9:y:1993:i:01:p:1-18_00
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    Cited by:

    1. Horowitz, Joel L., 2004. "Semiparametric models," Papers 2004,17, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    2. Beckert, Walter & McFadden, Daniel L., 2010. "Maximal Uniform Convergence Rates In Parametric Estimation Problems," Econometric Theory, Cambridge University Press, vol. 26(02), pages 469-500, April.
    3. Khan, Shakeeb, 2013. "Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions," Journal of Econometrics, Elsevier, vol. 172(1), pages 168-182.
    4. Lee, Sokbae & Seo, Myung Hwan, 2008. "Semiparametric estimation of a binary response model with a change-point due to a covariate threshold," Journal of Econometrics, Elsevier, vol. 144(2), pages 492-499, June.
    5. Hoderlein, Stefan & Sherman, Robert, 2015. "Identification and estimation in a correlated random coefficients binary response model," Journal of Econometrics, Elsevier, vol. 188(1), pages 135-149.
    6. Chen, Songnian & Zhang, Hanghui, 2015. "Binary quantile regression with local polynomial smoothing," Journal of Econometrics, Elsevier, vol. 189(1), pages 24-40.
    7. Jun, Sung Jae & Pinkse, Joris & Wan, Yuanyuan, 2015. "Classical Laplace estimation for n3-consistent estimators: Improved convergence rates and rate-adaptive inference," Journal of Econometrics, Elsevier, vol. 187(1), pages 201-216.

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