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Nonparametric maximum likelihood density estimation and simulation-based minimum distance estimators

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Abstract

Indirect inference estimators (i.e., simulation-based minimum distance estimators) in a parametric model that are based on auxiliary nonparametric maximum likelihood density estimators are shown to be asymptotically normal. If the parametricmodel is correctly specified, it is furthermore shown that the asymptotic variance-covariance matrix equals the inverse of the Fisher-information matrix. These results are based on uniform-in-parameters convergence rates and a uniform-inparameters Donsker-type theorem for nonparametric maximum likelihood density estimators.

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References

  1. F. Altissimo and A. Mele, “Simulated Nonparametric Estimation of Dynamic Models”, Rev. Econom. Stud. 76, 413–450 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Beran, Minimum Hellinger Distance Estimates for Parametric Models. Ann. Statist. 5, 445–463 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Berge, Topological Spaces (Oliver and Boyd, Edinburgh, 1963).

    MATH  Google Scholar 

  4. M. S. Birman and M. Z. Solomyak, “Piecewise-Polynomial Approximations of Functions of the ClassW α p ”, Math. USSR Sbornik 2, 295–317 (1967).

    Article  Google Scholar 

  5. M. Carrasco, M. Chernov, J.-P. Florens, and E. Ghysels, “Efficient Estimation of General Dynamic Models with a Continuum of Moment Conditions”, J. Econometrics 140, 529–573 (2007).

    Article  MathSciNet  Google Scholar 

  6. R. M. Dudley, Uniform Central Limit Theorems, in Cambridge Studies in Advanced Mathematics (Cambridge Univ. Press, Cambridge, 1999), Vol. 63.

    Book  MATH  Google Scholar 

  7. N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory (Wiley, New York, 1966).

    Google Scholar 

  8. J. D. Fermanian and B. Salanié, “A Nonparametric Simulated Maximum Likelihood Estimation Method”, Econometric Theory 20, 701–734 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Gach, Efficiency in Indirect Inference, Ph.D. Thesis (Dept. of Statist., Univ. of Vienna, 2010).

  10. F. Gach and B. M. Pötscher, “Nonparametric Maximum Likelihood Density Estimation and Simulation-Based Minimum Distance Estimators”,Working Paper, Univ. of Vienna. arXiv:1012.3851v1.

  11. R. Gallant and J. Long, “Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Squared”, Biometrika 84, 125–141 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Gouriéroux and A. Monfort, Simulation-Based Econometric Methods (Oxford Univ. Press, New York, 1996).

    Google Scholar 

  13. C. Gouriéroux, A. Monfort, and E. Renault, “Indirect Inference”, J. Appl. Econom. 8, 85–118 (1993).

    Article  Google Scholar 

  14. W. Jiang and B. Turnbull, “The Indirect Method: Inference Based on Intermediate Statistics — a Synthesis and Examples”, Statist. Sci. 19, 239–263 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Ledoux and M. Talagrand, Probability in Banach Spaces, in Results in Mathematics and Related Areas (3) (Springer, Berlin, 1991), Vol. 23.

    MATH  Google Scholar 

  16. J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. I., in Die Grundlehren der mathematischen Wissenschaften (Springer, New York-Heidelberg, 1972), Vol. 181.

    Google Scholar 

  17. G. G. Lorentz, M. v. Golitschek, and Y. Makovoz, Constructive Approximation: Advanced Problems (Springer, 1996).

  18. R. Nickl, “Donsker-Type Theorems for Nonparametric Maximum Likelihood Estimators”, Probab. Theory Rel. Fields 138, 411–449 (2007). Erratum ibidem 141, 331–332.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Nickl, “Uniform Central Limit Theorems for Sieved Maximum Likelihood and Trigonometric Series Estimators on the Unit Circle”, In: High Dimensional Probability V: The Luminy Volume, Ed. by C. Houdré, V. Koltchinskii, D. Mason, M. Peligrad, IMS Collections 5, 338–356 (2009).

  20. R. Nickl and B. M. Pötscher, “Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type”, J. Theoret. Probab. 20, 177–199 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Nickl and B. M. Pötscher, “Efficient Simulation-Based Minimum Distance Estimation and Indirect Inference”, Math. Meth. Statist. 19, 327–364 (2010).

    Article  Google Scholar 

  22. B.M. Pötscher and I. R. Prucha, Dynamic Nonlinear Econometric Models. Asymptotic Theory (Springer, Berlin, 1997).

    MATH  Google Scholar 

  23. A. Smith, “Estimating Nonlinear Time-Series Models Using Simulated Vector Autoregressions”, J. Appl. Econom. 8, 63–84 (1993).

    Article  Google Scholar 

  24. H. Triebel, Theory of Function Spaces, in Monographs in Mathematics (Birkhäuser, Basel, 1983), Vol. 78.

    Book  Google Scholar 

  25. S. van de Geer, “Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators”, Ann. Statist. 21, 14–44 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes (Springer, New York, 1996).

    MATH  Google Scholar 

  27. W. P. Ziemer, Weakly Differentiable Functions (Springer, New York, 1989).

    Book  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Dept. of Statist., Univ. of Vienna, Vienna, Austria

    F. Gach & B. M. Pötscher

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  1. F. Gach
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  2. B. M. Pötscher
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Correspondence to B. M. Pötscher.

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Gach, F., Pötscher, B.M. Nonparametric maximum likelihood density estimation and simulation-based minimum distance estimators. Math. Meth. Stat. 20, 288–326 (2011). https://doi.org/10.3103/S1066530711040028

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  • DOI: https://doi.org/10.3103/S1066530711040028

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