Abstract
Given a random sample from a parametric model, we show how indirect inference estimators based on appropriate nonparametric density estimators (i.e., simulation-based minimum distance estimators) can be constructed that, under mild assumptions, are asymptotically normal with variance-covariance matrix equal to the Cramér-Rao bound.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. (Elsevier, 2003).
F. Altissimo and A. Mele, “Simulated Nonparametric Estimation of Dynamic Models”, Rev. Economic Studies, 76, 413–450 (2009).
R. Beran, “MinimumHellingerDistance Estimates for ParametricModels”, Ann. Statist. 5, 445–463 (1977).
R. Beran and P.W. Millar, “Stochastic Estimation and Testing”, Ann. Statist. 15, 1131–1154 (1987).
P. Bickel and Y. Ritov, “Nonparametric Estimators that Can Be ‘Plugged-In’”, Ann. Statist. 31, 1033–1053 (2003).
M. Carrasco, M. Chernov, J. P. Florens, and E. Ghysels, “Efficient Estimation of General Dynamic Models with a Continuum ofMoment Conditions”, J. Econometrics 140, 529–573 (2007).
R.A. DeVore and G. G. Lorentz, Constructive Approximation (Springer, 1993).
D. L. Donoho and R. C. Liu, The ‘Automatic’ Robustness ofMinimum Distance Functionals”, Ann. Statist. 16, 552–586 (1988).
J. D. Fermanian and B. Salanié, “A Nonparametric Simulated Maximum Likelihood Estimation Method”, Econometric Theory 20, 701–734 (2004).
G. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed. (Wiley, 1999).
F. Gach, Efficiency in Indirect Inference, PhD Thesis (Univ. of Vienna, 2010).
F. Gallant and G. Tauchen, “WhichMoments to Match?”, EconometricTheory 12, 657–681 (1996).
R. Gallant and J. Long, “Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Squared”, Biometrika 84, 125–141 (1997).
E. Giné and V. Koltchinskii, “Concentration Inequalities and Asymptotic Results for Ratio Type Empirical Processes”, Ann. Probab. 34, 1143–1216 (2006).
E. Giné, R. Latała, and J. Zinn, “Exponential and Moment Inequalities for U-Statistics”, in: Progress in Probability, Vol. 47: High-dimensional Probability II, ed. by E. Giné, D. M. Mason, and J. A. Wellner (2000), pp. 13–38.
E. Giné and R. Nickl, “Uniform Central Limit Theorems for Kernel Density Estimators”, Probab. Theory Rel. Fields 141, 333–387 (2008).
E. Giné and R. Nickl, “An Exponential Inequality for the Distribution Function of the Kernel Density Estimator, with Applications to Adaptive Estimation”, Probab. Theory Rel. Fields, 143, 569–596 (2009a).
E. Giné and R. Nickl, “Uniform Limit Theorems forWavelet Density Estimators”, Ann. Probab., 37, 1605–1646 (2009b).
C. Gourieroux, A. Monfort, and E. Renault, “Indirect Inference”, J. Appl. Econometrics 8, 85–118 (1993).
C. Gourieroux and A. Monfort, Simulation-Based Econometric Methods (Oxford Univ. Press, 1996).
P. J. Huber, “Robust Statistics: A Review”, Ann. Math. Statist. 43, 1041–1067 (1972).
W. Jiang and B. Turnbull, “The IndirectMethod: Inference Based on Intermediate Statistics—A Synthesis and Examples”, Statist. Science 19, 239–263 (2004).
Lindsay, “Efficiency Versus Robustness: The Case for Minimum Hellinger Distance and Related Methods”, Ann. Statist. 22, 1081–1114 (1994).
G. G. Lorentz, M. v. Golitschek, and Y. Makovoz, Constructive Approximation: Advanced Problems (Springer, 1996).
P.W. Millar, “Robust Estimation viaMinimum Distance Methods”, Z. Wahrsch. und Verw.Gebiete 55, 73–89 (1981).
R. Nickl, “Donsker-Type Theorems for Nonparametric Maximum Likelihood Estimators”, Probab. Theory Rel. Fields 138, 411–449 (2007).
B. M. Pötscher and I. R. Prucha, Dynamic Nonlinear EconometricModels: Asymptotic Theory (Springer, 1997).
A. Yu. Shadrin, “The L ∞-Norm of the L 2-Spline Projector is Bounded Independently of the Knot Sequence: A Proof of de Boor’s Conjecture”, ActaMathematica 187, 59–137 (2001).
A. Smith, “Estimating Nonlinear Time-Series Models Using Simulated Vector Autoregressions”, J. Applied Econometrics 8, 63–84 (1993).
A.W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes with Applications to Statistics (Springer, 1996).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Nickl, R., Pötscher, B.M. Efficient simulation-based minimum distance estimation and indirect inference. Math. Meth. Stat. 19, 327–364 (2010). https://doi.org/10.3103/S1066530710040022
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530710040022