We prove that, under certain regularity conditions, the asymptotic distribution of the Koenker – Bassett estimator coincides with the asymptotic distribution of the integral of indicator process generated by a random noise weighted by the gradient of the regression function.
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References
J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, New York (1994).
N. N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer, Dordrecht (1999).
P. Doukhan, G. Oppenheim, and M. S. Takku, Theory and Applications of Long-Range Dependence, Birkhauser, Boston (2003).
U. Grenander and M. Rozenblatt, Statistical Analysis of Stationary Time Series, Wiley, New York (1957).
G. Bassett and R. Koenker, “Regression quantile,” Econometrica, 46, 33–50 (1978).
O. V. Ivanov and I. V. Orlovs’kyi, “Asymptotic normality of the Koenker–Bassett estimators in nonlinear regression models,” Teor. Imovirn. Mat. Statist., Issue 72, 30–41 (2005).
I. V. Orlovs’kyi, “Consistency of the Koenker–Bassett estimators in nonlinear regression models,” Nauk. Visti NTUU “KPI,” No. 3 (35), 144–150 (2004).
A. G. Kukush, J. Beirlant, and Y. Goegebeur, Nonparametric Estimation of Conditional Quantiles, Dept. Appl. Econ., Kath. Univ., Leuven (2005), Res. Rept. OR 0557.
R. I. Jennrich, “Asymptotic properties of nonlinear least-squares estimators,” Ann. Math. Statist., 40, 633–643 (1969).
J. Pfanzagl, On the measurability and consistency of minimum contrast estimates,” Metrika, 14, 249–272 (1969).
L. Schmetterer, Introduction to Mathematical Statistics, Springer, Berlin (1974).
A. V. Ivanov, Asymptotic Theory of Nonlinear Regression, Kluwer, Dordrecht (1997).
I. M. Savych, “Consistency of quantile estimates in regression models with strongly dependent noise,” Teor. Imovirn. Mat. Statist., No. 82, 128–136 (2010).
I. A. Ibragimov and Yu. A. Rozanov, Gaussian Random Processes, Springer, Berlin (1978).
A. S. Kholevo, “On the estimates of regression coefficients,” Teor. Veroyatn. Primen., 14, 78–101 (1969).
A. S. Kholevo, “On the asymptotic normality of the estimators of regression coefficients,” Teor. Veroyatn. Primen., 16, 724–728 (1971).
A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, Kluwer, Dordrecht (1989).
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).
A. V. Ivanov and I. N. Savych, “μ-Admissibility of the spectral density of a strongly dependent random noise in nonlinear regression models,” Nauk. Visti NTUU “KPI,” No. 1, 143–148 (2009).
A. V. Ivanov and I. N. Savych, “Asymptotic properties of the Koenker–Bassett estimator in the regression model with long-range dependence,” in: Internat. Conf. on “Modern Stochastics: Theory and Applications II” (07–10.09.2010), Kyiv (2010), p. 88.
P. J. Huber, Robust Statistics, Wiley, New York (1981).
P. J. Huber, “The behavior of maximum likelihood estimates under nonstandard conditions,” in: Proc. of the 5th Berkeley Symp. on Math. Statistics and Probability, Vol. 1, Univ. Calif. Press, Berkeley (1967), pp. 221–233.
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1962).
E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).
V. V. Anh, V. P. Knopova, and N. N. Leonenko, “Continuous-time stochastic processes with cyclical long-range dependence,” Austral. N. Z. J. Statist., 46, No. 2, 275–296 (2004).
S. R. Rao, Linear Statistical Inference and Its Applications, Wiley, New York (1965).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1030–1052, August, 2011.
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Ivanov, O.V., Savych, I.M. On the asymptotic distribution of the Koenker–Bassett estimator for a parameter of the nonlinear model of regression with strongly dependent noise. Ukr Math J 63, 1187–1212 (2012). https://doi.org/10.1007/s11253-012-0572-x
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DOI: https://doi.org/10.1007/s11253-012-0572-x