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Theory of Probability and Mathematical Statistics

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Asymptotic distributions of least squares estimators of the coefficients in the model of linear regression with nonlinear constraints and long-memory dependence

Author: E. M. Moldavs'ka
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal: Theor. Probability and Math. Statist. 75 (2007), 121-137
MSC (2000): Primary 62E20, 62F10; Secondary 60G18
Published electronically: January 24, 2008
MathSciNet review: 2321186
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Abstract: We consider least squares estimators for linear regression models with long-memory dependence, continuous time, and nonlinear inequality constraints imposed on the parameter. We study the solution of the problem of minimization of the least squares functional in the linear regression with a given (long) radius of dependence and nonlinear inequality constraints imposed on the parameter. We prove that the solution being appropriately centered and normalized converges in distribution to the solution of the quadratic programming problem. The latter solution is non-Gaussian in contrast to known results for long-memory dependence without constraints for which an analogous transform of the solution of the minimization problem is asymptotically Gaussian in many typical cases.

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  • 1. P. S. Knopov, On a nonstationary model of $ M$-estimators with discrete time, Teor. Imovir. Mat. Statist. 57 (1997), 60-66; English transl. in Theory Probab. Math. Statist. 57 (1998), 61-67. MR 1806885 (2003c:62061)
  • 2. A. S. Korkhin, Some properties of estimates for regression parameters with a priori inequality constraints, Kibernetika 6 (1985), 106-115. (Russian) MR 825195 (87c:62119)
  • 3. E. M. Moldavs'ka, Methods for the Identification of Parameters of Stochastic Systems with Weak and Strong Dependence, Candidate Dissertation, Kyiv University, Kyiv, 2003. (Ukrainian)
  • 4. R. Dahlhaus, Efficient location and regression estimation for long-range dependence regression model, Ann. Statist. 23 (1995), 1029-1047. MR 1345213 (96h:62035)
  • 5. R. L. Dobrushin and P. Major, Non-central limit theorems for non-linear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27-52. MR 550122 (81i:60019)
  • 6. J. Dupacova and R. Wets, Asymptotic behavior of statistical estimators and optimal solutions of stochastic optimization problems, Ann. Statist. 16 (1988), no 4, 1517-1549. MR 964937 (89i:62044)
  • 7. A. V. Ivanov and N. N. Leonenko, Asymptotic inferences for a nonlinear regression with strong dependence, Teor. Imovir. Mat. Statist. 63 (2000), 61-79; English transl. in Theory Probab. Math. Statist. 63 (2001), 65-85. MR 1870775 (2003a:62074)
  • 8. A. V. Ivanov and N. N. Leonenko, Asymptotic behavior of $ M$-estimators in continuous-time non-linear regression with long-range dependent errors, Random Oper. Stochastic Equations 10 (2002), no. 3, 201-222. MR 1923424 (2003k:62080)
  • 9. A. V. Ivanov and N. N. Leonenko, Asymptotic theory of nonlinear regression with long-range dependence, Math. Methods Statist. 13 (2004), no. 2, 153-178. MR 2090470 (2005h:62175)
  • 10. H. R. Künsch, J. Beran, and F. R. Hampel, Constraints under long-range correlations, Ann. Statist. 21 (1993), 943-964. MR 1232527 (94f:62134)
  • 11. N. N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer Academic Publishers, Dordrecht, 1999. MR 1687092 (2000k:60102)
  • 12. N. N. Leonenko and M. Bensic, On estimation of regression coefficients of long-memory random fields observed on the arrays, Random Oper. Stochastic Equations 6 (1998), no. 3, 237-252. MR 1610205 (99c:62275)
  • 13. N. N. Leonenko and E. M. Moldavskaya, Non-Gaussian scenarios in long-memory regression models with non-linear constraints, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2002), no. 2, 44-46. MR 1916769
  • 14. E. M. Moldavskaya, Theorem useful in proving of estimates consistency in long-memory regression models, Visn. Kyiv. Univ. Ser. Fiz.-Mat. Nauk (2002), no. 1, 58-65. MR 1934119
  • 15. N. K. Nagaraj and W. A. Fuller, Estimation of the parameters of linear time series models subject to nonlinear restrictions, Ann. Statist. 19 (1991), no. 3, 1143-1154. MR 1126318 (92h:62153)
  • 16. M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50 (1979), 53-83. MR 550123 (81i:60020)
  • 17. J. Wang, Asymptotics of least-squares estimators for constrained nonlinear regression, Ann. Statist. 24 (1996), no. 3, 1316-1326. MR 1401852 (97i:62072)
  • 18. Y. Yajima, Asymptotic properties of the LSE in a regression model with long-memory stationary errors, Ann. Statist. 19 (1991), 158-177. MR 1091844 (92d:62124)

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Additional Information

E. M. Moldavs'ka
Affiliation: Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Keywords: Long-memory (strong) dependence, linear regression, least squares estimators, inequality constraints, non-Gaussian distributions, asymptotic distribution, continuous time
Received by editor(s): October 23, 2004
Published electronically: January 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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