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

Moldavs’ka, E.

doi:10.1090/S0094-9000-08-00719-9

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
Journal: Theor. Probability and Math. Statist. 75 (2007), 121-137
MSC (2000): Primary 62E20, 62F10; Secondary 60G18
DOI: https://doi.org/10.1090/S0094-9000-08-00719-9
Published electronically: January 24, 2008
MathSciNet review: 2321186
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References

P. S. Knopov, On a nonstationary model of $M$-estimators with discrete time, Teor. Ĭmovīr. Mat. Stat. 57 (1997), 60–66 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 57 (1998), 61–67 (1999). MR 1806885
A. S. Korkhin, Some properties of estimates for regression parameters with a priori inequality constraints, Kibernetika (Kiev) 6 (1985), iv, 106–114 (Russian). MR 825195

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)

Rainer Dahlhaus, Efficient location and regression estimation for long range dependent regression models, Ann. Statist. 23 (1995), no. 3, 1029–1047. MR 1345213, DOI https://doi.org/10.1214/aos/1176324635

R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27–52. MR 550122, DOI https://doi.org/10.1007/BF00535673

Jitka Dupačová and Roger Wets, Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems, Ann. Statist. 16 (1988), no. 4, 1517–1549. MR 964937, DOI https://doi.org/10.1214/aos/1176351052

O. V. Īvanov and M. M. Leonenko, Asymptotic inferences for a nonlinear regression with strong dependence, Teor. Ĭmovīr. Mat. Stat. 63 (2000), 61–79 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 63 (2001), 65–85 (2002). MR 1870775

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, DOI https://doi.org/10.1515/rose.2002.10.3.201

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

H. Künsch, J. Beran, and F. Hampel, Contrasts under long-range correlations, Ann. Statist. 21 (1993), no. 2, 943–964. MR 1232527, DOI https://doi.org/10.1214/aos/1176349159

Nikolai Leonenko, Limit theorems for random fields with singular spectrum, Mathematics and its Applications, vol. 465, Kluwer Academic Publishers, Dordrecht, 1999. MR 1687092

N. N. Leonenko and M. Benšić, On estimation of regression coefficients of long memory random fields observed on the arrays, Random Oper. Stochastic Equations 6 (1998), no. 1, 61–76. MR 1610205, DOI https://doi.org/10.1515/rose.1998.6.1.61

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 2 (2002), 44–46 (English, with Ukrainian summary). MR 1916769

Elina M. Moldavskaya, Theorem useful in proving of estimates consistency in long-memory regression models, Vīsn. Kiïv. Unīv. Ser. Fīz.-Mat. Nauki 1 (2002), 58–65 (English, with English and Ukrainian summaries). MR 1934119

Neerchal K. Nagaraj and Wayne 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, DOI https://doi.org/10.1214/aos/1176348242

Murad S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 53–83. MR 550123, DOI https://doi.org/10.1007/BF00535674

Jinde Wang, Asymptotics of least-squares estimators for constrained nonlinear regression, Ann. Statist. 24 (1996), no. 3, 1316–1326. MR 1401852, DOI https://doi.org/10.1214/aos/1032526971

Yoshihiro Yajima, Asymptotic properties of the LSE in a regression model with long-memory stationary errors, Ann. Statist. 19 (1991), no. 1, 158–177. MR 1091844, DOI https://doi.org/10.1214/aos/1176347975