Asymptotic normality of regression estimators with long memory errors. (English) Zbl 0903.62022
Summary: This paper discusses asymptotic normality of certain classes of \(M\)- and \(R\)-estimators of the slope parameter vector in linear regression models with long memory moving average errors, extending recent results of H. L. Koul [ibid. 14, No. 2, 153-164 (1992; Zbl 0759.62023)] and H. L. Koul and K. Mukherjee [Probab. Theory Relat. Fields 95, No. 4, 535-553 (1993; Zbl 0794.60020)]. Like in the case of the long memory Gaussian errors, it is observed that all these estimators are asymptotically equivalent to the least squares estimator, a fact that is in sharp contrast with the i.i.d. errors case.
References:
| [1] | Avram, F.; Taqqu, M. S., Generalized powers of strongly dependent random variables, Ann. Probab., 15, 767-775 (1987) · Zbl 0624.60049 |
| [2] | Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201 |
| [3] | Beran, J., M-estimators for location for Gaussian and related processes with slowly decaying serial correlations, J. Amer. Statist. Assoc., 86, 704-708 (1991) · Zbl 0738.62082 |
| [4] | Beran, J., Statistical methods for data with long range dependence, Statist. Sci., 7, 404-427 (1992) |
| [5] | Dehling, H.; Taqqu, M. S., The empirical processes of some long range dependent sequences with application to U-statistics, Ann. Statist., 17, 1767-1783 (1989) · Zbl 0696.60032 |
| [6] | Giraitis, L.; Surgailis, D., Multivariate Appell polynomials and the central limit theorem, (Eberlein, E.; Taqqu, M. S., Dependence in Probability and Statistics (1986), Birkhäuser: Birkhäuser Boston), 27-51 · Zbl 0605.60031 |
| [7] | Giraitis, L.; Surgailis, D., A limit theorem for polynomials of linear process with long range dependence, Lithuanian J. Math., 29, 290-311 (1989) · Zbl 0714.60016 |
| [8] | Granger, C. W.J; Joyeux, R., An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal., 1, 15-29 (1980) · Zbl 0503.62079 |
| [9] | Hosking, J. R.M, Fractional differencing, Biometrika, 68, 165-176 (1981) · Zbl 0464.62088 |
| [10] | Ibragimov, I. A.; Linnik, Yu. V., Independent and Stationary Sequences of Random Variables (1971), Walters-Noordhoff: Walters-Noordhoff Groningen · Zbl 0219.60027 |
| [11] | Koul, H. L., M-estimators in linear models with long range dependent errors, Statist. Probab. Lett., 14, 153-164 (1992) · Zbl 0759.62023 |
| [12] | Koul, H. L.; Mukherjee, K., Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors, Probab. Theory Related Field, 95, 535-553 (1993) · Zbl 0794.60020 |
| [13] | Major, P., Multiple Wiener-Ito Integrals (1981), Springer: Springer New York · Zbl 0451.60002 |
| [14] | Surgailis, D., Convergence of sums of non-linear functions of moving averages to self-similar processes, Soviet Math. Doklady, 23, 247-250 (1981) · Zbl 0472.60021 |
| [15] | Taqqu, M. S., Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrsch. verw. Geb., 31, 287-302 (1975) · Zbl 0303.60033 |
| [16] | Taqqu, M. S.; Terrin, N., Convergence in distribution of sums of bivariate Appell polynomials with long range dependence, Probab. Theory Related Fields, 90, 153-165 (1991) · Zbl 0736.60035 |
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