## The Annals of Statistics

### Whittle estimator for finite-variance non-Gaussian time series with long memory

#### Abstract

We consider time series $Y_t = G(X_t)$ where $X_t$ is Gaussian with long memory and $G$ is a polynomial. The series $Y_t$ may or may not have long memory. The spectral density $g_\theta(x)$ of $Y_t$ is parameterized by a vector $\theta$ and we want to estimate its true value $\theta_0$ . We use a least-squares Whittle-type estimator $\hat{\theta}_N$ for $\theta_0$, based on observations $Y_1,\dots,Y_N$. If $Y_t$ is Gaussian, then $\sqrt{N}(\hat{\theta}_N-\theta_0)$ converges to a Gaussian distribution. We show that for non-Gaussian time series $Y_t$ , this $\sqrt{N}$ consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if $Y_t$ has short memory.

#### Article information

Source
Ann. Statist., Volume 27, Number 1 (1999), 178-203.

Dates
First available in Project Euclid: 5 April 2002

https://projecteuclid.org/euclid.aos/1018031107

Digital Object Identifier
doi:10.1214/aos/1018031107

Mathematical Reviews number (MathSciNet)
MR1701107

Zentralblatt MATH identifier
0945.62085

Subjects
Primary: 62E20: Asymptotic distribution theory 62F10: Point estimation
Secondary: 60G18: Self-similar processes

#### Citation

Giraitis, Liudas; Taqqu, Murad S. Whittle estimator for finite-variance non-Gaussian time series with long memory. Ann. Statist. 27 (1999), no. 1, 178--203. doi:10.1214/aos/1018031107. https://projecteuclid.org/euclid.aos/1018031107

#### References

• Avram, F. and Taqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 767-775.
• Beran, J. (1992). Statistical methods for data with long-range dependence (with discussion). Statist. Sci. 7 404-427.
• Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441.
• Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
• Dahlhaus, R. (1989). Efficient parameter estimation for self similar processes. Ann. Statist. 17 1749-1766.
• Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range dependent sequences with an application to U-statistics. Ann. Statist. 17 1767-1783.
• Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for non-linear functions of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27-52.
• Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517-532.
• Giraitis, L. and Koul, H. (1997). Estimation of the dependence parameter in linear regression with long-range dependent errors. Stochastic Process. Appl. 71 207-224.
• Giraitis, L., Leipus, R. and Surgailis, D. (1996). The change-point problem for dependent observations. J. Statist. Plann. Inference 53 297-310.
• Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Probab. Theory Related Fields 70 191-212.
• Giraitis, L. and Surgailis, D. (1986). Multivariate Appell polynomials and the central limit theorem. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 21-71. Birkh¨auser, Boston.
• Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and application to asymptotical normality of Whittle's estimate. Probab. Theory Related Fields 86 87-104.
• Giraitis, L. and Taqqu, M. S. (1997). Limit theorems for bivariate Appell polynomials I. Central limit theorems. Probab. Theory Related Fields 107 359-381.
• Giraitis, L. and Taqqu, M. S. (1998). Central limit theorems for quadratic forms with timedomain conditions. Ann. Probab. 26 377-398.
• Giraitis, L., Taqqu, M. S. and Terrin, N. (1998). Limit theorems for bivariate Appell polynomials II. Non-central limit theorems. Probab. Theory Related Fields 110 333-367.
• Hannan, E. J. (1973). The asymptotic theory of linear time series models. J. Appl. Probab. 10 130-145.
• Ho, H. C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long memory moving averages. Ann. Statist. 24 992-1024.
• Ho, H. C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Probab. 25 1636-1669.
• Koul, H. L. and Surgailis, D. (1997). Asymptotic expansion of M-estimators with long memory errors. Ann. Statist. 25 818-850.
• Robinson, P. M. (1994). Semiparametric analysis of long-memory time series. Ann. Statist. 22 515-539.
• Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630-1661.
• Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287-302.
• Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53-83.
• Taqqu, M. S. and Teverovsky, V. (1998). On estimating the intensity of long-range dependence in finite and infinite variance series. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications (R. Adler, R. Feldman and M. S. Taqqu, eds.) 177-217. Birkh¨auser, Boston.
• Taqqu, M. S., Teverovsky, V. and Willinger, W. (1995). Estimators for long-range dependence: an empirical study. Fractals 3 785-798. [Reprinted in Fractal Geometry and Analysis (C. J. G. Evertsz, H.-O. Peitgen and R. F. Voss, eds.) World Scientific, Singapore.]
• Zygmund, A. (1979). Trigonometric Series I, II. Cambridge Univ. Press.