Hostname: page-component-7dd5485656-hw7sx Total loading time: 0 Render date: 2025-10-30T00:29:43.571Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATES

Published online by Cambridge University Press:  04 November 2009

Weidong Liu  and
Wei Biao Wu
Weidong Liu*
Affiliation:
Zhejiang University
Wei Biao Wu
Affiliation:
University of Chicago
*
*Address correspondence to Weidong Liu, Department of Mathematics, Zhejiang University, Huangzhou, Zhejiang, China; e-mail: liuweidong99@gmail.com.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider nonparametric estimation of spectral densities of stationary processes, a fundamental problem in spectral analysis of time series. Under natural and easily verifiable conditions, we obtain consistency and asymptotic normality of spectral density estimates. Asymptotic distribution of maximum deviations of the spectral density estimates is also derived. The latter result sheds new light on the classical problem of tests of white noises.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 4 , August 2010 , pp. 1218 - 1245
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley.Google Scholar
Andrews, D.W.K. (1984) Nonstrong mixing autoregressive processes. Journal of Applied Probability 21, 930–934.Google Scholar
Bentkus, R.Y. & Rudzkis, R.A. (1982) On the distribution of some statistical estimates of spectral density. Theory of Probability and Its Applications 27, 795–814.CrossRefGoogle Scholar
Berman, S. (1962) A law of large numbers for the maximum of a stationary Gaussian sequence. Annals of Mathematical Statistics 33, 93–97.Google Scholar
Brillinger, D.R. (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56, 375–390.Google Scholar
Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holden-Day.Google Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed.Springer-Verlag.Google Scholar
Bühlmann, P. and Künsch, H.R. (1999) Block length selection in the bootstrap for time series. Computational Statistics and Data Analysis 31, 295–310.Google Scholar
Davidson, J. & de Jong, R.M. (2002) Consistency of kernel variance estimators for sums of semiparametric linear processes. Econometrics Journal 5, 160–175.CrossRefGoogle Scholar
de Jong, P. (1987) A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields 75, 261–277.CrossRefGoogle Scholar
Einmahl, U. & Mason, D.M. (1997) Gaussian approximation of local empirical processes indexed by functions. Probability Theory and Related Fields 107, 283–311.Google Scholar
Freedman, D. (1975) On tail probabilities for martingales. Annals of Probability 3, 100–118.CrossRefGoogle Scholar
Götze, F. & Tikhomirov, A.N. (1999) Asymptotic distribution of quadratic forms. Annals of Probability 27, 1072–1098.CrossRefGoogle Scholar
Grenander, U. & Rosenblatt, M. (1957) Statistical Analysis of Stationary Time Series. Wiley.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. Academic Press.Google Scholar
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.CrossRefGoogle Scholar
Mikosch, T. (1991) Functional limit theorems for random quadratic forms. Stochastic Processes and Their Applications 37, 81–98.Google Scholar
Nagaev, S.V. (1979) Large deviations of independent random variables. Annals of Probability 7, 745–789.CrossRefGoogle Scholar
Phillips, P.C.B., Sun, Y., & Jin, S. (2006) Spectral density estimation and robust hypothesis testing using steep origin kernels without truncation. International Economic Review 47, 837–894.Google Scholar
Phillips, P.C.B., Sun, Y., & Jin, S. (2007) Long run variance estimation and robust regression testing using sharp origin kernels with no truncation. Journal of Statistical Planning and Inference 137, 985–1023.Google Scholar
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer-Verlag.CrossRefGoogle Scholar
Priestley, M.B. (1981) Spectral Analysis and Time Series 1. Academic Press.Google Scholar
Priestley, M.B. (1988) Nonlinear and Nonstationary Time Series Analysis. Academic Press.Google Scholar
Rosenblatt, M. (1984) Asymptotic normality, strong mixing, and spectral density estimates. Annals of Probability 12, 1167–1180.Google Scholar
Rosenblatt, M. (1985) Stationary Sequences and Random Fields. Birkhäuser.CrossRefGoogle Scholar
Rudzkis, R. (1985) On the distribution of the maximum deviation of the Gaussian stationary time series spectral density estimate. Lithuanian Mathematical Journal 25, 118–130.Google Scholar
Shao, X. & Wu, W.B. (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 35, 1773–1801.Google Scholar
Song, W.M. & Schmeiser, B.W. (1995) Optimal mean-squared-error batch sizes. Management Science 41, 110–123.Google Scholar
ten Vregelaar, J.M. (1991) Note on the convergence to normality of quadratic forms in independent variables. Theory of Probability and Its Applications 35, 177–179.Google Scholar
Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford University Press.Google Scholar
Velasco, C. & Robinson, P.M. (2001) Edgeworth expansions for spectral density estimates and Studentized sample mean. Econometric Theory 17, 497–539.Google Scholar
Watson, G.S. (1954) Extreme values in samples from m-dependent, stationary, stochastic processes. Annals of Mathematical Statistics 25, 798–800.Google Scholar
Wiener, N. (1958) Nonlinear Problems in Random Theory. MIT Press.Google Scholar
Woodroofe, M. & Van Ness, J.W. (1967) The maximum deviation of sample spectral densities. Annals of Mathematical Statistics 38, 1558–1569.Google Scholar
Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences 102, 14150–14154.Google Scholar
Wu, W.B. & Shao, X. (2007) A limit theorem for quadratic forms and its applications. Econometric Theory 23, 930–951.CrossRefGoogle Scholar