Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-23T18:39:38.754Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

Part of: Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

V. V. Anh ,
N. N. Leonenko  and
L. M. Sakhno
V. V. Anh
Affiliation:
Program in Statistics and Operations Research, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia. Email address: v.anh@qut.edu.au
N. N. Leonenko
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, UK. Email address: leonenkon@cardiff.ac.uk
L. M. Sakhno
Affiliation:
Department of Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, 01033, Kyiv, Ukraine. Email address: lms@univ.kiev.ua
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides a quasi-likelihood or minimum-contrast-type method for the parameter estimation of random fields in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general kth order and allows for possible long-range dependence in the random fields. To avoid bias due to edge effects, data tapering is incorporated into the method. The suggested minimum contrast functional is linear with respect to the periodogram of kth order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretization is not required in the estimation of continuously observed random fields. The consistency and asymptotic normality of the resulting estimators are established. Illustrative applications of the method to some problems in mathematical finance and signal detection are given.

Keywords

Quasi-likelihood estimationminimum contrast estimatorhigher-order spectral densitieslong-range dependence

MSC classification

Primary: 62M09: Non-Markovian processes
Secondary: 62M40: Random fields; image analysis
Type
Part 2. Estimation methods
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 35 - 53
Copyright
Copyright © Applied Probability Trust 2004 

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.)

References

Anh, V. V., Heyde, C. C. and Leonenko, N. N. (2002). Dynamic models of long-memory processes driven by Lévy noise. J. Appl. Prob. 39, 730–747.Google Scholar
Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2002a). Minimum contrast estimation of random processes and fields based on information of the second and third order. Submitted.Google Scholar
Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2002b). Statistical inference using higher-order information. Submitted.Google Scholar
Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2003). On a class of minimum contrast estimators for fractional stochastic processes and fields. To appear in J. Statist. Planning Infer. Google Scholar
Bentkus, R. (1972a). Asymptotic normality of an estimate of the spectral function. Liet. Mat. Rink. 12, 5–18 (in Russian).Google Scholar
Bentkus, R. (1972b). On the error of the estimate of the spectral function of a stationary process. Liet. Mat. Rink. 12, 55–71 (in Russian).Google Scholar
Bentkus, R. and Maliukevicius, R. R. (1988a). Statistical estimation of a multidimensional parameter of a spectral density I. Lithuanian Math. J. 28, 115–126.Google Scholar
Bentkus, R. and Maliukevicius, R. R. (1988b). Statistical estimation of a multidimensional parameter of a spectral density II. Lithuanian Math. J. 28, 415–431.Google Scholar
Bentkus, R. and Rutkauskas, R. (1973). On the asymptotics of the first two moments of second order spectral estimators. Liet. Mat. Rink. 13, 29–45 (in Russian).Google Scholar
Brillinger, D. R. (1965). Introduction to polyspectra. Ann. Math. Statist. 36, 1351–1374.CrossRefGoogle Scholar
Brillinger, D. R. and Rosenblatt, M. (1967a). Asymptotic theory of kth order spectra. In Spectral Analysis of Time Series (Proc. Advanced Sem., Maidson, WI, 1966), ed. Harris, B., John Wiley, New York, pp. 153–188.Google Scholar
Brillinger, D. R. and Rosenblatt, M. (1967b). Computation and interpretation of kth order spectra. In Spectral Analysis of Time Series (Proc. Advanced Sem. Madison, WI, 1966), ed. Harris, B., John Wiley, New York, pp. 189–232.Google Scholar
Dahlhaus, R. (1983). Spectral analysis with tapered data. J. Time. Ser. Anal. 4, 163–175.Google Scholar
Dahlhaus, R. and Künsch, H. (1987). Edge effects and efficient parameter estimation for stationary random fields. Biometrika 74, 877–882.Google Scholar
Dunsmuir, W. and Hannan, E. J. (1976). Vector linear time series models. Adv. Appl. Prob. 8, 339–360.CrossRefGoogle Scholar
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.Google Scholar
Gao, J., Anh, V. V. and Heyde, C. C. (2002). Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency. Stoch. Process. Appl. 99, 295–321.Google Scholar
Gao, J., Anh, V. V, Heyde, C. C. and Tieng, Q. (2001). Parameter estimation of stochastic processes with long-range dependence and intermittency. J. Time Ser. Anal. 22, 517–535.Google Scholar
Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle's estimate. Prob. Theory Relat. Fields 86, 87–104.Google Scholar
Giraitis, L. and Taqqu, M. (1999). Whittle estimator for finite-variance non-Gaussian time series with long memory. Ann. Statist. 27, 178–203.Google Scholar
Guyon, X. (1982). Parameter estimation for a stationary process on a d-dimensional lattice. Biometrica 69, 95–105.Google Scholar
Guyon, X. (1995). Random Fields on a Network: Modelling, Statistics and Applications. Springer, New York.Google Scholar
Hannan, E. J. (1970). Multiple Time Series. Springer, New York.Google Scholar
Hannan, E. J. (1973). The asymptotic theory of linear time series models. J. Appl. Prob. 10, 130–145.Google Scholar
Heyde, C. C. (1997). Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer, New York.Google Scholar
Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 1234–1239.Google Scholar
Heyde, C. C. and Gay, R. (1993). Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence. Stoch. Process. Appl. 45, 169–182.Google Scholar
Ibragimov, I. A. (1963). On estimation of the spectral function of a stationary Gaussian process. Theory Prob. Appl. 8, 366–401.Google Scholar
Ibragimov, I. A. (1967). On maximum likelihood estimation of parameters of the spectral density of stationary time series. Theory Prob. Appl. 12, 115–119.Google Scholar
Ivanov, A. V. and Leonenko, N. N. (1989). Statistical Analysis of Random Fields. Kluwer, Dordrecht.Google Scholar
Leonenko, N. N. and Moldavs'Ka, E. M. (1999). Minimum contrast estimators of a parameter of the spectral density of continuous time random fields. Theory Prob. Math. Statist. 58, 101–112.Google Scholar
Rosenblatt, M. R. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston, MA.Google Scholar
Rudin, W. (1991). Functional Analysis , 2nd edn. McGraw-Hill, New York.Google Scholar
Swami, A., Giannakis, G. B. and Zhou, G. (1997). Bibliography on higher-order statistics. Signal Processing 60, 65–126.Google Scholar
Tukey, J. W. (1967). An introduction to the calculations of numerical spectrum analysis. In Spectral Analysis of Time Series (Proc. Advanced Sem., Madison, WI, 1966), ed. Harris, B., John Wiley, New York, pp. 25–46.Google Scholar