Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-17T16:50:56.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimated factorisation of the spectral density of a stationary point process

Published online by Cambridge University Press:  01 July 2016

John Rice
John Rice*
Affiliation:
University of California, San Diego
Get access Rights & Permissions [Opens in a new window]

Abstract

A statistical method for estimating the factorisation of the spectral density of a stationary point process is presented and asymptotic properties of the resulting estimates are derived. The estimated functions are of interest in the analysis of a self-exciting process and more generally in the problem of linear prediction, and can be viewed as an alternative second-order analysis of the process. Some examples are discussed.

Keywords

POINT PROCESSSPECTRAL ANALYSISFACTORISATION
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 4 , December 1975 , pp. 801 - 817
Copyright
Copyright © Applied Probability Trust 1975 

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

Bhansali, R. J. (1974) Asymptotic properties of the Wiener–Kolmogorov predictor. I. J. R. Statist. Soc. B 36, 61–73.Google Scholar
Bohman, H. (1960) Approximate Fourier analysis of distribution functions. Ark. Mat. 4, 99–157.Google Scholar
Brillinger, D. R. (1972) The spectral analysis of stationary interval functions. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 1, 483–513.Google Scholar
Brillinger, D. R. (1974) Cross-spectral analysis of processes with stationary increments including the stationary G/G/∞ queue. Ann. Prob. 2, 815–827.Google Scholar
Cooley, J. W. and Tukey, J. W. (1965) An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19, 297–301.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1970) Multivariate point processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 3, 401–448.Google Scholar
Hawkes, A. G. (1971) Point spectra of some mutually exciting point processes. J. R. Statist. Soc: B 33, 438–443.Google Scholar
Jowett, J. and Vere-Jones, D. (1972) The prediction of stationary point processes. In Stochastic Point Processes: Statistical Analysis, Theory, and Applications, ed. Lewis, P. A. W., Wiley, New York, 405–435.Google Scholar
Krein, M. G. (1962) Integral equations on a half-line with kernel depending on the difference of the argument. Amer. Math. Soc. Transl. (2) 22, 163–288.Google Scholar
Neyman, J. E. and Scott, E. L. (1958) A statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
Papangelou, P. (1972) Integrability of expected increments of point processes and a related random change of scale. Trans. Amer. Math. Soc. 165, 483–506.Google Scholar
Rice, J. (1973) Statistical Analysis of Self-exciting Point Processes and Related Linear Models. Ph.D. Thesis, University of California, Berkeley.Google Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 30, 1–45.Google Scholar
Wiener, N. (1949) Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Wiley, New York.Google Scholar