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Measuring the association of a time series and a point process

Published online by Cambridge University Press:  14 July 2016

J. S. Willie
J. S. Willie*
Affiliation:
Bell Laboratories
*
∗Postal address: 2F-18, Bell Laboratories, 11900 N. Pecos St., Denver, CO 80234, U.S.A.
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Abstract

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.

Keywords

ASYMPTOTIC NORMALITYCUMULANTNEUROPHYSIOLOGYNON-PARAMETRIC DENSITY ESTIMATIONSTATIONARY PROCESSSQUARE ROOT TRANSFORMATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 597 - 608
Copyright
Copyright © Applied Probability Trust 1982 

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References

Brillinger, D. R. (1972) The spectral analysis of stationary interval functions. In Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 483–513.Google Scholar
Brillinger, D. R. (1975a) Time Series: Data Analysis and Theory. Holt, New York.Google Scholar
Brillinger, D. R. (1975b) Statistical inference for stationary point process. In Stochastic Processes and Related Topics 1, ed. Puri, M. L., Academic Press, New York, 55–99.Google Scholar
Brillinger, D. R. (1977) Comparative aspects of the study of ordinary time series and of point processes. In Developments in Statistics 1, ed. Krishnaiah, P. R., Academic Press, New York, 33–133.Google Scholar
Bryant, H. L. and Segundo, J. P. (1976) Spike initiation by transmembrane current. J. Physiol. 260, 279–314.Google Scholar
Freeman, W. J. (1975) Mass Action in the Nervous System. Academic Press, New York.Google Scholar
Hagiwara, T. and Ohatake, M. (1972) Seismic activity associated with filling the reservoir behind the Kurobe Dam, Japan 1963–1970. Tectonophysics 15, 241–254.Google Scholar
Leonov, V. P. and Shiryaev, A. N. (1959) On a method of calculation of semi-invariants. Theory Prob. Appl. 4, 319–329.Google Scholar
Mann, H. B. and Wald, A. (1943) On stochastic limits and order relationships. Ann. Math. Statist. 14, 217–226.Google Scholar
Willie, J. S. (1980) Measuring the association of a time series and a point process. Unpublished technical memorandum.Google Scholar