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Testing the hypothesis that a point is Poisson

Published online by Cambridge University Press:  01 July 2016

Robert B. Davies
Robert B. Davies*
Affiliation:
D.S.I.R., Wellington, New Zealand
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Abstract

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

Keywords

ASYMPTOTIC EFFICIENCYCUMULANTCUMULANT SPECTRALOCAL EFFICIENCYLOCAL OPTIMALITYPERIODOGRAMPOINT PROCESSPOISSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 4 , December 1977 , pp. 724 - 746
Copyright
Copyright © Applied Probability Trust 1977 

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References

Brigham, E. O. (1974) The Fast Fourier Transform. Prentice Hall, Englewood Cliffs, N. J. Google Scholar
Brillinger, D. R. (1972) The spectral analysis of stationary interval functions. Proc. 6th Berkeley Symp. Math. Statist. Prob. 483–513.Google Scholar
Brillinger, D. R. (1975) Time Series Analysis, Data Analysis and Theory. Holt Rinehart and Winston, New York.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of a Series of Events. Methuen, London.Google Scholar
Davies, R. B. (1973) Asymptotic inference in stationary Gaussian time-series. Adv. Appl. Prob. 5, 469–497.Google Scholar
Davies, R. B. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247–254.Google Scholar
Hawkes, A. G. (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
Kendall, M. G. and Stuart, A. (1961) The Advanced Theory of Statistics, Vol 2: Inference and Relationship. Griffin, London.Google Scholar
Kuznetsov, P. I. and Stratonovich, R. L. (1956) On the mathematical theory of correlated random points. Izv. Akad. Nauk. SSSR Ser. Mat. 20, 167–178. Sel. Trans. Math. Statist. Prob. 7, (1968), Amer. Math. Soc.Google Scholar
Macchi, O. (1975) The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83–122.Google Scholar
Milne, R. K. and Westcott, M. (1972) Summary of “Further results for Gauss–Poisson processes”. In Stochastic Point Processes, ed. Lewis, P. A. W. Wiley, New York, 257–260.Google Scholar
Neyman, J. (1959) Optimal asymptotic tests of composite statistical hypotheses. In Probability and Statistics, The Harald Cramér Volume, ed. Grenander, U. Wiley, New York.Google Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrences. J. R. Statist. Soc. B 32, 1–62.Google Scholar
Yang, G. L. (1968) Contagion in stochastic models for epidemics. Ann. Math. Statist. 39, 1863–1889.Google Scholar