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A Second Order Cumulant Spectrum Test That a Stochastic Process is Strictly Stationary and a Step Toward a Test for Graph Signal Strict Stationarity

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  • Denisa Roberts
  • Douglas Patterson

Abstract

This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process in the frequency domain. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs in the principal domain. The test uses a window averaged sample estimate of the second order cumulant spectrum to build a test statistic with an asymptotic complex standard normal distribution. We derive the test statistic, study the properties of the test and demonstrate its application using 137Cs gamma ray decay data. Future areas of research include testing for strict stationarity of graph signals, with applications in learning convolutional neural networks on graphs, denoising, and inpainting.

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  • Denisa Roberts & Douglas Patterson, 2018. "A Second Order Cumulant Spectrum Test That a Stochastic Process is Strictly Stationary and a Step Toward a Test for Graph Signal Strict Stationarity," Papers 1801.06727, arXiv.org, revised Mar 2020.
    • Handle: RePEc:arx:papers:1801.06727
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    1. Hinich , Melvin J. & Rothman, Philip, 1998. "Frequency-Domain Test Of Time Reversibility," Macroeconomic Dynamics, Cambridge University Press, vol. 2(1), pages 72-88, March.
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    3. John U. Farley & Melvin J. Hinich, 1970. "Detecting "Small" Mean Shifts in Time Series," Management Science, INFORMS, vol. 17(3), pages 189-199, November.
    4. Marsaglia, George & Tsang, Wai Wan & Wang, Jingbo, 2003. "Evaluating Kolmogorov's Distribution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 8(i18).
    5. Hinich, Melvin A. & Wild, Phillip, 2001. "Testing Time-Series Stationarity Against An Alternative Whose Mean Is Periodic," Macroeconomic Dynamics, Cambridge University Press, vol. 5(3), pages 380-412, June.
    6. Ashley, Richard A. & Patterson, Douglas M., 2010. "Apparent Long Memory In Time Series As An Artifact Of A Time-Varying Mean: Considering Alternatives To The Fractionally Integrated Model," Macroeconomic Dynamics, Cambridge University Press, vol. 14(S1), pages 59-87, May.
    7. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2005. "Stationarity Tests Under Time-Varying Second Moments," Econometric Theory, Cambridge University Press, vol. 21(6), pages 1112-1129, December.
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