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 AM  Vol.12 No.11 , November 2021
Scanning for Clusters of Large Values in Time Series: Application of the Stein-Chen Method
Abstract: The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as It,It+1,It+2 = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold. The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model. Note that if {1 0 1} is known to not occur, for example, starting at position t = 1, then this information impacts the probability that {1 0 1} occurs starting at position t = 2 or t = 3, because the trials to obtain {1 0 1} are overlapping and thus not independent, so the Poisson distribution assumptions are not met. Nevertheless, the results shown in four examples demonstrate that Poisson-based approximation (that is strictly correct only for independent trials) can be remarkably accurate, and the SC method provides a bound on the total variation distance between the true and approximate PDF.
Cite this paper: Burr, T. and Henderson, B. (2021) Scanning for Clusters of Large Values in Time Series: Application of the Stein-Chen Method. Applied Mathematics, 12, 1031-1037. doi: 10.4236/am.2021.1211067.
References

[1]   R Core Team (2017) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria.
https://www.R-project.org/

[2]   Arratia, R., Goldstein, L. and Gordon, L. (1990) Poisson Approximation and the Chen-Stein Methods. Statistical Science, 5, 403-434.
https://doi.org/10.1214/ss/1177012015

[3]   Shumway, R. and Stoffer, D. (2016) Time Series Analysis and Its Applications with R Examples. 4th Edition, Springer, Pittsburgh.
https://doi.org/10.1007/978-3-319-52452-8

[4]   Sahatsathatsana, C. (2017) Applications of the Stein-Chen Method for the Problem of Coincidences. International Journal of Pure and Applied Mathematics, 116, 49-59.
https://doi.org/10.12732/ijpam.v116i1.5

[5]   Kim, S. (2000) A Use of the Stein-Chen Method in Time Series Analysis. Journal of Applied Probability, 37, 1129-1136.
https://doi.org/10.1239/jap/1014843092

[6]   Aleksandrov, B., Weis, C. and Jentsch, C. (2021) Goodness-of-Fit Tests for Poisson Count Time Series Based on the Stein-Chen Identity. Statistica Neerlandica, 1-30.
https://doi.org/10.1111/stan.12252

[7]   Weis, C. and Aleksandrov, B. (2020) Computing (Bivariate) Poisson Moments Using Stein-Chen Identities. The American Statistician, 1-6.
https://doi.org/10.1080/00031305.2020.1763836

 
 
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