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Binary segmentation and Bonferroni-type bounds
Kybernetika, Tome 47 (2011) no. 1, pp. 38-49 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We introduce the function $Z(x; \xi, \nu) := \int_{-\infty}^x \varphi(t-\xi)\cdot \Phi(\nu t)\ \text{d}t$, where $\varphi$ and $\Phi$ are the pdf and cdf of $N(0,1)$, respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables of a certain type. We show three applications of the method – (a) calculation of critical values of the segmentation statistic, (b) evaluation of its efficiency and (c) evaluation of an estimator of a point of change in the mean of time series.
We introduce the function $Z(x; \xi, \nu) := \int_{-\infty}^x \varphi(t-\xi)\cdot \Phi(\nu t)\ \text{d}t$, where $\varphi$ and $\Phi$ are the pdf and cdf of $N(0,1)$, respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables of a certain type. We show three applications of the method – (a) calculation of critical values of the segmentation statistic, (b) evaluation of its efficiency and (c) evaluation of an estimator of a point of change in the mean of time series.
Classification : 05A20, 62E17
Keywords: Bonferroni inequality; segmentation statistic; Z-function 
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     author = {\v{C}ern\'y, Michal},
     title = {Binary segmentation and {Bonferroni-type} bounds},
     journal = {Kybernetika},
     pages = {38--49},
     year = {2011},
     volume = {47},
     number = {1},
     mrnumber = {2807862},
     zbl = {1209.62014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a2/}
}
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Černý, Michal. Binary segmentation and Bonferroni-type bounds. Kybernetika, Tome 47 (2011) no. 1, pp. 38-49. http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a2/

[1] Antoch, J., Hušková, M.: Permutation tests in change point analysis. Statist. Probab. Lett. 53 (2001), 37–46. | DOI | MR | Zbl

[2] Antoch, J., Hušková, M., Jarušková, D.: Off-line statistical process control. In: Multivariate Total Quality Control, Physica-Verlag, Heidelberg 2002, 1–86. | MR | Zbl

[3] Antoch, J., Hušková, M., Prášková, Z.: Effect of dependence on statistics for determination of change. J. Statist. Plan. Infer. 60 (1997), 291–310. | DOI | MR

[4] Antoch, J., Jarušková, D.: Testing a homogeneity of stochastic processes. Kybernetika 43 (2007), 415–430. | MR | Zbl

[5] Černý, M., Hladík, M.: The regression tolerance quotient in data analysis. In: Proc. 28th Internat. Conf. on Mathematical Methods in Economics 2010 (M. Houda and J. Friebelová, eds.), University of South Bohemia, České Budějovice 1 (2010), 98–104.

[6] Chen, X.: Inference in a simple change-point problem. Scientia Sinica 31 (1988), 654–667. | MR

[7] HASH(0x2428320). K. Dohmen: Improved inclusion-exclusion identities and Bonferroni inequalities with applications to reliability analysis of coherent systems.

[8] HASH(0x24285c0). Internet: citeseer.ist.psu.edu/550566.html.

[9] Dohmen, K., Tittman, P.: Inequalities of Bonferroni-Galambos type with applications to the Tutte polynomial and the chromatic polynomial. J. Inequal. in Pure and Appl. Math. 5 (2004), art. 64. | MR

[10] Gombay, E., Horváth, L.: Approximations for the time of change and the power function in change-point models. J. Statist. Plan. Infer. 52 (1996), 43–66. | MR

[11] Galambos, J.: Bonferroni inequalities. Ann. Prob. 5 (1997), 577–581. | MR

[12] Galambos, J., Simonelli, I.: Bonferroni-type Inequalities with Applications. Springer Verlag, Berlin 1996. | MR | Zbl

[13] Hladík, M., Černý, M.: New approach to interval linear regression. In: 24th Mini-EURO Conference On Continuous Optimization and Information-Based Technologies in The Financial Sector MEC EurOPT 2010, Selected Papers (R. Kasımbeyli et al., eds.), Technika, Vilnius (2010), 167–171.

[14] Sen, A., Srivastava, M.: On tests for detecting change in mean. Ann. Statist. 3 (1975), 98–108. | DOI | MR | Zbl

[15] Worsley, K.: Testing for a two-phase multiple regression. Technometrics 25 (1983), 35–42. | DOI | MR | Zbl