Geodesic
Binary segmentation and Bonferroni-type bounds
Kybernetika, Tome 47 (2011) no. 1, pp. 38-49.

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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.
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},
     publisher = {mathdoc},
     volume = {47},
     number = {1},
     year = {2011},
     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/