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Binary segmentation and Bonferroni-type bounds

Michal Černý — 2011

Kybernetika

We introduce the function $Z (x; ξ, ν) : = \int_{- \infty}^{x} ϕ (t - ξ) \cdot Φ (ν t) d t$ , where $ϕ$ and $Φ$ 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...

Goffin's algorithm for zonotopes

Michal Černý — 2012

Kybernetika

The Löwner-John ellipse of a full-dimensional bounded convex set is a circumscribed ellipse with the property that if we shrink it by the factor $n$ (where $n$ is dimension), we obtain an inscribed ellipse. Goffin’s algorithm constructs, in polynomial time, a tight approximation of the Löwner-John ellipse of a polyhedron given by facet description. In this text we adapt the algorithm for zonotopes given by generator descriptions. We show that the adapted version works in time polynomial in the size...

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