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Binary segmentation and Bonferroni-type bounds

Michal Černý (2011)

Kybernetika

We introduce the function Z ( x ; ξ , ν ) : = - x ϕ ( t - ξ ) · 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...

Broken Circuits in Matroids-Dohmen’s Inductive Proof

Wojciech Kordecki, Anna Łyczkowska-Hanćkowiak (2013)

Discussiones Mathematicae Graph Theory

Dohmen [4] gives a simple inductive proof of Whitney’s famous broken circuits theorem. We generalise his inductive proof to the case of matroids

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