×

Distribution of the length of the longest significance run on a Bernoulli net and its applications. (English) Zbl 1118.62308

Summary: We consider the length of the longest significance run in a (two-dimensional) Bernoulli net and derive its asymptotic limit distribution. Our results can be considered as generalizations of known theorems in significance runs. We give three types of theoretical results: (1) reliability-style lower and upper bounds, (2) Erdös-Rényi law, and (3) the asymptotic limit distribution. To understand the rate of convergence to the asymptotic distributions, we carry out numerical simulations. The convergence rates in a variety of situations are presented. To understand the relation between the length of the longest significance run(s) and the success probability \(p\), we propose a dynamic programming algorithm to implement simultaneous simulations. Insights from numerical studies are important for choosing the values of design parameters in a particular application, which motivates this article. The distribution of the length of the longest significance run in a Bernoulli net is critical in applying a multiscale methodology in image detection and computational vision. Approximation strategies to some critical quantities are discussed.

MSC:

62E20 Asymptotic distribution theory in statistics
90C39 Dynamic programming
65C60 Computational problems in statistics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI