A martingale approach to scan statistics

Abstract

Abstract. Scan statistics are commonly used in biology, medicine, engineering and other fields where interest is in the probability of observing clusters of events in a window at an unknown location. Due to the dependent nature of the number of events in a large number of overlapping window locations, even approximate solutions for the simplest scan statistics may require elaborate calculations. We propose a new martingale method which allows one to approximate the distribution for a wide variety of scan statistics, including some for which analytical results are computationally infeasible.

On occurrence of patterns in Markov chains: Method of gambling teams

“…Because of the practical importance the occurrence of patterns has been studied extensively by many different techniques. Here is a list of some recent works in the area: Antzoulakos (2001), Chang (2005), Fu (2001), Fu and Chang (2002), Fu and Lou (2006), Hirano and Aki (2003), Pozdnyakov et al (2005), Rukhin (2002) and Stefanov (2000Stefanov ( , 2003. Useful reviews of different approaches can be found in two recent books: Balakrishnan and Koutras (2002) and Fu and Lou (2003).…”

Martingale Methods for Patterns and Scan Statistics

“…The initial arguments of Pozdnyakov et al (2005) in favor of the shifted exponential approximation were predominantly empirical, but subsequently a more theoretical motivation has emerged from work of Fu and Lou (2006, p. 307) which shows that for large n one has…”

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