Titlebook: Scan Statistics; Joseph Glaz,Joseph Naus,Sylvan Wallenstein Book 2001 Springer Science+Business Media New York 2001 Poisson process.Random

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Springer Series in Statisticshttp://image.papertrans.cn/s/image/861130.jpg

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https://doi.org/10.1007/978-1-4757-3460-7Poisson process; Random variable; Scan Statistics; calculus; clustering; statistics

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Retrospective Scanning of Events Over Timeod of time. The researchers ask whether the cluster is unusual in the sense that it is unlikely to have arisen by chance if events were distributed independently and completely at random over time. The relative frequency of large clusters that occur purely by chance is one tool to answer such questions.

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The Generalized Birthday Problemve possible outcomes of a trial as . and .. The maximum number within any m contiguous trials within the . trials, denoted .′., is called the .. For the special case where .′. = ., a . of length . has occurred within the . trials. For the general case where .′.= ., a quota of at least . successes within . consecutive trials has occurred.

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Powerthe computation of power, which requires obtaining the distribution under other alternatives. Unlike other parts of the book, we will attempt to treat nearly simultaneously the cases previously discussed: unconditional scan, conditional scan, and discrete.

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Prospective Scanning of Events Over TimeChapter 2 deals with the scan statistic given that a fixed number . of events have occurred at random in a time period. This conditional (on .) case is called retrospective because in typical applications the researcher scans and makes inferences about . events that have occurred.

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Higher-Dimensional ScansThe previous chapters describe one-dimensional scan statistics for clusters of events over time, or over a sequence of trials. The present chapter describes and applies scan statistics for two and higher dimensions.

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Scanning , Uniform Distributed Points: Exact ResultsLet .1, .2, ... , .., be independent and identically distributed (i.i.d.) random variables, with common density .(x)=1, 0≤x≤1, .(x)=0, elsewhere. Let ..(.) denote the number of X’s in the interval .+.).

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