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August 2000 $r$-scan statistics of a marker array in multiple sequences derived from a common progenitor
Chingfer Chen, Samuel Karlin
Ann. Appl. Probab. 10(3): 709-725 (August 2000). DOI: 10.1214/aoap/1019487507


This study is motivated by problems of molecular sequence comparisons for biological traits conserved or lost over evolution time.A marker of interest is distributed in the genome of the ancestor and inherited among $l$ offspring species which descend from this common ancestor. Each marker will be retained or lost during the evolution of the descendent species. The objective of the analysis here is to ascertain probabilities of clustering or overdispersion of the marker array among the sequences of the descendent species. Limiting distributions for the extremal $r$-scan statistics (defined in text) of the trait distributed among the $l$ dependent offspring processes are derived by adapting the Chen–Stein Poisson approximation method. Results that accommodate new occurrences of the trait (gene) arising from duplications and transposition occurrences are also described.The $r$-scan statistical analysis is further applied to a multi sequence combined Poisson model where ${B_1,\dots, B_l}$ are generated from $m$ independent Poisson processes ${A_1,\dots, A_m}$ such that $B_k = \bigcup_{i\epsilonZ_k}A_i$, where ${Z_k}_1\leqk\leql$ are subsets of ${1, 2,\dots,m}$.


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Chingfer Chen. Samuel Karlin. "$r$-scan statistics of a marker array in multiple sequences derived from a common progenitor." Ann. Appl. Probab. 10 (3) 709 - 725, August 2000.


Published: August 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1084.92506
MathSciNet: MR1789977
Digital Object Identifier: 10.1214/aoap/1019487507

Primary: 60E05
Secondary: 60G50

Keywords: Asymptotic distributions , Chen-Stein Poisson approximation , Poisson processes , r-scan statistics , total variation distance

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 3 • August 2000
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