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September, 1986 An Extreme Value Theory for Sequence Matching
Richard Arratia, Louis Gordon, Michael Waterman
Ann. Statist. 14(3): 971-993 (September, 1986). DOI: 10.1214/aos/1176350045

Abstract

Consider finite sequences $X_1, X_2,\cdots, X_m$ and $Y_1, Y_2,\cdots, Y_n$ where the letters ${X_i}$ and ${Y_i}$ are chosen i.i.d. on a countable alphabet with $p=P(X_1=Y_1)\in(0,1)$ We study the distribution of the longest contiguous run of matches between the X's and Y's allowing at most k mismatches. The distribution is closely approximated by that of the maximum of (1 - p)mn i.i.d. negative binomial random variables. The latter distribution is in turn shown to behave like the integer part of an extreme value distribution. The expectation is approximately $\log(qmn)+k\log\log(qmn)+k\log(q/p)-\log(k!)+\gamma\log(e)-\frac{1}{2}$, where q = 1 - p, log denotes logarithm base 1/p, and y is the Euler constant. The variance is approximated by $(\pi\log(e))2/6+\frac{1}{2}$. The paper concludes with an example in which we compare segments taken from the DNA sequence of the bacteriophage lambda.

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Richard Arratia. Louis Gordon. Michael Waterman. "An Extreme Value Theory for Sequence Matching." Ann. Statist. 14 (3) 971 - 993, September, 1986. https://doi.org/10.1214/aos/1176350045

Information

Published: September, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0602.62015
MathSciNet: MR856801
Digital Object Identifier: 10.1214/aos/1176350045

Subjects:
Primary: 62E20
Secondary: 62P10

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 3 • September, 1986
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