In the minimax problem the goal is to find procedures

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In the minimax problem, the goal is to find procedures that would min- imize ESADD ( T ) and SADD ( T ) subject to E T γ , i.e., in the class C γ . Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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40 A. G. Tartakovsky Note that ideally one would prefer to find a procedure that would min- imize E ν ( T ν | T > ν ) for all ν 0, when the frequency of false alarms is kept at a desired level. However, no such uniformly optimal procedure exists in the class C γ , so we have to resort to minimax or other settings. In the i.i.d. setting, Moustakides (1986) showed that Page’s (1954) CUSUM procedure is strictly minimax with respect to Lorden’s measure of delay to detection ESADD given by (2.2). The CUSUM procedure is based on a maximum likelihood principle: max k 0 Λ k n , which leads (after taking logarithms) to the detection statistic W n = (0 , W n 1 + Z n ) + , n 1 , W 0 = 0 , (2.4) where, as usual, x + = max(0 , x ) denotes a positive part of x , W n = log(max k 0 Λ k n ) and Z n = log L n is the log-likelihood ratio (LLR) for the n th observation. The procedure stops and declares that the change has occurred at T CS ( h ) = min { n 1: W n h } , min { } = , (2.5) i.e., as soon as the detection statistic W n exceeds h , a positive detection threshold preset so as to achieve the desired level of false alarms γ > 1. This can be done by choosing h = h γ = log γ since E T CS ( h ) e h for any h > 0 (Lorden’s, 1971). This choice of the detection threshold guarantees T CS ( h ) C γ , but this lower bound is extremely conservative. For relatively large values of γ a connection of h and γ is given in Tartakovsky (2005): E T CS ( h ) v 1 e h as h → ∞ , where 0 < v < 1 is a computable constant depending on the model. Thus, setting h = log( ) guarantees E T CS ( h ) γ . Hereafter x b y b means that lim b →∞ ( x b /y b ) = 1, which can also be written as x b = y b (1 + o (1)), where o (1) 0 as b → ∞ . A competitive detection procedure, the so-called Shiryaev–Roberts (SR) procedure, is based on the averaging of the likelihood ratio over the uniform prior distribution rather than maximizing it over the unknown changepoint. Specifically, define R n = n k =1 n i = k L i , T SR ( A ) = min { n 1: R n A } , (2.6) where A is a positive threshold that controls the FAR. A connection between A and the ARL2FA E T SR ( A ) is given in Pollak (1987): E T SR ( A ) A for every A > 0 and E T SR ( A ) ζ 1 A as A → ∞ , where the constant 0 < ζ < 1 is subject of renewal theory. Hence, taking A γ = γζ yields ARL2FA ( T SR ) γ for sufficiently large γ .
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