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Generalized Group Testing Procedures

Steven F. Arnold
The Annals of Statistics
Vol. 5, No. 6 (Nov., 1977), pp. 1170-1182
Stable URL: http://www.jstor.org/stable/2958650
Page Count: 13
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Generalized Group Testing Procedures
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Abstract

A person wishes to determine which, if any, of n = Πk j=1 aj i.i.d. random variables, X(i1,⋯, ik), ij = 1,⋯, aj, lie in some specified set A. Such observations will be called unsafe. It is assumed that the density of the X's is known and that Yj(i1,⋯, ij), the sum of all the X's whose first j indices are i1,⋯, ij, can be measured as easily as the individual X's. In this paper, search procedures of the following form are studied. The person first measures Y0, the sum of all the X's. On the basis of Y0, he decides whether to stop, and classify all the X's as safe, or to continue and measure Y1(1),⋯, Y1(a1 - 1) (and hence know Y1(a1) = Y0 - ∑a1-1 i=1 Y1(i)). If he has decided to continue, he measures Y1(j). For each of (Y0, Y1(j)), he must decide whether to stop and classify as safe all X's whose first index is j, or to continue and measure Y2(j, 1),⋯, Y2(j, a2 - 1) (and hence know Y2(j, a2)). He continues in this fashion until each X has either been classified safe or has been observed. Unlike most group testing problems, he is not restricted to procedures that will locate all the unsafe observations. Instead there is a loss function L(x) measuring the loss if X(i1,⋯, ik) = x and is not observed. Let V1 be the expected loss of a procedure (summed over all the X's), and let V2 be the expected number of measurements. For each 0 ≤ p ≤ 1, a class of rules D(p) is defined such that if a procedure is in D(p), it minimizes pV1 + (1 - p)V2, and conversely, if a procedure minimizes pV1 + (1 - p)V2, then there is a rule in D(p) that leads to the same decisions a.e. The union of the D(p) is shown to be an essentially complete class of rules. A simpler form for the rules in D(p) is derived for the case where the loss function is nondecreasing. More specific calculations are given for the case where the X's are normally distributed, and L(x) is the indicator function for the set {x ≥ d}.

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