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Binomial Group-Testing with an Unknown Proportion of Defectives

Milton Sobel and Phyllis A. Groll
Technometrics
Vol. 8, No. 4 (Nov., 1966), pp. 631-656
DOI: 10.2307/1266636
Stable URL: http://www.jstor.org/stable/1266636
Page Count: 26
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Binomial Group-Testing with an Unknown Proportion of Defectives
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Abstract

The binomial group-testing problem is extended to the case in which the common probability p of a unit being defective is unknown. A Bayes "non-mixing" procedure R(1) is derived and compared with other procedures, in particular with the corresponding procedure R1 that requires the knowledge of p and with another procedure based on continually revising the maximum likelihood estimate of p; the latter is called an empirical Bayes solution. Finally, a Bayes procedure derived in the Appendix allows "mixing" and this is conjectured to be the unrestricted Bayes solution; the improvements due to "mixing" are shown to be small in Table III. Several applications of the general problem of group-testing are discussed in the introduction and this virtually constitutes a general review of the subject. After the procedure R(1) is defined a detailed illustration is given in section 2.1 showing how to carry out this procedure with the use of Tables I and II. Lower bounds for the Bayes risk associated with any group testing procedure for unknown p are given in Table III; one of these bounds, based on information theory, is derived in section 4.

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