Hypothesis Test for Upper Bound on the Size of Random Defective Set

Let 1 <= s < t, N >= 2 be fixed integers and a complex electronic circuit of size t is said to be an s-active, s << t, and can work as a system block if not more than s elements of the circuit are defective. Otherwise, the circuit is said to be an s-defective and should be replaced by a similar s-active circuit. Suppose that there exists a possibility to run N non-adaptive group tests to check the s-activity of the circuit. As usual, we say that a (disjunctive) group test yields the positive response if the group contains at least one defective clement. In this paper, we will interpret the unknown set of defective elements as a random set and discuss upper bounds on the error probability of the hypothesis test for the null hypothesis {H-0 : the circuit is s-active} versus the alternative hypothesis {H-1 : the circuit is s-defective}. Along with the conventional decoding algorithm based on the known random set of positive responses and disjunctive s-codes, we consider a T-weight decision rule which is based on the simple comparison of a fixed threshold T, 1 <= T < N. with the known random number of positive responses p, 0 <= p <= N.