Buying with Exact Confidence

Abstract

We derive some results which may be helpful to buyers of software testing for faults, or to buyers of large lots screening for defectives. Suppose that a fixed but unknown number $n$ of faults or defectives remain before testing. In the testing phase they are observed at random times, $X_1, X_2, \cdots, X_n$, which are order statistics corresponding to $n$ i.i.d. random variables. Since testing is usually an ongoing activity, this distribution is typically known. Under this assumption we derive a stopping criterion that guarantees, for any specified level $\alpha$ and integer $m$, that for all $n > m$, with probability exactly $1 - \alpha$, when stopping occurs, the software has no more than $m$ faults remaining. We study various properties of this stopping rule, both finite and asymptotic, and show that it is optimal in a certain sense. We modify a conservative stopping rule proposed by Marcus and Blumenthal to make it exact, and we give some numerical comparisons.