## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 2, Number 3 (1992), 752-765.

### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 2, Number 3 (1992), 752-765.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005658

**Digital Object Identifier**

doi:10.1214/aoap/1177005658

**Mathematical Reviews number (MathSciNet)**

MR1177908

**Zentralblatt MATH identifier**

0753.62051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 62G30: Order statistics; empirical distribution functions 62L15: Optimal stopping [See also 60G40, 91A60] 62N99: None of the above, but in this section

**Keywords**

Software testing optimal stopping sampling inspection

#### Citation

Dalal, S. R.; Mallows, C. L. Buying with Exact Confidence. Ann. Appl. Probab. 2 (1992), no. 3, 752--765. doi:10.1214/aoap/1177005658. https://projecteuclid.org/euclid.aoap/1177005658