The Annals of Statistics

Optimal stopping of sequential size-dependent search

Issa Fakhre-Zakeri and Eric Slud

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In many areas of application, one searches within finite populations for items of interest, where the probability of sampling an item is proportional to a random size attribute from an i.i.d. superpopulation of attributes which may or may not be observable upon discovery. Here we treat the problem of asymptotically optimal stopping rules for size-dependent searches of this type, as the size of the underlying population grows, where the loss function includes an asymptotically smooth time-dependent cost, a constant cost per item sampled and a cost per undiscovered item which may depend on the size attribute of the undiscovered item. Under some regularity and convexity conditions related to the asymptotic expected loss, we characterize asymptotically optimal rules even when the initial population size and the distribution of size attributes are unknown. We direct especial attention to applications in software reliability, where the items of interest are software faults ("bugs"). In this setting, the size attributes will not be observable when faults are found, and, in addition, our search model allows new bugs to be introduced into the software when faults are detected "imperfect debugging"). Our results extend those of Dalal and Mallows and Kramer and Starr, and are illustrated in the perfect-debugging case on a previously analyzed dataset of Musa.

Article information

Ann. Statist., Volume 24, Number 5 (1996), 2215-2232.

First available in Project Euclid: 20 November 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Asympotically optimal stopping rule exponential order statistic model imperfect debugging loss function mixture model sequential search size-dependent successive sampling


Fakhre-Zakeri, Issa; Slud, Eric. Optimal stopping of sequential size-dependent search. Ann. Statist. 24 (1996), no. 5, 2215--2232. doi:10.1214/aos/1069362318.

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