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Optimal stopping of sequential size-dependent search. (English) Zbl 0867.62069

Summary: 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 S. R. Dalal and C. L. Mallows [J. Am. Stat. Assoc. 83, No. 403, 872-879 (1988; Zbl 0662.62084)] and M. Kramer and N. Starr [Sequential Anal. 9, No. 1, 59-80 (1990; Zbl 0694.62036)] and are illustrated in the perfect-debugging case on a previously analyzed dataset.

MSC:

62L15 Optimal stopping in statistics
62N05 Reliability and life testing
62D05 Sampling theory, sample surveys
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