In multiple comparisons, a key problem is to estimate or bound the probability of a union of events, and inclusion-exclusion plays an important role in attacking such problems. Naiman and Wynn (1997) introduced the notion of an abstract tube and described why it is relevant and useful this context. This notion will be reviewed and key properties of abstract tubes will be described. In particular, associated with any abstract tube is an inclusion-exclusion identity and corresponding truncation inequalities. Classical inclusion-exclusion arises as a special case, but there are theorems to the effect that these inequalities are typically weaker than can be obtained when a smaller tube is used instead. Recent new abstract tubes, some due to Dohmen (1999A, 1999B) and others building on the work of Naiman and Wynn (1992), all with applications to multiple comparisons and reliability will be presented.

References:

1. Dohmen, K. (1999A).``An improvement of the inclusion-exclusion principle.'' Arch. Math. (Basel) 72 no. 4, 298--303.
2. Dohmen, K. (1999B).``Improved inclusion-exclusion identities and inequalities based on a particular class of abstract tubes.'' Electron. J. Probab. 4 no. 5, 12 pp. (electronic)
3. Naiman, D. and Wynn, H.P. (1992).``Inclusion-exclusion-Bonferroni identities and inequalities for discrete tube-like problems via Euler characteristics.'' Annals of Statistics 20 43-76.
4. Naiman, D. and Wynn, H.P. (1997). ``Abstract Tubes, Improved Inclusion-Exclusion Identities and Inequalities, and Importance Sampling.'' Annals of Statistics 25 1954-1983.