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Matched pairs in independent Poisson processes. J Appl Probab 2016. [DOI: 10.1017/s0021900200033477] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
Given two independent Poisson processes 𝒜 and ℬ with rates respectivelyαandβ(α<β) we match each point of 𝒜 with the closest point of ℬ that has not already been matched. The points of 𝒜 are taken in random order. It is shown that the point process of unmatched points of ℬ is a renewal process with the same interval distribution as the busy period of anM/M/1 queue. The distribution and moments of the distance between a typical point of 𝒜 and the corresponding matched point of ℬ are obtained. Variants of the matching process in which the assigned point of ℬ must lie to the right of the point of 𝒜, and in which the matching distance must be less than a fixed tolerance are studied. The use of matched samples to control for bias in observational studies is discussed.
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