Advances and Applications in Statistics
Volume 16, Issue 2, Pages 163 - 190
(June 2010)
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ON THE DUALITY OF THE POISSON RATE
John J. Hsieh
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Abstract: The intensity or transition rate of the Poisson process has dual interpretations as an absolute rate and also as a relative rate. The dual interpretations of the Poisson rate is the consequence of the independent increments property of the Poisson processes. The stationary increments property that the homogeneous Poisson process possesses, but the nonhomogeneous Poisson process does not possess, plays no part in the dual interpretations of the Poisson rate. The two different interpretations of the Poisson rate can give rise to very different expressions which lead to useful results. The absolute rate interpretation equates the Poisson rate to the sum of the density functions of all Poisson jump times, while the relative rate interpretation equates the Poisson rate to the conditional hazard function of a jump time whose distribution is truncated below the preceding jump epoch. Many results relating Poisson processes may be more expediently derived from choosing one or the other of the two different interpretations. From the relative rate formula so obtained we derive the conditional, joint and marginal distributions of the jump times and the interjump sojourns as well as the distribution of the counting process. From the absolute rate formula we describe the absolute rates involved in Poisson thinning, decomposition, translation and filtering, and derive the distributions of the current life and the excess life as well as a renewal equation for the nonhomogeneous Poisson process. |
Keywords and phrases: independent increments, nonhomogeneous Poisson process, counting process, jump time, interjump sojourn, absolute rate, relative rate, intensity process, Poisson thinning, decomposition, translation and filtering. |
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