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• W156793992 abstract "Randomized Intensity Estimates for Transformed Poisson Processes Introduction to the problem Suppose N is a stationary Poisson process on the line with intensity A, observed from time 0 to time T. The transformed process, Af, given by Mit) := N(t/), is a Poisson process with intensity 1. M may equivalently be specified by prescribing that N has a point at t iff. M has a point at t X A. It was shown in [1] that if A is estimated by maximum likelihood, the estimated trans- formed process Mit) behavior. := N(t/) is not a Poisson process, but instead exhibits self-correcting That is, the conditional intensity A„(t) of M is less than 1 when Mit) > t (i.e. when there have been more points than expected), and is greater than 1 when Mit) < t (when there have been fewer than expected). [1] also shows by simulations that, for small samples, this difference between M and a Poisson process can be rather significant. The question addressed here is: can this problem be remedied somehow? Is there some other way to estimate A, so that M will closer approximate the standard Poisson process? One naive attempt may be to multiply the estimate A by some factor. No such procedure can be very effective. The main problem is that for a Poisson process, the number of points in the interval [0,T] may be quite different from its expected value, A X T. But when A is estimated by MLE, the transformed process M is guaranteed to contain NiT) points in a region of size NiT). Multiplying A by some factor simply forces NiT) to be that factor of the size of the transformed region, and does not allow the ratio of the number of points to the size of the transformed region to vary randomly in the way that the ratio N(T)/T varies for the standard Poisson process. Randomized Bayes estimates Another approach is to let the estimate A be randomized. That is, conditional on the observation of the point process N on [0, T], generate a random variable A which may depend on TV. One way to do this involves a Bayesian approach. and for x in Place a uniform prior on R for A, let P(A = xN(T) = i) be equal to the posterior probability P(A = xN = It works out then that J P(N o = zX = y)dy i)." @default.
• W156793992 created "2016-06-24" @default.
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• W156793992 date "2011-10-25" @default.
• W156793992 modified "2023-09-27" @default.
• W156793992 title "Randomized Intensity Estimates for Transformed Poisson Processes" @default.
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