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Let's suppose we deal with a non-homogeneous Poisson process having intensity function λ(t), t ≥ 0. The event times X1, X2, … of such a process are dependent random variables, described by a conditional distribution. I need to apply the (empirical) probability integral transformation in order to run Goodness-of-Fit tests (test of homogeneity). Can we apply the (empirical) probability integral transformation to X1, X2, …? As far as I know, this transformation can be applied to i.i.d. observations on X. However, in this case, we deal with event times that depend on event times of other events.

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  • $\begingroup$ Check the copula keyword. $\endgroup$
    – Xi'an
    Apr 7 '20 at 10:30
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Yes, you can simply use the conditional predictive distribution $F_t$ at each time point $t$ and plug in the actual observation $X_t$. (As an example, I did so in the context of density forecasting for retail sales in Kolassa, 2016.)

Note that a uniform histogram resulting from the PIT is necessary, but not sufficient, to deduce that your densities are calibrated (Gneiting et al., 2007)).

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  • $\begingroup$ Thank you Stephan. However, I cannot use the explicit formulation of the conditional intensity describing the non-homogeneous Poisson process. I can only use the empirical CDF. $\endgroup$
    – Angela
    Apr 7 '20 at 10:45

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