# Can we apply the Probability Integral Transform to Dependent Random Variables?

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.

• Check the copula keyword. Apr 7 '20 at 10:30

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.)