# probability – Order of non-homogeneous Poisson process rate functions by specific time (end)

The larger problem that I'm trying to solve is the following:

Given $$k$$ lists of arrival times (all times less than or equal to an end time $$T$$) derived from $$k$$ non-homogeneous Poisson processes with unknown rate functions $$lambda_1 (t), ldots, lambda_k (t)$$, I want to order the processes in descending order of $$lambda_1 (T), ldots, lambda_k (T)$$

My current thinking was to predict the following time of arrival (perhaps the expected arrival time?) for each of the $$k$$ PPSN and order them based on that, but this logic may be circular?

I found work on PSPS arrival time forecasting (eg Goulding et al., 2016, Shen & Huang, 2008 and Weinberg et al., 2006), but these documents seem to be mainly focused on the estimation of PPSN parameters, which does not interest me (directly?): I only care about the relative extent of their rate functions to an end ), which, I imagine, might not require as many calculations. Any pointers or tips would be greatly appreciated