Imagine we have three independent random variables, distributed according to pdfs $a(t), b(t), c(t)$, with cdfs $A(t), B(t), C(t)$. In my case the distributions are lognormal.
However, our observations are not independent draws from these variables. Instead, the observations are "competing" with each other, as in a race: on each draw, a sample is taken from each and we only get to see the "winning" (lowest-valued) of the three samples. We also know which of the samples won.
From our observed data we can build empirical pdfs and cdfs. The observed data will be distributed as
$a'(t) \propto a(t)(1-B(t))(1-C(t))$
$b'(t) \propto (1-A(t))b(t)(1-C(t))$
$c'(t) \propto (1-A(t))(1-B(t))c(t)$
If we estimate these "mutually censored" distributions from the data, or if we just use the raw observations themselves, is there then a good way of estimating the original pdfs?
(Related topics I have searched: "progressive censoring", "competing risks". Haven't found a general famework for this yet.)
