I am working with years data and I want to see how likely it is for something to have happened for the first time in, say, 1997 given other dates as well. Let me explain.
I have something (a lab observation of a specific molecule) that has happened 10 times (in possibly different years) and I have all those years. x <- c(1997, 1998, 1998, 1998, 2000, 2005, 2006, 2006, 2010, 2019) Here I see the first occurrence is in 1997 and I wonder how likely is it that it happened then (so in a way, was that expected).
I am interested in the first occurrence only (so 1997).
I have a distribution (a much much larger pool of years, ~120k) from which I can resample my 10 occurence dates. These ~120k observations are actually all lab observations of the same type as mine (so they use similar techniques but different molecules). [This sampling distribution is very negatively skewed and between 1960 and 2020.]
I want to know what's the probability of really having the first occurrence in 1997, so I thought I'd do this:
I resample from 3) 1000 times (10 years each time) and record the earliest year of occurence every time. So now I have a sample of size 1000 with the first years. I call this sample F. [* I'm not sure I can assume normality for the sample F, given the skewness of the initial sampling distribution, so I decided I need another method. This is why I went for the bootstrap.]
What I did next is resample from F with replacement and every time calculated the rank-turned-probability (probability) for 1997. This gives me a "sample" for the probability of having the first observation in 1997. And then I thought I can take the mean of those probabilities as the estimate for "the probability of observing 1997".
However, I didn't find any explicit basis to do that. It kind of looks like bootstrapping but bootstrapping, as I've seen it, has only been used for estimating confidence intervals around a metric (but is this a metric?). Is there a resource I can consult to confirm my method or to adjust it in order to make it correct? Or is this the right approach at all?
Any guidance would be greatly appreciated.