Short version: is there any statistical method that can be used to develop estimates of abundance, spatial distribution, and/or rate of spread of a species when my dataset only includes presence data and not absence? I am not dealing with fossils in the strict sense, but I speculate that if such a method exists it might be found in the field of paleontology.
Long version: I recently undertook a small pilot project trying to gather data on the distribution of a newly introduced species of moth in the US, and possibly to project backwards and estimate the date and site of introduction. We only became aware of this species in 2009, but I searched insect photography databases (mostly contributed by amateurs: bugguide.net, mothphotographersgroup.org, etc.) and found that there were many photos of this beast, dating back to 2004, and covering 8 or 9 states in the southeastern U.S.
It got me thinking about how such a dataset could be used. These photos are functionally exactly like a fossil record, in that positive data points are sparse but intrinsically very informative because they represent a definitive date and locality for the target organism. On the other hand, true negative data is even more sparse (meaning that data exists, showing that the organism wasn't present at a certain place and time). I have a lot of a third kind of "data," and I'm not sure if it can be used at all: the absence of positive data.
On one hand, if no positive data exists for a certain site, it says as much about the conditions for "fossil" (or photo) creation and subsequent collection as it does about the distribution of the species. For fossils I imagine the conditions are geological/topographical, whereas for photos they relate to the density of potential photographers, and the likelihood that they'd post a photo where I can find it.
On the other hand, no data is not the same as no information, right? If I standardize for human population density, citizen scientists across the US should be equivalent in their probability of encountering a moth and photographing it. If moth density per photographer is high, there will be a greater chance of a photo being taken. If moth density is low, there is less chance of a photo. If the moth is truly absent, there is zero chance of a photo. Put differently, I don't think it is random* that all the photos are from southeastern states, and that more photos were found from some states than others among that group. I just don't know if there is a way to statistically support this based on the type of data that I have.
*Full disclosure: I know it isn't random. I didn't want to add another level of complication to my question, but the distribution of its host tree is well documented and represents an absolute boundary for this moth.