I've been thinking about this problem and found this old thread. The problem is of trying to estimate a superpopulation size when Markovian emigration is present for the RD model of Kendall, Nichols, and Hines 1997. Cooch stated that it is possible to get there but I have yet to see anyone explain how. I couldn't find anything doing some lit searches. The formula by Murray Efford is the only attempt I've seen. So I thought I'd post my thought process about it here to see if anyone can point out the flaws or what I'm missing exactly.
I’ll distinguish between two subpopulations here, which sum to total the superpopulation,
. One is the “local” population, which is
. The other is the portion of the superpopulation unavailable for capture,
. The superpopulation during the following period is the sum of individuals that entered the local population (from somewhere in the superpopulation) (
), individuals that were unavailable (
), and individuals that were recruited into the local (
) and unavailable
populations.
That is,
,
where
and
.
This leaves us with the problem of how to estimate recruitment (new individuals that were not in the superpopulation during the previous time period). Recruitment is not modeled directly by the Kendall et al. 1997 method. However, this should just be the number of individuals of the sub population minus the contribution from the previous subpopulation
(
).
If we calculate a superpopulation size assuming random temporary emigration, the following relationship appears to hold:
. The consequence of this, I think, is that the ratio of recruitment into the superpopulation is assumed to be equivalent to the ratio of recruitment into the local population. This may be a strong assumption, but if we make it for the Markovian case, we can solve for
. However, we need to assume what the superpopulation size was to have a starting point (
). Would this work for periods where gamma' happens to equal gamma" and we can use the N/(1-gamma) estimator? Perhaps this would work using something like Murray Efford's solution?
If the Markovian migration parameters are constant and superpopulation size is constant, isn't the equilibrium probability of being available given by
Pr(available) = Pr(entering available subpopn) / (Pr(entering available subpop) + Pr(leaving available subpop))
or (1 - GammaPrime) / (1 - GammaPrime + GammaDoublePrime) (using RMark parameter names)?
That would lead to a super population estimate of Ni-hat/Pr(available).
What is odd is that in screwing around with some estimates from a model of real data, changing the
around doesn't seem to affect the subsequent estimates by much at all. It may not be even appropriate to change a single value to see how it affects the rest because I would guess that there is some interdependence among the model parameter estimates.
In any event, this is kind of where I'm at with it and would love to see if anyone (particularly the experts) has any additional insight or could even spell out a more general approach to this problem when you have time-varying Markovian temporary emigration and want to get at an estimate of the superpopulation.