by **gwhite** » Thu May 23, 2019 4:46 pm

The Schnabel estimator is a completely different estimator from the Huggins estimator -- it is not a maximum likelihood estimator, and hence does not share all the good characteristics of an MLE. Further, I'm not sure that the Schnabel estimator conditions on just the encounter histories where animals are captured at least once like the Huggins estimator does.

You can also run the full likelihood estimator, and see you will again get a very similar estimate but not identical to the Huggins estimator. The full likelihood estimator does not condition on encounter histories with the probability that animals are captured >0. The all zero encounter history is part of the likelihood.

The estimator from the literature most similar to the M(t) Huggins estimator is the Darroch estimator, although it is exactly the full likelihood M(t) estimator. These subtleties escape users until you actually write out the likelihoods.

You might ask why the Huggins estimator is used and recommended instead of the full likelihood estimator -- because 1) individual covariates can be used with the Huggins estimator as all individual in the sample were captured to measure the covariate value (which is not the case for uncaptured individuals with the all zero encounter history), and 2) the slight loss in efficiency is negligible compared to the full likelihood estimator. Further, there are fewer numerical problems with optimizing the Huggins estimator compared to the full likelihood estimator (although this issue is pretty much not an issue now days). The Otis et al. monograph and Program CAPTURE used the full likelihood version for all likelihood estimators in the program. The idea of using individual covariates was not discussed at the time. Huggins and Alho brought this idea to the literature.

Gary

An appropriate quote:

“The most misleading assumptions are the ones you don't even know you're making...”

-- Douglas Adams, English author, scriptwriter, essayist, humorist, satirist and dramatist.