On the near-singularity of models for animal recovery data

Certain probability models sometimes provide poor descriptions when fitted to data by maximum likelihood. We examine one such model for the survival of wild animals, which is fitted to two sets of data. When the model behaves poorly, its expected information matrix, evaluated at the maximum likelihood estimate of parameters, has a 'small' smallest eigenvalue. This is due to the fitted model being similar to a parameter-redundant submodel. In this case, model parameters that are precisely estimated have small coefficients in the eigenvector corresponding to the smallest eigenvalue. Approximate algebraic expressions are provided for the smallest eigenvalue. We discuss the general applicability of these results.