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In general we have a model (g) that predicts the number of recaptured tags as a function of a number of parameters (phi). The predicted recaptures (C) can be related to the observed recaptures (C hat) by a “likelihood” function of some sort. The most familiar of these is of course the mean squared error. The values of the parameters that maximize L are the maximum likelihood estimators.
Note the subscript k indicates that there can be several models with different sets of parameters.
The advantages of MLE are that the framework is completely general hypothesis testing is simplified using likelihood ratio tests and in principle the standard deviations of the parameter estimates can be calculated.
The likelihood function attempts to accommodate the observational error in the data.
I’ll take up process error when I discuss models of fishing mortality.
For further reading about maximum likelihood estimation, refer to:
Kendall, M. and A. Stuart. 1979. The advanced theory of statistics. Volume 2. Inference and relationship. Chapter 18.