Abstract||Two dimensional stocks of fish can be assessed with methods that mimic the analysis of research survey data but that use commercial catch-effort data. This finite population approach has scarcely been used in fisheries science though it brings about very large sample sizes of local fish density with models of only moderate levels of complexity. The extracted information about the status of the stock can be interpreted as biomass indices.
Statistical inference on finite populations has been the locus of a highly specialized branch of sampling-distribution inference, unique because observable variables are not considered as random variables. If statistical inference is defined as "the identification of distinct sets of plausible and implausible values for unobserved quantities using observations and probability theory" then it is shown that Godambe's paradox implies that the classical finite populations approach is inherently contradictory as a technique of statistical inference. The demonstration is facilitated by the introduction of an extended canonical form of an experiment of chance, that apart from the three components identified by Birnbaum, also contains the time at which the experiment is performed. Realization of the time random variable leaves the likelihood function as sole data-based mathematical tool for statistical inference, in contradiction with sampling-distribution inference and in agreement with direct-likelihood and Bayesian inference.
A simple mathematical model is introduced for biomass indices in the spatial field defined by the fishing grounds. It contains three unknown parameters, the natural mortality rate, the probability of observing the stock in the area covered by the fishing grounds, and mean fish density in the sub-areas where the stock was present. A new theory for the estimation of mortality rates is introduced, using length frequency data, that is based on the population ecology analogue of Hamilton-Jacobi theory of classical mechanics. The family of equations require estimations of population growth, individual growth, and recruitment pattern. Well known or new techniques are used for estimating parameters of these processes. Among the new techniques, a likelihood-based geostatistical model to estimate fish density is proposed and is now in use in fisheries science (Roa-Ureta and Niklitschek, 2007, ICES Journal of Marine Science 64:1723-1734), as well as a new method to estimate individual growth parameters (Roa-Ureta, In Press, Journal of Agricultural, Biological, and Environmental Statistics).
All inference is done only using likelihood functions and approximations to likelihood functions, as required by the Strong Likelihood principle and the direct-likelihood school of statistical inference. The statistical model for biomass indices is a hierarchical model with several sources of data, hyperparameters, and nuisance parameters. Even though the level of complexity is not low, a full Bayesian formulation is not necessary. Physical factors, mathematical manipulation, profile likelihoods and estimated likelihoods are used for the elimination of nuisance parameters. Marginal normal and multivariate normal likelihood functions, as well as the functional invariance property, are used for the hierarchical structure of estimation. In this manner most sources of information and uncertainty in the data are carried over up the hierarchy to the estimation of the biomass indices.