A theoretical approach is described for determining the joint distribution of the parameters of the von Bertalanffy growth model from statistical moments of length. The approach extends the work of K. J. Sainsbury, who had demonstrated that different mean parameter estimates are obtained by assuming that the von Bertalanffy equation applies to individual fish rather than to groups of fish. Sainsbury articulated the goal of studying the joint probability distributions of K and L∞ in animal populations and developed a maximum likelihood procedure for estimating the parameters of particular distributional forms describing K and L∞, which were assumed for mathematical convenience to be statistically independent. The primary goal of the present paper is to provide a framework for future research in generalizing Sainsbury's approach by considering (K, L∞) to be a random vector described by a joint probability density function and by allowing broader classes of distributions to be considered. Minimum cross-entropy (MCE) inversion, an information–theoretic methodology for approximating probability distributions, is shown to be effective in selecting a reasonable and unique joint distribution corresponding to observable length moments. Appealing features of the MCE methodology include the ability to include prior knowledge of uncertain applicability and the capacity of the resulting approximate distribution to represent potential stochastic dependencies between the von Bertalanffy parameters. Several numerical examples, using simulated and historical data, are presented to illustrate how information about the variation and covariation of L∞ and K can be inferred from a minimal set of length moments. The directions developed in this paper are far from a practical and useful methodology. The MCE inversion procedure is a "method of moments," with no statistical assessment of reliability. Further research is needed to make this promising pdf approximation scheme better suited for real fisheries problems.
Joint distribution of Busemann functions in the exactly solvable corner growth model
