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The Chapman-Richards Distribution and its Relationship to the Generalized Beta

Forest Ecosystems 2019年第3期6卷 219-235页

作者： Jeffrey H.Gove Thomas B.Lynch Mark J.Ducey USDA Forest Service Northern Research Station271 Mast Road Durham 03824NHUSA Professor Emeritus Department of Natural Resource Ecology and ManagementOklahoma State UniversityAg Hall Room 008CStillwater 74078OKUSA Department of Natural Resources and the Environment University of New Hampshire114 James Hall Durham 03824NHUSA

Background: The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions of the Chapman-Richards growth function, constant mortality and recruitment into the mathematical form of the distribution. Therefore, unlike 'assumed' distribution models, it is intrinsically linked with the underlying vital rates for the forest area under consideration. Methods: It is shown that the Chapman-Richards distribution can be recast as a subset of the generalized beta distribution of the first kind, a rich family of assumed probability distribution models with known properties. These known properties for the generalized beta are then immediately available for the Chapman-Richards distribution, such as the form of the compatible basal area-size distribution. A simple two-stage procedure is proposed for the estimation of the model parameters and simulation experiments are conducted to validate the procedure for four different possible distribution shapes. Results: The simulations explore the efficacy of the two-stage estimation procedure;these cover the estimation of the growth equation and mortality-recruitment derives from the equilibrium assumption. The parameter estimates are shown to depend on both the sample size and the amount of noise imparted to the synthetic measurements. The results vary somewhat by distribution shape, with the smaller, noisier samples providing less reliable estimates of the vital rates and final distribution forms. Conclusions: The Chapman-Richards distribution in its original form, or recast as a generalized beta form, presents a potentially useful model integrating vital rates and stand diameters into a flexible family of resultant distributions shapes. The data requirements are modest, and parameter estimation is straightforward provided the minimal recommended sample sizes are obtained.

关键词： Diameter distributions Chapman-Richards growth Generalized beta distribution of the first kind Maximum likelihood McKendrick-Von foerster equation Physiologically structured population model Size-structured distributions

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