Use of Gumbel and Weibull functions to model extreme values of diameter distributions in forest stands

Abstract

& Context Families of the Gumbel (type I), Fréchet (type II) and Weibull (type III) distributions can be combined in the generalized extreme value (GEV) family of distributions. Maximum and minimum values of diameters in forest stands can be used in forest modelling, mainly to define parameters of the functions used in diameter class models as well as in some practical cases, such as modelling maximum diameters for sawing and processing purposes. & Aims The purpose of this study was to examine and compare two extreme value distribution functions (the Gumbel and the Weibull functions) in modelling the distribution of the minimum and the maximum values of representative sets of tree diameter samples. Both of these functions were applied to the lower and upper values of the diameter distributions of the main forest species in northwest Spain: Quercus robur L., Betula pubescens Ehrh., Pinus radiata D. Don, Pinus pinaster Ait. and Pinus sylvestris L. & Methods Parameters of the Gumbel function were estimated using the mode and the moments of the distributions, and parameters of the Weibull function were estimated using the moments method.& Results In general, the Weibull distribution was the most suitable model for describing the maximum diameters. The mode method of the Gumbel yielded the best results for minimum diameters of birch and Monterrey pine. The Gumbel distribution, fitted by either the mode-or momentsbased methods, proved more suitable than the Weibull distribution for describing the minimum diameters in maritime pine and Scots pine stands. & Conclusion In some cases, better results were obtained with the Gumbel than the Weibull distribution for describing the distribution of extreme diameter values in forest stands in northwest Spain. This is the first example of the application of the Gumbel distribution in forest modelling.

Diameter Structure, Spatial Pattern, and Management Scenarios of Acapu Trees: A Case Study

“…Several types of distribution can describe the diametric structure of trees. Among the most prominent in the literature, and used in this paper, are Log-Normal, Weibull-3P, Gamma-2P, Burr-3P, and Johnson SB (e.g., Lima et al, 2014, Gorgoso-Varela & Rojo-Alboreca, 2014, Lima et al, 2017. The parameters of these probability distribution functions were estimated using the maximum likelihood estimator (MLE).…”

ForestFit: An R package for modeling plant size distributions

“…The Weibull distribution is perhaps the most conspicuous distribution used for modeling diameter distributions due to its ability to represent common shapes in diameter distribution data [1,9,10,21,23,28]. ForestFit offers several methods for estimating the parameters of two-and threeparameter Weibull distributions, which are listed in Table 1 along with relevant sources for further details regarding each estimation method.…”

Optimal bandwidth selection for recursive Gumbel kernel density estimators

“…1. The results given in Table 3 are based on the Plug-In estimator of I and I (see, (9), (10), (16) and (17) for the estimation of I and I and (14) and (19) for the plug-In MWISE). This estimators are very useful when we use real data since we do not need to know the density of our sample.…”

“…Many approaches have been considered in the parametric framework, we can cite the work of [17], in which the model parameters were estimated using an inferential procedure based on likelihood, we can list also the work of [9], in which the parameters of the Gumbel function were estimated using the mode and the moments of the distributions. It should be noted that all these methods involve the estimation of unknown parameters.…”