Constraint Shock in Diameter Distribution Models: The Cost of Ignoring Negative Bounds

Abstract

Diameter distributions are invariably fitted with constrained zero or positive lower bounds to prevent negative diameters. Goodwin (2021) found that the bound for the 3-parameter Weibull could be negative for negatively skewed data, and that constraining it to zero reduced model accuracy. The loss of model accuracy due to a bound constraint is referred to here as constraint shock. This article shows that the 4-parameter Kumaraswamy and Johnson’s SB distributions can also have negative lower bounds and exhibit constraint shock. A 3-step parameter recovery method was used to fit these distributions to plots in unthinned mixed species eucalypt plantations and results were compared with the less flexible Weibull. Based on Kolmogorov-Smirnov statistics, mean and maximum constraint shock for the 4-parameter distributions were 19% and 62%, respectively, compared with 19% and 51% for the Weibull, which indicated that constraint shock was not affected by model flexibility. Constraint shock was largely avoided by truncating and normalizing distributions with negative bounds. This work introduces a paradigm shift in diameter distribution modelling and adds clarity to a field that has not previously recognized the existence of constraint shock. Study Implications: Constraint shock is the loss of accuracy incurred by constraining the negative lower bound of a distribution to a nonnegative value. For a plantation eucalypt dataset in which 83% of plot diameters were negatively skewed, maximum constraint shock was 62% for the Kumaraswamy and Johnson’s SB distributions, and 51% for the less flexible 3-parameter Weibull, based on Kolmogorov-Smirnov statistics. Using distribution expectations of kurtosis, skewness, maximum diameter, and minimum diameter—all independent of bound parameters—a tractable 3-step parameter recovery method is described for unconstrained, constrained, and truncated 3- and 4-parameter distributions. This work introduces a paradigm shift in the treatment of distribution bounds that will result in substantial model improvements for negatively skewed data.