Bayesian inference for Johnson's SB and Weibull distributions

Abstract

ABSTRACT The four-parameter Johnson's SB (JSB) and three-parameter Weibull distributions have received significant attention in the field of forestry for characterising diameters at breast height (DBH). This study suggests the Bayesian method for estimating parameters of the JSB distribution. The maximum likelihood approach uses iterative methods such as a Newton–Raphson (NR) algorithm for maximising the logarithm of the likelihood function. However, there is no guarantee that the NR method converges. Through simulation, this study verified that the NR method for estimating the parameters of the JSB distribution sometimes fails to converge. Further, the Bayesian estimators presented herein were shown to be robust with respect to the initial values and estimate the parameters of the JSB distribution efficiently. The performance of the JSB and three-parameter Weibull distributions was compared in a Bayesian paradigm when these models were fitted to DBH data of three plots randomly selected from a study established in 107 plots of mixed-age ponderosa pine (Pinus ponderosa Dougl. ex Laws.) with scattered western juniper (Juniperus occidentalis Hook.) at the Malheur National Forest on the south end of the Blue Mountains near Burns, Oregon, USA. The Bayesian paradigm demonstrated that JSB was superior to the three-parameter Weibull for characterising the DBH distribution when these models were fitted to the DBH data of the three plots. Moreover, the Bayesian approach outperformed the moment, conditional maximum likelihood, and two-percentile methods when the JSB distribution was fitted to DBH data of three plots and all 107 plots simultaneously.