Effects of measurement error on Monte Carlo integration estimators of tree volume: sample diameters measured at random upper-stem heights

A study of the effects of measurement error was conducted on importance sampling and control variate sampling estimators of tree stem volume in which sample diameters are measured at randomly selected upper-stem heights. It was found that these estimators were unbiased in the presence of additive mean zero and multiplicative mean one measurement error applied to random samples of upper-stem diameter squared. However, biases due to measurement error are present if additive or multiplicative error is applied to upper-stem diameter rather than to upper-stem diameter squared. This is significant, as it appears that most of the previous studies on the magnitude of upper-stem diameter measurement error implicitly assume that the mean error is centered around the diameter rather than about the square of the diameter. Application of typical upper-stem measurement error obtained from previous studies to bias formulae derived here indicates that the bias could be a concern for small trees and with additive measurement error within ranges found in previous studies. Formulae for the variances of importance sampling and control variate sampling are derived, which include the contribution of both measurement error and sampling error. Results from previous studies of Monte Carlo integration estimator sampling error are combined with results from studies of upper-stem measurement error to obtain estimates of the typical magnitude of the contribution of measurement error to total estimator variability. Increases in upper-stem sample size may be warranted due to the impact of measurement error if precise estimates of stem volume at the individual-tree level are desired.