Estimation and inference in mixed effect regression models using shape constraints, with application to tree height estimation

Estimation of tree height given diameter is an important part of the forest inventory analysis of the US Forest Service. Existing methods use parametric models to estimate the curve. We propose a semiparametric model in which log (height) is a smooth, increasing and concave function of diameter, with a random‐plot component and fixed effect covariates. Large sample properties and inference methods that work well in practice are derived. Proposed inference methods use approximate normal distributions for the fixed effects and a likelihood ratio test for the significance of the random effect. A closed form approximate prediction method is provided and overall it outperformed competitors for both a simulation and a real data application. The methods are implemented by the cgamm routine in the R package cgam and can be used for a wide range of mixed model applications.