Fast grid search and bootstrap‐based inference for continuous two‐phase polynomial regression models

Two‐phase polynomial regression models (Robison, 1964; Fuller, 1969; Gallant and Fuller, 1973; Zhan et al., 1996) are widely used in ecology, public health, and other applied fields to model nonlinear relationships. These models are characterized by the presence of threshold parameters, across which the mean functions are allowed to change. That the threshold is a parameter of the model to be estimated from the data is an essential feature of two‐phase models. It distinguishes them, and more generally, multiphase models, from the spline models and has profound implications for both computation and inference for the models. Estimation of two‐phase polynomial regression models is a nonconvex, nonsmooth optimization problem. Grid search provides high‐quality solutions to the estimation problem, but is very slow when performed by brute force. Building upon our previous work on piecewise linear two‐phase regression models estimation, we develop fast grid search algorithms for two‐phase polynomial regression models and demonstrate their performance. Furthermore, we develop bootstrap‐based pointwise and simultaneous confidence bands for mean functions. Monte Carlo studies are conducted to demonstrate the computational and statistical properties of the proposed methods. Three real datasets are used to help illustrate the application of two‐phase models, with special attention on model choice.