Quick and easy one-step parameter estimation in differential equations

Differential equations are customarily used to describe dynamic systems. Existing methods for estimating unknown parameters in those systems include parameter cascade, which is a spline-based technique, and pseudo-least-squares, which is a local-polynomial-based two-step method. Parameter cascade is often referred to as a ‘one-step method’, although it in fact involves at least two stages: one to choose the tuning parameter and another to select model parameters. We propose a class of fast, easy-to-use, genuinely one-step procedures for estimating unknown parameters in dynamic system models. This approach does not need extraneous estimation of the tuning parameter; it selects that quantity, as well as all the model parameters, in a single explicit step, and it produces root-n-consistent estimators of all the model parameters. Although it is of course not as accurate as more complex methods, its speed and ease of use make it particularly attractive for exploratory data analysis.