Nonparametric and Parametric Estimation for a Linear Germination-Growth Model

Seeds are planted on the interval [0, L] at various locations. Each seed has a location x and a potential germination time tε [0, [infinity]), and it is assumed that the collection of such (x, t) pairs forms a Poisson process in [0, L] x [0, [infinity]) with intensity measure dxdΛ(t). From each seed that germinates, an inhibiting region grows bidirectionally at rate 2v. These regions inhibit germination of any seed in the region with a later potential germination time. Thus, seeds only germinate in the uninhibited part of [0, L]. We want to estimate Λ on the basis of one or more realizations of the process, the data being the locations and germination times of the germinated seeds. We derive the maximum likelihood estimator of v and a nonparametric estimator of Λ and describe methods of obtaining parametric estimates from it, illustrating these with reference to gamma densities. Simulation results are described and the methods applied to some neurobiological data. An Appendix outlines the S-PLUS code used.