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 t epsilon [0,infinity], and it is assumed that the collection of such (x,t) pairs forms a Poisson proce ss in [0,L] x [0,infinity] with intensity measure dxd Lambda(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 p art of [0,L]. We want to estimate Lambda on the basis of one or more realiz ations of the process, the data being the locations and germination times o f the germinated seeds. We derive the maximum likelihood estimator of v and a nonparametric estimator of Lambda and describe methods of obtaining para metric estimates from it, illustrating these with reference to gamma densit ies. Simulation results are described and the methods applied to some neuro biological data. An Appendix outlines the S-PLUS code used.