Estimation and model selection in constrained deconvolution.

We analyze in detail the estimation problem associated with the following problem. Given n noisy measurements (yi, i = 1, ..., n) of the response of a system to an input (A(t) where t indicates time), obtain an estimate of A(t) given a known K(t) (the unit impulse response function of the system) under the model: yi = integral of 0(ti) A(s)K(ti - s)ds + epsilon i where epsilon 1, ... ,epsilon n are independent identically distributed random variables with mean zero and common finite variance. In the solution to the problem, the unknown function is represented by a spline function, and the problem is recast in terms of (inequality constrained) linear regression. The main issues addressed are: (a) the comparison of different nonparametric regression methods in this context, and (b) how to do model selection, i.e., given a (finite) set of candidate spline functions, select the (possibly unique) best one using some (statistically based) selection criteria. Different spline candidate sets, and different asymptotic and resampling-based statistical selection criteria are compared by means of simulations. Due to the particular nature of the estimation problem, modifications to the criteria are suggested. Applications to simulated and real pharmacokinetics data are reported.