Nonparametric adaptive inference of birth and death models in a large population limit

Abstract

Motivated by improving mortality tables from human demography databases, we investigate statistical inference of an age-evolving population model alimented by time inhomogeneous mortality and fertility. Asymptotics are taken as the size of the population grows within a fixed time horizon: the observation gets closer to the solution of the McKendrick–Von Foerster equation, and the difficulty lies in controlling simultaneously the stochastic approximation to the limiting PDE in a suitable sense together with an appropriate parametrisation of the anisotropic solution. In this setting, we prove a concentration inequality that enables us to implement the Goldenshluger–Lepski algorithm and derive oracle inequalities. We obtain minimax optimality and adaptation over a range of anisotropic Hölder smoothness classes.