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

## TL;DR

This paper develops a nonparametric adaptive statistical method for inferring age-dependent birth and death rates in large populations, using stochastic models and PDE analysis to improve mortality data estimation.

## Contribution

It introduces a novel adaptive inference framework for stochastic population models, achieving minimax optimality and handling anisotropic smoothness in demographic data.

## Key findings

- Established new concentration inequalities for stochastic PDE approximation.
- Implemented Goldenshluger-Lepski algorithm for adaptive estimation.
- Achieved minimax optimality over broad smoothness classes.

## Abstract

Motivated by improving mortality tables from human demography databases, we investigate statistical inference of a stochastic age-evolving density of a population alimented by time inhomogeneous mortality and fertility. Asymptotics are taken as the size of the population grows within a limited time horizon: the observation gets closer to the solution of the Von Foerster Mc Kendrick 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 new concentration inequalities that enable us to implement the Goldenshluger-Lepski algorithm and derive oracle inequalities. We obtain minimax optimality and adaptation over a wide range of anisotropic H\"older smoothness classes.