Optimal adaptive estimation of the relative density

This paper deals with the classical statistical problem of comparing the probability distributions of two real random variables $$X$$ X and $$X_0$$ X 0 , from a double independent sample. While most of the usual tools are based on the cumulative distribution functions $$F$$ F and $$F_0$$ F 0 of the variables, we focus on the relative density, a function recently used in two-sample problems, and defined as the density of the variable $$F_0(X)$$ F 0 ( X ) . We provide a nonparametric adaptive strategy to estimate the target function. We first define a collection of estimates using a projection on the trigonometric basis and a preliminary estimator of $$F_0$$ F 0 . An estimator is selected among this collection of projection estimates, with a criterion in the spirit of the Goldenshluger–Lepski methodology. We show the optimality of the procedure both in the oracle and the minimax sense: the convergence rate for the risk computed from an oracle inequality matches with the lower bound that we also derived. Finally, some simulations illustrate the method. Copyright Sociedad de Estadística e Investigación Operativa 2015