Optimal adaptive estimation of the relative density

Gaëlle Chagny () and Claire Lacour ()

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2015, vol. 24, issue 3, 605-631

Abstract: 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

Keywords: Nonparametric estimation; Model selection; Relative density; Two-sample problem; 62G05; 62G07; 62G30 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11749-015-0426-6

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