Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 4, 734--763

Adaptive Estimation of the Conditional Density in the Presence of Censoring

Elodie Brunel, Universit\'e Montpellier 2, Montpellier, France
Fabienne Comte, Universit\' Paris Descartes, Paris, France
Claire Lacour, Universit\'e Paris-Sud 11, Orsay cedex, France

SUMMARY. Consider an i.i.d. sample $(X_i,Y_i)$, $i=1, \dots, n$, of observations and denote by $\pi(x,y)$ the conditional density of $Y_i$ given $X_i=x$. We provide an adaptive nonparametric strategy to estimate $\pi$. We prove that our estimator achieves optimal rates of convergence in a context of anisotropic function classes. We prove that our procedure can be adapted to positive censored random variables $Y_i$'s, i.e., when only $Z_i=\inf(Y_i, C_i)$ and $\delta_i=\1_{\{Y_i\leq C_i\}}$ are observed, for an i.i.d. censoring sequence $(C_i)_{1\leq i\leq n}$ independent of $(X_i,Y_i)_{1\leq i\leq n}$. Simulation experiments illustrate the method.

AMS (2000) subject classification. Primary 62N02, 62G07.

Key words and phrases. Adaptive estimation, censored data, conditional density, nonparametric methods.

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