Adaptive estimation of the conditional density in presence of censoring.

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 realizes a global squared-bias/variance compromise 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.