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Adaptive estimation of the conditional density in presence of censoring. (English) Zbl 1193.62055
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 $$\delta_i=\mathbb1_{\{Y_i\leq C_i\}}$$ 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.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62N01 Censored data models 65C60 Computational problems in statistics (MSC2010)
censored data