Minimax estimation of the conditional cumulative distribution function under random censorship

Abstract : Consider an i.i.d. sample $(X_i,Y_i)$, $i=1, \dots, n$ of observations and denote by $F(x,y)$ the conditional cumulative distribution function of $Y_i$ given $X_i=x$. We provide a data driven nonparametric strategy to estimate $F$. We prove that, in term of the integrated mean square risk on a compact set, our estimator performs a squared-bias variance compromise. We deduce from this an upper bound for the rate of convergence of the estimator, in a context of anisotropic function classes. A lower bound for this rate is also proved, which implies the optimality of our estimator. Then 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.
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Sankhya: The Indian Journal of Statistics, 2010, A (Part 2), pp.293-330
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Elodie Brunel, Fabienne Comte, Claire Lacour. Minimax estimation of the conditional cumulative distribution function under random censorship. Sankhya: The Indian Journal of Statistics, 2010, A (Part 2), pp.293-330. <hal-00274637>

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