Minimax estimation of the conditional cumulative distribution function

Abstract

Consider an i.i.d. sample (Xi, Yi), i = 1, . . . , n of observations and denote by F (y|x) the conditional cumulative distribution function of Yi given Xi = 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 Yi's, i.e. when only Zi = inf(Yi, Ci) and δi = ½ {Y i ≤C i } are observed, for an i.i.d. censoring sequence (Ci) 1≤i≤n independent of (Xi, Yi) 1≤i≤n . Simulation experiments and a real data example illustrate the method.

Semiparametric topographical mixture models with symmetric errors

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Statistical modelling of wildfire size and intensity: a step toward meteorological forecasting of summer extreme fire risk

“…In this study we use parametric distributions as they provide a simple framework to model fire risk with a limited number of coefficients, which can be of interest for the implementation of a fire risk forecast system. Non-parametric estimations of the conditional distributions of the fire variables with respect to the meteorological covariates could be performed (Brunel et al, 2010). This would lead to longer computation times but probably more accurate estimations of the conditional distributions and associated probabilities.…”

Adaptive estimation of the baseline hazard function in the Cox model by model selection, with high-dimensional covariates

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