Abstract
It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kernel-type Inverse Probability of Censoring Weighted estimators of the multivariate regression function \citeDebbarhViallon3. In this paper, we get the exact rate of strong uniform consistency for such estimators. Our uniform limit laws especially lead to the construction of asymptotic simultaneous 100% confidence bands for the true regression function.
Highlights
Consider a triple (Y, C, X) of random variables defined in IR+ × IR+ × IRd, d ≥ 2, where Y is the variable of interest, C a censoring variable and X = (X1, . . . , Xd) a vector of concomitant variables
In this paper, following the ideas initiated by [28], we use a nonparametric version of particular synthetic data estimators, commonly referred to as Inverse Probability of Censoring Weighted [I.P.C.W.] estimators
Assumption (C.4) is classical when dealing with kernel type estimators of the regression function
Summary
Following the ideas introduced in [10], we make use of the marginal integration method, coupled with initial kernel-type I.P.C.W. estimators to provide an estimator for the additive censored regression function. This combination leads to estimators for which the theory is easier to derive, which was wanted here, given the technicalities in the proof, even in this simplified setting (note that, as already mentioned, extensions to other synthetic data estimators can be obtained; see Paragraph 3.1). This kind of bands may be complementary to the more classical (1 − α) × 100% pointwise confidence intervals derived from CLT type results (see Section 4)
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