Local Variance Estimation for Censored Observations

This talk is motivated by a problem frequently considered in medical science.
The Robert Bosch hospital in Stuttgart treats different patients, collects data about these patients and makes them available to Universit¨at Stuttgart for research.
Assume now that we have information about patients which is collected during the treatment of a cancer. On the basis of these data the statistician helps physicians predict whether the cancer may come back after the treatment,
e.g., predict the survival time of a new patient in dependence on different factors (”predictors”).
Mathematically speaking, the task is to estimate the mean survival time Y given a realization x of the d-dimensional predictor vector X :
E{Y |X = x} =: m(x). The quality of the prediction by m can be globally defined by the (minimum) mean squared error and locally by the local variance E 2(x) := E{(Y − m(X))2|X = x}.
The aim of the talk is to give estimators of the local variance function and to
show the common properties of convergence and its rate. In particular, we deal with local averaging (partitioning, nearest neighbor) and least squares
estimation methods.
A feature that complicates the analysis is that the follow-up program of the patients may be incomplete. After a certain censoring time, there is no information any longer about the patient. We estimate the local variance also under censoring, using in addition the product-limit estimator.