  Nonparametric estimation of the conditional expectation $E(Y | U)$ of an
outcome $Y$ given a covariate vector $U$ is of primary importance in many
statistical applications such as prediction and personalized medicine. In some
problems, there is an additional auxiliary variable $Z$ in the training dataset
used to construct estimators, but $Z$ is not available for future prediction or
selecting patient treatment in personalized medicine. For example, in the
training dataset longitudinal outcomes are observed, but only the last outcome
$Y$ is concerned in the future prediction or analysis. The longitudinal
outcomes other than the last point is then the variable $Z$ that is observed
and related with both $Y$ and $U$. Previous work on how to make use of $Z$ in
the estimation of $E(Y|U)$ mainly focused on using $Z$ in the construction of a
linear function of $U$ to reduce covariate dimension for better estimation.
Using $E(Y|U) = E\{E(Y|U, Z)| U\}$, we propose a two-step estimation of inner
and outer expectations, respectively, with sufficient dimension reduction for
kernel estimation in both steps. The information from $Z$ is utilized not only
in dimension reduction, but also directly in the estimation. Because of the
existence of different ways for dimension reduction, we construct two
estimators that may improve the estimator without using $Z$. The improvements
are shown in the convergence rate of estimators as the sample size increases to
infinity as well as in the finite sample simulation performance. A real data
analysis about the selection of mammography intervention is presented for
illustration.
