  We present single imputation method for missing values which borrows the idea
of data depth---a measure of centrality defined for an arbitrary point of a
space with respect to a probability distribution or data cloud. This consists
in iterative maximization of the depth of each observation with missing values,
and can be employed with any properly defined statistical depth function. For
each single iteration, imputation reverts to optimization of quadratic, linear,
or quasiconcave functions that are solved analytically by linear programming or
the Nelder-Mead method. As it accounts for the underlying data topology, the
procedure is distribution free, allows imputation close to the data geometry,
can make prediction in situations where local imputation (k-nearest neighbors,
random forest) cannot, and has attractive robustness and asymptotic properties
under elliptical symmetry. It is shown that a special case---when using the
Mahalanobis depth---has direct connection to well-known methods for the
multivariate normal model, such as iterated regression and regularized PCA. The
methodology is extended to multiple imputation for data stemming from an
elliptically symmetric distribution. Simulation and real data studies show good
results compared with existing popular alternatives. The method has been
implemented as an R-package. Supplementary materials for the article are
available online.
