  Kendall's tau and conditional Kendall's tau matrices are multivariate
(conditional) dependence measures between the components of a random vector.
For large dimensions, available estimators are computationally expensive and
can be improved by averaging. Under structural assumptions on the underlying
Kendall's tau and conditional Kendall's tau matrices, we introduce new
estimators that have a significantly reduced computational cost while keeping a
similar error level. In the unconditional setting we assume that, up to
reordering, the underlying Kendall's tau matrix is block-structured with
constant values in each of the off-diagonal blocks. Consequences on the
underlying correlation matrix are then discussed. The estimators take advantage
of this block structure by averaging over (part of) the pairwise estimates in
each of the off-diagonal blocks. Derived explicit variance expressions show
their improved efficiency. In the conditional setting, the conditional
Kendall's tau matrix is assumed to have a constant block structure,
independently of the conditioning variable. Conditional Kendall's tau matrix
estimators are constructed similarly as in the unconditional case by averaging
over (part of) the pairwise conditional Kendall's tau estimators. We establish
their joint asymptotic normality, and show that the asymptotic variance is
reduced compared to the naive estimators. Then, we perform a simulation study
which displays the improved performance of both the unconditional and
conditional estimators. Finally, the estimators are used for estimating the
value at risk of a large stock portfolio; backtesting illustrates the obtained
improvements compared to the previous estimators.
