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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 andcan be improved by averaging. Under structural assumptions on the underlyingKendall's tau and conditional Kendall's tau matrices, we introduce newestimators that have a significantly reduced computational cost while keeping asimilar error level. In the unconditional setting we assume that, up toreordering, the underlying Kendall's tau matrix is block-structured withconstant values in each of the off-diagonal blocks. Consequences on theunderlying correlation matrix are then discussed. The estimators take advantageof this block structure by averaging over (part of) the pairwise estimates ineach of the off-diagonal blocks. Derived explicit variance expressions showtheir improved efficiency. In the conditional setting, the conditionalKendall's tau matrix is assumed to have a constant block structure,independently of the conditioning variable. Conditional Kendall's tau matrixestimators are constructed similarly as in the unconditional case by averagingover (part of) the pairwise conditional Kendall's tau estimators. We establishtheir joint asymptotic normality, and show that the asymptotic variance isreduced compared to the naive estimators. Then, we perform a simulation studywhich displays the improved performance of both the unconditional andconditional estimators. Finally, the estimators are used for estimating thevalue at risk of a large stock portfolio; backtesting illustrates the obtainedimprovements compared to the previous estimators.

Fast estimation of Kendall's Tau and conditional Kendall's Tau matrices under structural assumptions