Conditional Estimation of Multivariate Dependence Using Projection-Based Bergsma–Dassios τ∗ and Nonparametric Hotelling’s T2

Abstract This study develops a conditional estimation framework for a projection-based multivariate extension of the Bergsma–Dassios τ ∗ dependence measure. The proposed approach combines a fourth-order U-statistic estimator of multivariate dependence with a nonparametric Hotelling-type quadratic statistic. First, the unbiasedness and variance properties of the projection-based estimator are established. Subsequently, conditional estimation is investigated through the Rao–Blackwell principle. Under an unrestricted multivariate normal model with unknown mean vector and covariance matrix, the sample mean and sample scatter matrix are shown to constitute a complete sufficient statistic. An explicit conditional representation of the Rao–Blackwell estimator is then obtained through a uniform distribution on an appropriate Stiefel manifold. The Lehmann–Scheffé theorem establishes the resulting estimator as the unique uniformly minimum variance unbiased estimator (UMVUE) of the specified multivariate dependence parameter. Variance decomposition results are also derived to compare the original U-statistic, the estimator conditioned on the nonparametric Hotelling statistic, and the fully Rao–Blackwellized estimator.