  Distance covariance and distance correlation are scalar coefficients that
characterize independence of random vectors in arbitrary dimension. Properties,
extensions, and applications of distance correlation have been discussed in the
recent literature, but the problem of defining the partial distance correlation
has remained an open question of considerable interest. The problem of partial
distance correlation is more complex than partial correlation partly because
the squared distance covariance is not an inner product in the usual linear
space. For the definition of partial distance correlation we introduce a new
Hilbert space where the squared distance covariance is the inner product. We
define the partial distance correlation statistics with the help of this
Hilbert space, and develop and implement a test for zero partial distance
correlation. Our intermediate results provide an unbiased estimator of squared
distance covariance, and a neat solution to the problem of distance correlation
for dissimilarities rather than distances.
