  We initiate the study of goodness-of-fit testing when the data consist of
positive definite matrices. Motivated by the recent appearance of the cone of
positive definite matrices in numerous areas of applied research, including
diffusion tensor imaging, models of the volatility of financial time series,
wireless communication systems, and the analysis of polarimetric radar images,
we apply the method of Hankel transforms of matrix argument to develop
goodness-of-fit tests for Wishart distributions with given shape parameter and
unknown scale matrix. We obtain the limiting null distribution of the test
statistic and the corresponding covariance operator. We show that the
eigenvalues of the operator satisfy an interlacing property, and we apply our
test to some financial data. Moreover, we establish the consistency of the test
against a large class of alternative distributions and we derive the asymptotic
distribution of the test statistic under a sequence of contiguous alternatives.
We establish the Bahadur and Pitman efficiency properties of the test statistic
and we show the validity of a modified Wieand condition.
