  In the statistical literature, as well as in artificial intelligence and
machine learning, measures of discrepancy between two probability distributions
are largely used to develop measures of goodness-of-fit. We concentrate on
quadratic distances, which depend on a non-negative definite kernel. We propose
a unified framework for the study of two-sample and k-sample goodness of fit
tests based on the concept of matrix distance. We provide a succinct review of
the goodness of fit literature related to the use of distance measures, and
specifically to quadratic distances. We show that the quadratic distance
kernel-based two-sample test has the same functional form with the maximum mean
discrepancy test. We develop tests for the $k$-sample scenario, where the
two-sample problem is a special case. We derive their asymptotic distribution
under the null hypothesis and discuss computational aspects of the test
procedures. We assess their performance, in terms of level and power, via
extensive simulations and a real data example. The proposed framework is
implemented in the QuadratiK package, available in both R and Python
environments.
