  In this paper new two-dimensional goodness of fit tests are proposed. They
are of supremum-type and are based on different types of characterizations. For
the first time a characterization based on independence of two statistics is
used for goodness-of-fit testing. The asymptotics of the statistics is studied
and Bahadur efficiencies of the tests against some close alternatives are
calculated. In the process a theorem on large deviations of Kolmogorov-type
statistics has been extended to the multidimensional case.
