  We apply the method of Hankel transforms to develop goodness-of-fit tests for
gamma distributions with given shape parameter and unknown rate parameter,
thereby extending results of Baringhaus and Taherizadeh (2010) on the
exponential distributions. We derive the limiting null distribution of the test
statistic as an integrated squared Gaussian process, obtain the corresponding
covariance operator, and oscillation properties of its eigenfunctions. We show
that the eigenvalues of the operator satisfy an interlacing property, and we
apply that property in approximating critical values of the test statistic in
one of the two applications to data considered. Further, we establish the
consistency of the test. In studying properties of the test statistic under a
variety of contiguous alternatives, we obtain the asymptotic distribution of
the test statistic for gamma alternatives with varying rate or shape parameters
and for a class of contaminated gamma models. We investigate the approximate
Bahadur slope of the test statistic under local alternatives and we establish
the validity of the Wieand condition, under which the approaches through the
approximate Bahadur efficiency and the Pitman efficiency are in accord.
