Large deviations for a class of scale-free statistics under Gamma distribution

We study large deviations for sums of functions from Gamma-distributed random values normalized by their sample mean. In particular, large deviations are worked out for several well-known scale-free exponentiality tests such as Moran test and Liliefors test. We also obtain some new results concerning a class of goodness-of-fit tests for the uniform distribution based on spacings.