  We consider goodness-of-fit tests for uniformity of a multinomial
distribution by means of tests based on a class of symmetric statistics,
defined as the sum of some function of cell-frequencies. We are dealing with an
asymptotic regime, where the number of cells grows with the sample size. Most
attention is focused on the class of power divergence statistics. The aim of
this article is to study the intermediate asymptotic relative efficiency of two
tests, where the powers of the tests are asymptotically non-degenerate and the
sequences of alternatives converge to the hypothesis, but not too fast. The
intermediate asymptotic relative efficiency of the chi-square test wrt an
arbitrary symmetric test is considered in details.
