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We consider goodness-of-fit tests with i.i.d. samples generated from acategorical distribution $(p_1,...,p_k)$. For a given $(q_1,...,q_k)$, we testthe null hypothesis whether $p_j=q_{\pi(j)}$ for some label permutation $\pi$.The uncertainty of label permutation implies that the null hypothesis iscomposite instead of being singular. In this paper, we construct a testingprocedure using statistics that are defined as indefinite integrals of somesymmetric polynomials. This method is aimed directly at the invariance of theproblem, and avoids the need of matching the unknown labels. The asymptoticdistribution of the testing statistic is shown to be chi-squared, and its poweris proved to be nearly optimal under a local alternative hypothesis. Variousdegenerate structures of the null hypothesis are carefully analyzed in thepaper. A two-sample version of the test is also studied.

Goodness-of-Fit Tests for Random Partitions via Symmetric Polynomials