Generalized Error Exponents for Small Sample Universal Hypothesis Testing

The small sample universal hypothesis testing problem is investigated in this paper, in which the number of samples n is smaller than the number of possible outcomes m . The goal of this paper is to find an appropriate criterion to analyze statistical tests in this setting. A suitable model for analysis is the high-dimensional model in which both n and m increase to infinity, and n=o(m) . A new performance criterion based on large deviations analysis is proposed and it generalizes the classical error exponent applicable for large sample problems (in which m=O(n) ). This generalized error exponent criterion provides insights that are not available from asymptotic consistency or central limit theorem analysis. The following results are established for the uniform null distribution: 1) The best achievable probability of error P_{e} decays as P_{e}=\exp \{-(n^{2}/m) J (1+o(1))\} for some J>0 . 2) A class of tests based on separable statistics, including the coincidence-based test, attains the optimal generalized error exponents. 3) Pearson's chi-square test has a zero generalized error exponent and thus its probability of error is asymptotically larger than the optimal test.