14 research outputs found
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 is smaller than the number of
possible outcomes . The goal of this work 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 and increase to
infinity, and . 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 ). 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:
(i) The best achievable probability of error decays as
for some .
(ii) A class of tests based on separable statistics, including the
coincidence-based test, attains the optimal generalized error exponents.
(iii) Pearson's chi-square test has a zero generalized error exponent and
thus its probability of error is asymptotically larger than the optimal test.Comment: 43 pages, 4 figure