The sparse-sample goodness of fit problem is considered, in which the number of
samples n is smaller than the size of the alphabet m.   A convenient setting
for
analysis is the high-dimensional model in which both $n$ and $m$ tend to
infinity, and $n=o(m)$.    A new performance criterion is introduced, based on
large deviation analysis, which generalizes the classical error exponent
applicable for large sample problems (in which $m=O(n)$). The results are:
(i)  The best achievable probability of error $P_e$ decays as
$-log(P_e)=(n^2/m)(1+o(1)) J$  for some $ J>0$.
(ii)  The coincidence-based test has nonzero generalized error exponent $J$ 
while the widely-used Pearson's chi-square test has a zero generalized error
exponent.