Quantization complexity and training sample size in detection

For the
k
-hypothesis detection problem, it is shown that, among the
k
-classes of probability density functions with
m
fixed quantiles, histograms achieve the least favorable performance as measured by the probability of correct detection and Chernoff distance. It is assumed that the
m
cell probabilities are estimated using
n
training samples per class. With the aid of the estimated cell probabilities, new observations are processed. A distribution-free upper bound to the probability of
\epsilon
-deviation between the actual probability of correct detection and the theoretical (known quantiles) probability is derived as a function of
(m,n,\epsilon,k,u_{o})
, where
u_{o}
is a uniform upper bound to the true class densities. The bound converges exponentially to zero as
n \rightarrow \infty
. Exponential convergence is obtained by choosing
m = n^{\alpha}, 0 < \alpha < 1
. Hence, the rule
m = n^{\alpha}
answers the long standing question of how to relate
m
and
n
in a distribution-free manner. The question of the optimal choice of a is also discussed.