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 . 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.