The Average Performance of the Minimax Code

The minimax distribution P_F ^N defined by the probabilities p_i,F ^N = 1/lambda_N(i-1) ^(i-1)/iⁱ plays an important role in the context of encoding finite monotone sources with unknown probabilities. The Shannon code for this distribution is a suboptimal minimax code. The average performance of this suboptimal code is considered in this paper. In order to evaluate the average performance of the minimax code, we compare it with the minave code which minimizes the average redundancy over all monotone sources with N symbols. To achieve this, we study the informational divergence of the minave and minimax distributions and prove that it asymptotically grows as -2.275+log log N. The log log N degradation in average performance should not be regarded as a drawback of the minimax code because the minimum average codeword length over the class of monotone sources grows as log N, which asymptotically dominates log log N