  This work contains two main contributions concerning the expurgation of
hierarchical ensembles for the asymmetric broadcast channel. The first is an
analysis of the optimal maximum likelihood (ML) decoders for the weak and
strong user. Two different methods of code expurgation will be used, that will
provide two competing error exponents. The second is the derivation of
expurgated exponents under the generalized stochastic likelihood decoder (GLD).
We prove that the GLD exponents are at least as tight as the maximum between
the random coding error exponents derived in an earlier work by Averbuch and
Merhav (2017) and one of our ML-based expurgated exponents. By that, we
actually prove the existence of hierarchical codebooks that achieve the best of
the random coding exponent and the expurgated exponent simultaneously for both
users.
