  The classical theory of rank-based inference is entirely based either on
ordinary ranks, which do not allow for considering location (intercept)
parameters, or on signed ranks, which require an assumption of symmetry. If the
median, in the absence of a symmetry assumption, is considered as a location
parameter, the maximal invariance property of ordinary ranks is lost to the
ranks and the signs. This new maximal invariant thus suggests a new class of
statistics, based on ordinary ranks and signs. An asymptotic representation
theory \`{a} la H\'{a}jek is developed here for such statistics, both in the
nonserial and in the serial case. The corresponding asymptotic normality
results clearly show how the signs add a separate contribution to the
asymptotic variance, hence, potentially, to asymptotic efficiency. As shown by
Hallin and Werker [Bernoulli 9 (2003) 137--165], conditioning in an appropriate
way on the maximal invariant potentially even leads to semiparametrically
efficient inference. Applications to semiparametric inference in regression and
time series models with median restrictions are treated in detail in an
upcoming companion paper.
