  Competing styles of Statistical Mechanics have been introduced as practical
succedaneous to the conventional well established Boltzmann-Gibbs statistical
mechanics, when in the use of the latter the researcher is impaired in his/her
capacity for satisfying the Criteria of Efficiency and/or Sufficiency in
statistics [Fisher, 1922], that is, a failure in the characterization (presence
of fractality, scaling, etc.) of the system related to some aspect relevant to
the given physical situation. To patch this limitation on the part of the
observer, in order to make predictions on the values of observables and
response functions, are introduced unconventional approaches. We present a
detailed description of their construction and a clarification of its scope and
interpretation. Also, resorting to the use of the particular case of Renyi's
unconventional statistics is built a nonequilibrium ensemble formalism. The
unconventional distribution functions of fermions and bosons are obtained, and
in a follow-up article [cond-mat/0306247] we describe applications to the study
of experimental results in semiconductor physics and in electro-chemistry
involving nanometric scales and fractal-like structures, and some additional
theoretical analysis is added. PACS: 05.70.Ln, 82.20.Mj, 82.20.Db Keywords:
Nonequilibrium Ensemble Formalism; Generalized Informational Entropies;
Generalized Statistics; Nonextensive Statistics; Renyi Statistics; Escort
Probability.
