  The general formalisms of the $q$-dual statistics, the Boltzmann-Gibbs
statistics, and three versions of the Tsallis statistics known as Tsallis-1,
Tsallis-2, and Tsallis-3 statistics have been considered in the canonical
ensemble. We have rigorously proved that the probability distribution of the
Tsallis-1 statistics is invariant under the uniform energy spectrum translation
at a fixed temperature. This invariance demonstrates that the formalism of the
Tsallis-1 statistics is consistent with the fundamentals of the equilibrium
statistical mechanics. The same results we have obtained for the probability
distributions of the Tsallis-3 statistics, Boltzmann-Gibbs statistics, and
$q$-dual statistics. However, we have found that the probability distribution
of the Tsallis-2 statistics, the expectation values of which are not consistent
with the normalization condition of probabilities, is indeed not invariant
under the overall shift in energy as expected.
