In this Letter, we show that the Shore-Johnson axioms for the maximum entropy principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought. Apart from a formal side of the proof where a one-parameter class of admissible entropies is identified, we substantiate our point by analyzing the effect of weak correlations and by discussing two pertinent examples: two-qubit quantum system and transverse-momentum behavior of hadrons in high-energy proton-proton collisions.