Zipf-Mandelbrot Distribution
X ~ ZM(N, q, s)
Power-Law / Rank-Frequency Family • Published July 22, 2026
Mandelbrot's plateau-correcting generalization of the already-published Zipf distribution: a single shift parameter q flattens the top-rank probabilities to fix Zipf's well-documented underfit of real word-frequency data, while leaving the power-law tail slope unchanged, verified via exact q=0 collapse to Zipf and a 2-million-draw simulation, a demonstrated weak joint identifiability between q and s that makes maximum-likelihood recovery visibly less tight than Zipf's own single-parameter fit, and simulation via the same inverse-transform construction used on the Zipf sheet.
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About This Reference Sheet
Mandelbrot's plateau-correcting generalization of the already-published Zipf distribution: a single shift parameter q flattens the top-rank probabilities to fix Zipf's well-documented underfit of real word-frequency data, while leaving the power-law tail slope unchanged, verified via exact q=0 collapse to Zipf and a 2-million-draw simulation, a demonstrated weak joint identifiability between q and s that makes maximum-likelihood recovery visibly less tight than Zipf's own single-parameter fit, and simulation via the same inverse-transform construction used on the Zipf sheet.
Support
x ∈ {1, 2, ..., N}
Parameters
N ∈ ℕ (number of ranked items), q ≥ 0 (plateau parameter), s > 0 (exponent)