ZipfMandelbrot

ZipfMandelbrot refers to the Zipf–Mandelbrot law, a generalization of Zipf's law that introduces a shift in rank through a parameter q. Proposed by Benoit Mandelbrot to improve fits for word-frequency data and other rank-size phenomena, the model accommodates systematic deviations observed for the most frequent items and at higher ranks.

Mathematically, let r denote rank (r = 1 for the top item). The frequency or probability of the r-th item is proportional to (r + q)^{-s}, with s > 0 and q > -1. In a finite sample of size N, p_r = (r + q)^{-s} / sum_{n=1}^N (n + q)^{-s}. In the infinite limit, normalization uses the Hurwitz zeta function Z(s, q) = sum_{n=0}^\infty (n + q)^{-s}, giving p_r = (r + q)^{-s} / Z(s, q).

Relation to Zipf's law is direct: when q = 0, the model reduces to Zipf's law, p_r ∝ r^{-s}. The parameter s controls the tail decay, while q shifts the distribution to the right, often yielding a better fit to empirical data than Zipf's law alone, especially for the top ranks.

Applications extend beyond linguistics to city-size distributions, income distributions, bibliometrics, and web or information-traffic data. Parameter estimation is typically done via maximum likelihood or non-linear regression on observed frequencies; q and s together determine the fit and the inferred heaviness of the tail.

Limitations include potential overfitting, sensitivity to sample size and data quality, and reduced universality across domains. Nevertheless, ZipfMandelbrot remains a commonly used generalized model for rank-frequency and rank-size phenomena.