Moment-based estimation for the Zipf law

Abstract Zipf’s law appears across many disciplines, but real datasets typically provide only frequency counts, and empirical ranks derived from noisy samples can differ from the true ranks, biasing common estimators of the Zipf exponent. Approximate Bayesian computation (ABC) mitigates this bias but relies on repeated accept–reject simulation with potentially large runtime. We propose a moment-based estimator that infers the Zipf exponent from type–token statistics alone, avoiding rank matching and accept–reject sampling. The estimator matches an empirical type–token curve constructed from a single sample to the theoretical expected number of distinct types, reducing estimation to a one-dimensional optimization. In simulations spanning Zipf exponents 1.10–2.00 and sample sizes of 100, 1000, and 10,000, the mean squared error of both methods generally increases with the exponent; the moment-based estimator is most competitive at the smallest exponents, while ABC is more accurate over most of the grid. Averaged over the simulation grid, it is roughly three orders of magnitude faster than ABC in geometric mean runtime. Additional simulations reveal sensitivity to finite support, Zipf–Mandelbrot deviations, and short-range dependence.