Approximation of the truncated Zeta distribution and Zipf's law
This provides a practical tool for researchers and practitioners in fields like linguistics and data science who rely on Zipf's law, though it is incremental as it builds on existing integral methods.
The authors tackled the lack of a closed-form expression for Zipf's law by proposing three approximations for the truncated Zeta distribution, with the trapezoidal approximation achieving errors as low as 0.1% for certain parameter ranges.
Zipf's law appears in many application areas but does not have a closed form expression, which may make its use cumbersome. Since it coincides with the truncated version of the Zeta distribution, in this paper we propose three approximate closed form expressions for the truncated Zeta distribution, which may be employed for Zipf's law as well. The three approximations are based on the replacement of the sum occurring in Zipf's law with an integral, and are named respectively the integral approximation, the average integral approximation, and the trapezoidal approximation. While the first one is shown to be of little use, the trapezoidal approximation exhibits an error which is typically lower than 1\%, but is as low as 0.1\% for the range of values of the Zipf parameter below 1.