A Crucial Parameter for Rank-Frequency Relation in Natural Languages

$f\propto r^{-\alpha }\cdot (r+\gamma {)}^{-\beta }$ has been empirically shown more precise than a naïve power law $f\propto r^{-\alpha }$ to model the rank-frequency ($r$-$f$) relation of words in natural languages. This work shows that the only crucial parameter in the formulation is $\gamma$, which depicts the resistance to vocabulary growth on a corpus. A method of parameter estimation by searching an optimal $\gamma$ is proposed, where a ``zeroth word’’ is introduced technically for the calculation. The formulation and parameters are further discussed with several case studies.