Papers · William Alexander Aldred

Zipf's Exponent Is an Estimator's Opinion: Three Standard Fits, One Book, Priced

by William Alexander Aldred — Independent Researcher

Publicly deposited November 1, 2026 · Zenodo · 7 pp · DOI: 10.5281/zenodo.23866354

Abstract

People quote Zipf's law as a fact about language: the exponent is about 1. But nobody reads an exponent off a book. A method produces it. This paper takes one book — Moby Dick, in the Project Gutenberg edition — counts its words once, and runs three standard ways of fitting the exponent on that one count table. The two methods that return a usable exponent do not agree: ordinary least squares on the rank–frequency curve gives 1.163, and a maximum-likelihood power-law fit gives 1.092. That gap, 0.070, is larger than the biggest formal error either method quotes for itself, 0.041. The third method, least squares on the frequency-of-frequencies curve, does something worse than disagree. Run over its full range, it returns a slope that converts to no positive exponent at all. And inside the likelihood method alone, one choice — where the fit starts — moves the answer from 1.092 to 1.408, on a fit the data plainly reject. On this book, the exponent is partly a choice. This paper prices the choice. It claims nothing about any other book.

The result, in plain language

People quote Zipf's law as a fact about language: the exponent is about 1. But nobody reads an exponent off a book. A method produces it. This paper takes one book — Moby Dick, in the Project Gutenberg edition — counts its words once, and runs three standard ways of fitting the exponent on that one count table. The two methods that return a usable exponent do not agree: ordinary least squares on the rank–frequency curve gives 1.163, and a maximum-likelihood power-law fit gives 1.092. That gap, 0.070, is larger than the biggest formal error either method quotes for itself, 0.041.

The third method, least squares on the frequency-of-frequencies curve, does something worse than disagree. Run over its full range, it returns a slope that converts to no positive exponent at all. And inside the likelihood method alone, one choice — where the fit starts — moves the answer from 1.092 to 1.408, on a fit the data plainly reject. On this book, the exponent is partly a choice. This paper prices the choice. It claims nothing about any other book.

Program note

Paper 147 of the November 1, 2026 day charge — batch 1 of the writing bench, deposited the day it was written, its grades printed on its face.

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The program