In 2013, Mark Pagel, Quentin D. Atkinson, Andreea S. Calude, and Andrew Meade published statistical evidence that attempts to overcome these objections. According to their earlier work, most words exhibit a "half-life" of between 2,000 and 4,000 years, consistent with existing theories of linguistic replacement. However, they also identified some words – numerals, pronouns, and certain adverbs – that exhibit a much slower rate of replacement with half-lives of 10,000 to 20,000 or more years. Drawing from research in a diverse group of modern languages, the authors were able to show the same slow replacement rates for key words regardless of current pronunciation. They conclude that a stable core of largely unchanging words is a common feature of all human discourse, and model replacement as inversely proportional to usage frequency.