Statistical Implications Of Turing'S Formula

Turing's formula is, perhaps, the only known method for estimating beyond the range of known data without making any parametric or semi-parametric assumptions. This book presents a unified and broad presentation of Turing’s formula and its connections to statistics. Topics with applications in a variety of different fields of study are included such as information theory; statistics; probability; computer science inclusive of artificial intelligence, machine learning; big data; biology; ecology; and genetics. The author provides an accessible and clear introduction to Turing's formula and features examinations of many core statistical issues within modern data science from Turing's perspective. Recently, a clear and systematic description of Turing's formula and its statistical properties and implications has evolved, and the newly found statistical implications have brought about substantial gains for some of the most important areas of modern data science. A systematic approach to long-standing problems, such as entropy and mutual information estimation, diversity index estimation, domains of attraction on general alphabets, and tail probability estimation are discussed in light of the most up-to-date understanding of Turing's formula. Featuring numerous exercises and examples throughout, the author presents a summary of the known properties of Turing's formula and explains how and when it works well; discusses the approach that led to Turing's formula in order to estimate a variety of quantities, all of which mainly come from information theory, but are also important for machine learning and for ecological applications; and uses Turing's formula to estimate the tail index of certain heavy tailed distributions.

Zhiyi Zhang, PhD, is Professor of Mathematics and Statistics at The University of North Carolina at Charlotte. He is an active consultant in both industry and government on a wide range of statistical issues, and his current research interests include Turing's formula and its statistical implications; probability and statistics on countable alphabets; nonparametric estimation of entropy and mutual information; tail probability and biodiversity indices; and applications involving extracting statistical information from low-frequency data space. He earned his PhD in Statistics from Rutgers University.