# Charles Joel Stone

## Professor of Statistics, Emeritus

Charles (Chuck) Joel Stone, emeritus Professor of Statistics at the University of California, Berkeley, died on April 16, 2019, at the age of 82. He is survived by his wife, Barbara, and two sons (and their spouses and children). Chuck received many honors during his career, including a Guggenheim fellowship in 1980. He was a fellow of the Institute of Mathematical Statistics and an inaugural fellow of the American Mathematical Society (2012). He was elected to the National Academy of Sciences in 1993. Among other honors, he gave the prestigious 1994 IMS Wald Lectures. According to the Mathematics Genealogy Project, Chuck had 14 Ph.D. students and 166 academic descendants.

Chuck graduated from North Hollywood High School in Los Angeles and was an undergraduate at the California Institute of Technology. He received his Ph.D. from Stanford University’s Department of Mathematics in 1961 under the supervision of Samuel Karlin. Chuck’s first academic appointment began in 1962 at the Department of Mathematics at Cornell University (N.Y.). He left for UC Los Angeles (UCLA), in 1964, first as a visitor and then was appointed to the faculty in the Department of Mathematics. He remained at UCLA for 17 years until he departed for the Department of Statistics at UC Berkeley. The culmination of that period was perhaps the book with his frequent collaborator Sidney Port, *Brownian Motion and Classical Potential Theory* (Academic Press, 1978). Three fundamental problems of electrostatics (and, more generally, potential theory) are already distinguishable in the work of Carl Friedrich Gauss in 1840: the Dirichlet-Poisson problem, the equilibrium problem, and the balayage problem. It was realized over a century later that there is an intimate connection between all three topics and the properties of Brownian motion. As the book’s review by Frank Knight observes, “The book under review is a straightforward presentation of classical potential theory making full use of the connection with Brownian motion. As far as probability is concerned, the watchword seems to be economy of means. By making skilled use of symmetry, the strong-Feller property, and continuity of path, much of the general methodology of probabilistic potential theory is neatly avoided. What has long been the treasured material of a few experts is thus at last made available to anyone with a course in modern analysis.” Chuck and Sidney Port also co-authored a much-celebrated trilogy of undergraduate books on probability and statistics with Paul Hoel.

In addition to further research into potential theory, other notable work of this period included investigations of local limit theorems, weak convergence of stochastic processes, and renewal theory. Major papers were “Infinitely divisible processes and their potential theory”. *Ann. Inst. Fourier (Grenoble)* **21** (1971), no. 2, 157–275; ibid. **21** (1971), no. 4, 179–265; “On local and ratio limit theorems”. 1967 *Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2* pp. 217–224 Univ. California Press; “Weak convergence of stochastic processes defined on semi-infinite time intervals”. *Proc. Amer. Math. Soc.* **14** (1963), 694–696; and “On characteristic functions and renewal theory”. *Trans. Amer. Math. Soc.* **120** (1965), 327–342.

It was during Chuck’s time at UCLA that his interests migrated largely to statistics (although a formal Department of Statistics at UCLA was not founded until long after Chuck left). A particular interest was nonparametric statistics, that is, statistics devoid of the usual normal (Gaussian) assumptions. He authored a much-cited discussion paper in *The Annals of Statistics* in 1978, titled “Consistent Nonparametric Regression.” This paper grew out of Chuck’s wanting to put the popular nearest-neighbor technology on a sound theoretical footing.

Chuck’s interest in statistics covered many areas. For example, in the early 1980s he wrote two landmark papers regarding optimal rates of convergence for statistical estimators. His results carefully took into account dependence on the dimensions of spaces in which predictors and outcomes lie, and the derivative being estimated.

A number of Chuck’s later efforts concerned log-splines and their applications to regression (including time series) and survival analysis. Many of the papers in this long series were co-authored. Among the co-authors are former students Charles Kooperberg (now of the University of Washington), Mark Hansen (of UC Davis), and Young Truong (of the University of North Carolina). A summary of that line of inquiry was the subject of Chuck’s Wald Lectures.

During his years in Los Angeles, Chuck consulted for Technology Services Corporation in Santa Monica, along with Leo Breiman, a UCLA colleague, who also later became a UC Berkeley Professor of Statistics. Based on this work, Chuck and Leo co-authored a 1978 technical report, *Parsimonious Binary Classification Trees,* which has since become something of a cult classic. The technical report was published in greatly expanded 1984 book, titled *Classification and Regression Trees*, with Breiman and two other co-authors, Jerome Friedman and Richard Olshen (both of Stanford). This book may be the single item for which Chuck is most remembered. Its algorithms are for “classification,” “probability class estimation,” and “regression.” They, and perhaps especially a computer program for their implementation, became known as CART. One of Chuck’s principal contributions to the technical report and to the book was “CART pruning,” an intricate scheme for validating the algorithms and enabling them to be computationally feasible with the computers widely available at the time of publication. The use of the graphs associated with mean-square error as it varies with complexity is now standard in many areas. The CART book was the first mathematically and computationally rigorous treatment of approaches which now are commonplace and have found wide application. It has many examples, real and contrived. Some of these examples have become benchmarks for subsequent technologies in the field of “machine learning”. The CART ideas are part of almost every serious statistical curriculum worldwide. There are numerous computer programs, available both freely and commercially, that incorporate extensions of CART.

Chuck’s true academic devotion was to his students. He was a committed and much-loved teacher. His philosophy of statistics and many mathematical details are summarized in a single-authored book, titled *An Introduction to Probability and Mathematical Statistics,* that was first published in 2000. Many former students have attested to Chuck’s remarkable counseling during extensive office hours, his intense teaching style, and the care he took to help students whenever possible. While all of us teach, and many of us teach well, we think it fair to say that hardly anyone took the time and expressed more concern for students than did Chuck.

Richard A. Olshen

Peter Bickel

Steven N. Evans

2020