Pressley Warwick Millar
Professor of Statistics, Emeritus
Warry received his A.B. from Brown University in 1961 and his Ph.D. from the University of Illinois at Urbana-Champaign in 1967. His dissertation “Martingale Integrals”, advised by Donald Burkholder, led to publications in the late 1960s on stochastic integrals with respect to martingales and processes with independent increments. He was appointed Assistant Professor of Statistics at Berkeley in 1967, became Professor in 1977, and remained on the Department faculty until his retirement in 2003. His remarkable talents for analysis and for exposition infused his research, teaching and service to the University.
In the early 1970s, Warry investigated a number of aspects of the sample paths of stochastic processes with stationary independent increments, including how they exit from an interval, how their excursions away from a point are initiated, and how they behave at instants of local minima. Around that time David Williams obtained results on path decompositions of Brownian motion and one-dimensional diffusion processes. These stimulated much interest in the probability community about how the path of a more general Markov process could be decomposed into components before and after suitable random times, such as the time some function of the process attains its minimum over the lifetime of the process.
Warry's survey paper “Random times and decomposition theorems” Probability Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill. (1976), 91–103 provided a scholarly review of results on this problem which were known at that time. His paper “Germ sigma fields and the natural state space of a Markov process” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 39 (1977), no. 2, 85–101 then provided a definitive account of the behavior of a process with stationary independent increments around suitable random times. In “A path decomposition for Markov processes” Ann. Probability 6 (1978), no. 2, 345–348 he established a very general path decomposition for a right-continuous strong Markov process, at the time at which a function of its current value and its left limit achieves its last ultimate minimum.
Warry’s research included three papers with his then Department colleague Rudolf Beran which lie at the intersection of mathematical statistics, probability theory, and algorithmic feasibility. The initial paper “Confidence Sets for a Multivariate Distribution” Ann. Statist. 14 (1986), 431-443 developed confidence sets for an unknown multivariate distribution characterized by all of its half-space probabilities. Up to then the half-space generalization of the Kolmogorov-Smirnov statistic, introduced in 1954 by Wolfowitz, had lacked applications because of daunting technical obstacles. By the 1980s empirical process theory determined its asymptotic distribution implicitly. Nonparametric bootstrap theory provided the required confidence set radius in an explicit though still analytically intractable form. Finally, Monte Carlo approximations to both the half-space statistic and its theoretical bootstrap distribution made computationally feasible the desired confidence sets. Immediate consequences were elegant “slice” confidence sets for unknown distributions supported on manifolds, such as spheres, that are embedded within a Euclidean space.
Second, the paper “Stochastic Estimation and Testing” Ann. Statist. 15 (1987), 1131-1154 justified randomized approximations to complex statistics and their theoretical bootstrap distributions. These made available a range of previously inaccessible statistical procedures. Third was the paper “Minimum Distance Estimation in Random Coefficient Regression Models” Ann. Statist. 22 (1994), 1976-1992 which developed nonparametric estimation of the unknown coefficient distributions in a random coefficient linear regression model. The problem arises in econometric modeling, generalizes fitting of statistical mixture models, and is mathematically linked to reconstructing medical images from noisy tomographic data. The minimum distance approach converted inversion of a noisy Radon transform to minimization with multiple local minima.
During his career at Berkeley, Warry had 15 Ph.D. students, many of whom went on to distinguished academic careers. Those students include Richard Bass (1977) now Emeritus at University of Connecticut, Carl Mueller (1979) now at University of Rochester, Davar Khoshnevisan (1989) now at University of Utah, Maria Eulália Vares (1980) now at Universidade Federal do Rio de Janeiro, Brazil, and Guenther Walther (1994) now at Stanford.
In the 1970s Warry developed a number of outstanding graduate courses on advanced topics in probability and stochastic processes, including Markov processes and Gaussian processes. Later he developed a course about probability and inference in Banach spaces, a very abstract niche area at the time which has now become important for functional data analysis. His clear, beautifully structured, classroom lectures seemingly relied on a few words written on a scrap of paper. The audience treasured their notes.
His undergraduate teaching and mentoring were likewise lucid, generous and effective. While serving throughout his career on the major Department Committees and in 1976-78, 1986-87, 1988-89 as the Statistics Department Vice-Chair, his ability to analyze and express in writing the essence of complex matters, notably faculty advancement cases, was repeatedly of great value.
Warry's lifelong interests outside academia included raising his family, fine furniture carpentry, running, culinary arts, summers at his cabin in New Hampshire, modern dance, and ballet. Upon retiring from Berkeley he withdrew to fully engaged private life, otium privatum.
—Rudolf Beran, UC Davis, and James Pitman, UC Berkeley
