Statistical Science

A Conversation with James Hannan

Dennis Gilliland and R. V. Ramamoorthi

Full-text: Open access

Abstract

Jim Hannan is a professor who has lived an interesting life and one whose fundamental research in repeated games was not fully appreciated until late in his career. During his service as a meteorologist in the Army in World War II, Jim played poker and made weather forecasts. It is curious that his later research included strategies for repeated play that apply to selecting the best forecaster.

James Hannan was born in Holyoke, Massachusetts on September 14, 1922. He attended St. Jerome’s High School and in January 1943 received the Ph.B. from St. Michael’s College in Colchester, Vermont. Jim enlisted in the US Army Air Force to train and serve as a meteorologist. This took him to army airbases in China by the close of the war. Following discharge from the army, Jim studied mathematics at Harvard and graduated with the M.S. in June 1947. To prepare for doctoral work in statistics at the University of North Carolina that fall, Jim went to the University of Michigan in the summer of 1947. The routine admissions’ physical revealed a spot on the lung and the possibility of tuberculosis. This caused Jim to stay at Ann Arbor through the fall of 1947 and then at a Veterans Administration Hospital in Framingham, Massachusetts to have his condition followed more closely. He was discharged from the hospital in the spring and started his study at Chapel Hill in the fall of 1948. There he began research in compound decision theory under Herbert Robbins. Feeling the need for teaching experience, Jim left Chapel Hill after two years and short of thesis to take a three year appointment as an instructor at Catholic University in Washington, DC. When told that renewal was not coming, Jim felt pressure to finish his degree. His 1953 UNC thesis contains results in compound decision theory, a density central limit theorem for the generalized binomial and exact and asymptotic distributions associated with a Kolmogorov statistic. He was encouraged to apply to the Department of Mathematics at Michigan State University and came as assistant professor in the fall of 1953. In the next few years, he accomplished his work on repeated games. The significance of the work was rediscovered by the on-line learning communities in computer science in the 1990s and the term Hannan consistency was coined. His retirement came in 2002 after a long career that included major contributions to compound and empirical Bayes decision theory and other areas. He and his colleague Václav Fabian co-authored Introduction to Probability and Mathematical Statistics (Wiley 1985).

A Hannan strategy is a strategy for the repeated play of a game that at each stage i plays a smoothed version of a component Bayes rule versus the empirical distribution Gi-1 of opponent’s past plays. [Play against the unsmoothed version is often called (one-sided) fictitious play.] As in compound decision theory, performance is measured in terms of modified regret, that is, excess of average risk across stages i=1, …, n over the component game Bayes envelope R evaluated at Gn. Hannan, James F., Approximation to Bayes Risk in Repeated Play, Contributions to the Theory of Games 3 97–139, Princeton University Press, is a paper rich with bounds on modified regrets. A Hannan consistent strategy is one where limsup (modified regret) is not greater than zero. In the 1990s, greater recognition of Hannan’s work began to emerge; the term Hannan consistency may have first appeared in Hart and Mas-Colell [J. Econom. Theory 98 (2001) 26–54].

Early on, only his students and a few others were aware of the specifics of his findings. The failure of others to recognize the specific results in the 1957 paper may be due to the cryptic writing style and notation of the author. The strategy for selecting forecasters in Foster and Vohra [8] [Operations Research 41 (1993) 704–709] is an unrecognized Hannan-strategy as is the strategy in Feder et al. [7] [IEEE Trans. Inform. Theory 38 (1992) 1258–1270]. Gina Kolata’s New York Times article, “Pity the Scientist who Discovers the Discovered” (February 5, 2006) uses the original Hannan discoveries as an example, although referring to him as a “statistician named James Hanna.”

In May 1998, the Department of Statistics and Probability hosted a Research Meeting in Mathematical Statistics in Honor of Professor James Hannan. Many came to honor Jim; the speakers included Václav Fabian, Stephen Vardeman, Suman Majumdar, Richard Dudley, Yoav Freund, Dean Foster, Rafail Khasminskii, Herman Chernoff, Michael Woodroofe, Somnath Datta, Anton Schick and Valentin Petrov.

Jim was ever generous in giving help to students. He enjoyed improving results and was very reluctant to submit research results until much effort was made to improve them. Jim directed or co-directed the doctoral research of twenty students: William Harkness (1958), Shashikala Sukatme (1960), John Van Ryzin (1964), Dennis Gilliland (1966), David Macky (1966), Richard Fox (1968), Allen Oaten (1969), Jin Huang (1970), Vyagherswarudu Susarla (1970), Benito Yu (1971), Radhey Singh (1974), Yoshiko Nogami (1975), Stephen Vardeman (1975), Somnath Datta (1988), Jagadish Gogate (1989), Chitra Gunawardena (1989), Mostafa Mashayekhi (1990), Suman Majumdar (1992), Jin Zhu (1992) and Zhihui Liu (1997). Most pursued academic careers and some ended up at research universities including Pennsylvania State, Columbia, Michigan State, UC-Santa Barbara, Guelph, SUNY-Binghamton, Iowa State, Louisville, Nebraska-Lincoln and Connecticut-Stamford.

It was in the Army in 1944 that Jim read his first statistics book. It was War Department Education Manual EM 327, An Introduction to Statistical Analysis, by C. H. Richardson, Professor of Mathematics, Bucknell University, published by United States Armed Forces Institute, Madison, Wisconsin (CQ).

Article information

Source
Statist. Sci. Volume 25, Number 1 (2010), 126-144.

Dates
First available in Project Euclid: 3 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.ss/1280841737

Digital Object Identifier
doi:10.1214/09-STS283

Mathematical Reviews number (MathSciNet)
MR2741818

Zentralblatt MATH identifier
1328.01056

Keywords
Hannan consistency repeated games compound decision theory empirical Bayes

Citation

Gilliland, Dennis; Ramamoorthi, R. V. A Conversation with James Hannan. Statist. Sci. 25 (2010), no. 1, 126--144. doi:10.1214/09-STS283. https://projecteuclid.org/euclid.ss/1280841737.


Export citation

References

  • [1] Blackwell, D. (1956). An analog of the minimax theorem for vector payoffs. Pacific J. Math. 6 1–8.
  • [2] Blackwell, D. (1956). Controlled random walks. In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. III 336–338. Noordhoff, Groningen.
  • [3] Datta, S. (1991). Asymptotic optimality of Bayes compound estimators in compact exponential families. Ann. Statist. 19 354–365.
  • [4] Datta, S. (1991). On the consistency of posterior mixtures and its applications. Ann. Statist. 19 338–353.
  • [5] Efron, B. and Morris, C. (1972). Limiting the risk of Bayes and empirical Bayes estimators. II. The empirical Bayes case. J. Amer. Statist. Assoc. 67 130–139.
  • [6] Fabian, V. and Hannan, J. (1985). Introduction to Probability and Mathematical Statistics. Wiley, New York.
  • [7] Feder, M., Merhav, N. and Gutman, M. (1992). Universal prediction of individual sequences. IEEE Trans. Inform. Theory 38 1258–1270.
  • [8] Foster, D. P. and Vohra, R. V. (1993). A randomization rule for selecting forecasters. Operations Research 41 704–709.
  • [9] Gilliland, D. C. and Hannan, J. (1986). The finite state compound decision problem, equivariance and restricted risk components. In Adaptive Statistical Procedures and Related Topics 129–145. IMS, Hayward, CA.
  • [10] Gilliland, D. C., Hannan, J. and Huang, J. S. (1976). Asymptotic solutions to the two state component compound decision problem, Bayes versus diffuse priors on proportions. Ann. Statist. 4 1101–1112.
  • [11] Gilliland, D. C. and Hannan, J. F. (1969). On an extended compound decision problem. Ann. Math. Statist. 40 1536–1541.
  • [12] Hannan, J. (1960). Consistency of maximum likelihood estimation of discrete distributions. In Contributions to Probability and Statistics 249–257. Stanford Univ. Press, Stanford, CA.
  • [13] Hannan, J. F. and Van Ryzin, J. R. (1965). Rate of convergence in the compound decision problem for two completely specified distributions. Ann. Math. Statist. 36 1743–1752.
  • [14] Hannan, J. (1956). The dynamic statistical decision problem when the component involves a finite number, m, of distributions (abstract). Ann. Math. Statist. 27 212.
  • [15] Hannan, J. (1957). Approximation to Bayes risk in repeated play. In Contributions to the Theory of Games, vol. 3. Annals of Mathematics Studies 39 97–139. Princeton Univ. Press, Princeton, NJ.
  • [16] Hannan, J. and Huang, J. S. (1972). A stability of symmetrization of product measures with few distinct factors. Ann. Math. Statist. 43 308–319.
  • [17] Hannan, J. F. and Robbins, H. (1955). Asymptotic solutions of the compound decision problem for two completely specified distributions. Ann. Math. Statist. 26 37–51.
  • [18] Hannan, J. F. (1953). Asymptotic solutions of compound decision problems. Ph.D. thesis, Univ. North Carolina.
  • [19] Hart, S. and Mas-Colell, A. (2001). A general class of adaptive strategies. J. Econom. Theory 98 26–54.
  • [20] Hoeffding, W. (1962). Book review of Wilks, mathematical statistics. Ann. Math. Statist. 33 1467–1473.
  • [21] James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. Fourth Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley.
  • [22] Kolata, G. (2006). Pity the scientist who discovers the discovered. New York Times, Feb 5.
  • [23] Majumdar, S. (2007). Uniform L1 posterior consistency in compact Gaussian shift experiments. J. Statist. Plann. Inference 137 2102–2114.
  • [24] Majumdar, S., Gilliland, D. and Hannan, J. (1999). Bounds for robust maximum likelihood and posterior consistency in compound mixture state experiments. Statist. Probab. Lett. 41 215–227. Special issue in memory of V. Susarla.
  • [25] Mashayekhi, M. (1993). On equivariance and the compound decision problem. Ann. Statist. 21 736–745.
  • [26] Mertens, J. F., Sorin, S. and Zamir, S. (1994). Repeated games. Discussion Paper 9420-9421-9422, CORE.
  • [27] Robbins, H. (1951). Asymptotically subminimax solutions of compound statistical decision problems. In Proc. Second Berkeley Sympos. Math. Statist. Probab. 131–148. Univ. California Press, Berkeley.
  • [28] Robbins, H. (1956). An empirical Bayes approach to statistics. In Proc. Third Berkeley Sympos. Math. Statist. Probab. 1 157–163. Univ. California Press, Berkeley.
  • [29] Robinson, J. (1951). An iterative method of solving a game. Ann. of Math. (2) 54 296–301.
  • [30] Rustichini, A. (1999). Minimizing regret: The general case. Games Econom. Behav. 29 224–243. Learning in games: A symposium in honor of David Blackwell.
  • [31] Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proc. Third Berkeley Sympos. Math. Statist. Probab. 197–206. Univ. California Press, Berkeley.
  • [32] Van Ryzin, J. (1966). The sequential compound decision problem with m×n finite loss matrix. Ann. Math. Statist. 37 954–975.
  • [33] Van Ryzin, J. R. (1966). The compound decision problem with m×n finite loss matrix. Ann. Math. Statist. 37 412–424.
  • [34] Vardeman, S. B. (1975). O(N1/2) convergence in the finite state restricted risk compnent sequence compound decision problems. Ph.D. thesis, Michigan State Univ.
  • [35] Vardeman, S. B. (1982). Approximation to minimum k-extended Bayes risk in sequences of finite state decision problems and games. Bull. Inst. Math. Acad. Sinica 10 35–52.
  • [36] Wilks, S. S. (1962). Mathematical Statistics. Wiley, New York.
  • [37] Young, H. P. (2004). Strategic Learning and Its Limits. Oxford Univ. Press.
  • [38] Zhang, C.-H. (2003). Compound decision theory and empirical Bayes methods. Ann. Statist. 31 379–390. Dedicated to the memory of Herbert E. Robbins.