Editing Herbert Robbins

{{short description|American mathematician}}
{{more footnotes|date=May 2022}}
{{Infobox scientist
| name              = Herbert Ellis Robbins
| image             = 1966-HerbertRobbins.jpg
| image_size        = 250px
| alt               = 
| caption           = Herbert Robbins visiting Purdue in 1966
| birth_date        = {{Birth date|1915|1|12}}
| birth_place       = [[New Castle, Pennsylvania|New Castle]], Pennsylvania, US
| death_date        = {{Death date and age|2001|2|12|1915|1|12}}
| death_place       = [[Princeton, New Jersey|Princeton]], New Jersey, US
| fields            = 
| workplaces        = {{ublist|[[University of North Carolina at Chapel Hill|University of North Carolina]]|[[Columbia University]]|[[Rutgers University]]}}
| alma_mater        = [[Harvard University]]
| doctoral_advisor  = [[Hassler Whitney]]
| doctoral_students = {{ublist|[[Raghu Raj Bahadur]]|[[Cyrus Derman]]|[[David Siegmund]]|[[Herbert Wilf]]|[[Gopinath Kallianpur]]}}
| thesis_title      = On the Classification of the Maps of a 2-Complex into a Space
| thesis_year       = 1938
| thesis_url        = https://www.jstor.org/stable/1990025
| notable_students  = 
| known_for         = 
| author_abbrev_bot = 
| author_abbrev_zoo = 
| influences        = 
| influenced        = 
| awards            = 
| signature         = <!--(filename only)-->
| signature_alt     = 
| footnotes         = 
| spouse            = 
}}
'''Herbert Ellis Robbins''' (January 12, 1915 – February 12, 2001) was an American  [[mathematician]] and [[statistician]]. He did research in [[topology]], [[measure theory]], [[statistics]], and a variety of other fields.  
 
He was the co-author, with [[Richard Courant]], of ''[[What is Mathematics?]]''.  The [[Robbins lemma]], used in [[empirical Bayes method]]s, is named after him.  [[Robbins algebra]]s are named after him because of a conjecture (since proved) that he posed concerning [[Boolean algebra (structure)|Boolean algebras]]. The [[Robbins' theorem]], in [[graph theory]], is also named after him, as is the [[Ear decomposition|Whitney–Robbins synthesis]], a tool he introduced to prove this theorem. The well-known unsolved problem of minimizing in sequential selection the expected rank of the selected item under full information, sometimes referred to as the fourth [[secretary problem]], also bears his name: [[Robbins' problem (of optimal stopping)]].

==Biography==
Robbins was born in [[New Castle, Pennsylvania|New Castle]], [[Pennsylvania]].

As an undergraduate, Robbins attended [[Harvard University]], where [[Marston Morse]] influenced him to become interested in mathematics.  Robbins received a [[doctorate]] from Harvard in 1938 under the supervision of [[Hassler Whitney]] and was an instructor at [[New York University]] from 1939 to 1941.  After [[World War II]], Robbins taught at the [[University of North Carolina at Chapel Hill]] from 1946 to 1952, where he was one of the original members of the department of mathematical statistics, then spent a year at the [[Institute for Advanced Study]].  In 1953, he became a professor of mathematical statistics at [[Columbia University]].  He retired from full-time activity at Columbia in 1985 and was then a professor at [[Rutgers University]] until his retirement in 1997.  He has  567 descendants listed at the  [http://genealogy.math.ndsu.nodak.edu/id.php?id=7781 Mathematics Genealogy Project].

In 1955, Robbins introduced [[empirical Bayes method]]s at the Third Berkeley Symposium on Mathematical Statistics and Probability.  Robbins was also one of the inventors of the first [[stochastic approximation]] algorithm, the Robbins–Monro method,  and worked on the theory of [[power-one test]]s and [[optimal stopping]]. In 1985, in the paper   "Asymptotically efficient adaptive allocation rules", with [[Tze Leung Lai|TL Lai]],  he constructed uniformly convergent population selection policies for the [[multi-armed bandit]] problem that possess the fastest rate of convergence to the population with highest mean, for the case that the population reward distributions are the one-parameter exponential family. These policies were simplified  in 
the 1995 paper "Sequential choice from several populations", with [[MN Katehakis|Michael Katehakis]].

He was a member of the [[National Academy of Sciences]] and the [[American Academy of Arts and Sciences]] and was past president of the [[Institute of Mathematical Statistics]].

==Selected writings==
;Books by Herbert Robbins
* ''What is Mathematics? An Elementary Approach to Ideas and Methods'', with [[Richard Courant]], London: Oxford University Press, 1941.
* "Great Expectations: The Theory of Optimal Stopping", with Y. S. Chow and [[David Siegmund]]  Boston: Houghton Mifflin, 1971.
* "Introduction to Statistics", with  John Van Ryzin,  Science Research Associates, 1975.

;Articles (selection)
* A theorem on graphs with an application to a problem on traffic control, ''American Mathematical Monthly'', vol. '''46''' (1939), pp.&nbsp;281–283.
* The [[central limit theorem]] for dependent [[random variable]]s, with [[Wassily Hoeffding]], ''Duke Mathematical Journal'', vol. '''15''' (1948), pp.&nbsp;773–780.
* A [[stochastic approximation]] method, with Sutton Monro, ''Annals of Mathematical Statistics'', vol. '''22''', no. 3 (September 1951), pp.&nbsp;400–407.
* Some aspects of the sequential design of experiments, in "Bulletin of the American Mathematical Society", vol. 58, 1952.
*  Two-stage procedures for estimating the difference between means, with Ghurye, SG, "Biometrika", 41(1), 146–152, 1954.
* The strong law of large numbers when the first moment does not exist, with [[Cyrus Derman|C. Derman]], in the ''Proceedings of the National Academy of Sciences of the United States of America'',  vol. 41, 1955.
* An empirical Bayes approach to statistics, in ''Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability'', Jerzy Neyman, ed., vol. 1,  Berkeley, California: University of California Press, 1956, pp.&nbsp;157–163.
* On the asymptotic theory of fixed-width sequential confidence intervals for the mean, with Chow, Y.S., "The Annals of Mathematical Statistics", 36(2), 457–462, 1965.
* Statistical methods related to the law of the iterated logarithm, "The Annals of Mathematical Statistics", 41(5), 1397–1409, 1970.
* Optimal stopping, "The American Mathematical Monthly", 77(4), 333–343, 1970.
* A convergence theorem for nonnegative almost supermartingales and some applications, with  [[David Siegmund]], "Optimizing methods in statistics", 233–257, 1971.
* Sequential tests involving two populations, with  [[David Siegmund]],  "Journal of the American Statistical Association, 132–139, 1974.
*A class of dependent random variables and their maxima, with Lai, T.L.  "Probability Theory and Related Fields", 42(2), 89–111, 1978
* Asymptotically efficient adaptive allocation rules with TL Lai, in "Advances in applied mathematics", vol. 6, 1985.
* Sequential choice from several populations with [[Michael N. Katehakis|M. N. Katehakis]], in the ''Proceedings of the National Academy of Sciences of the United States of America'',  vol. 92, 1995.

==References==
{{Reflist}}
* "The Contributions of Herbert Robbins to Mathematical Statistics", Tze Leung Lai and David Siegmund, ''Statistical Science'' '''1''', #2 (May 1986), pp.&nbsp;276–284. [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.ss/1177013708&view=body&content-type=pdf_1 Euclid]
* [http://isi.cbs.nl/NLet/memoriam01-3.htm In Memoriam], ''[[International Statistical Institute|ISI]] Newsletter'' '''25''', #3 (2001)
* [https://www.nytimes.com/2001/02/15/national/15ROBB.html '"Herbert Robbins, Statistician Who Fueled Interest in Math, Dies at 86"], ''NY Times'', Feb.15, 2001.
* "What is known about Robbins' Problem?", [[F. Thomas Bruss]], ''Journal of Applied Probability'' ''Volume'' '''42''', #1 (2005). pp.&nbsp;108&ndash;120 [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jap/1110381374 Euclid]
* "A continuous-time approach to Robbins' problem of minimizing the expected rank", [[F. Thomas Bruss]] and Yves Coamhin Swan, ''Journal of Applied Probability'', Volume 46 #1, 1&ndash;18, (2009).

==External links==
*{{MathGenealogy |id=7781}}
* {{MacTutor Biography|id= Robbins}}
* [https://findingaids.library.columbia.edu/ead/nnc-rb/ldpd_6913637/ Herbert Robbins Papers] at the Columbia University Rare Book and Manuscript Library, New York, NY
* [http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/robbins-herbert.pdf Tze Leung Lai and David Siegmund, "Herbert Robbins", Biographical Memoirs of the National Academy of Sciences (2018)]

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{{DEFAULTSORT:Robbins, Herbert}}
[[Category:1915 births]]
[[Category:2001 deaths]]
[[Category:People from New Castle, Pennsylvania]]
[[Category:20th-century American mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:Mathematics popularizers]]
[[Category:Harvard University alumni]]
[[Category:Institute for Advanced Study visiting scholars]]
[[Category:Presidents of the Institute of Mathematical Statistics]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:Columbia University faculty]]
[[Category:American mathematical statisticians]]