After graduating, Chernoff had been intending to study mathematics at graduate level but he received a letter asking him to contribute to the war effort by working as a junior physicist for the U.S. Navy. After eighteen months working at the Dahlgren Proving Ground on electronic equipment, he entered Brown University to undertake research in applied mathematics. He took courses by Bers, Feller, Loewner, Tamarkin, and others, and wrote a Master's dissertation Complex Solutions of Partial Differential Equations under Bergman's supervision. This work became his first mathematics paper when published by the American Journal of Mathematics in 1946. Before he could continue with his research, however, he was drafted into the Army shortly after the end of the war :-
... after three days basic training, [I] operated as a clerk in the night shift at the separation centre for returning army men. It was in a way the first vacation I had had in a long time, because I worked about six hours in the evening and had the whole day free. On the weekends, I rode to Brown University and I was actually taking a couple of reading courses while I was there.This worked well, but then he was sent for real basic training and told he was going to be assigned a post abroad. By this time he had been awarded an NSF predoctoral fellowship so he requested a discharge from the Army. This was granted and, after just over three months in the Army, he was able to return to Brown University.
By this time Chernoff had decided that his talents did not lay in applied mathematics but, having seen more statistical applications during his military service, he decided to change topics. After a few more months at Brown he went to North Carolina to take a summer course in statistics held in Raleigh with lectures given by R A Fisher, William Cochran and J Wolfowitz. Wald was at North Carolina and Chernoff prepared by reading papers by Wald on sequential analysis. He then asked Wald if he would supervise his Ph.D. This was a somewhat strange request since he was a registered student at Brown University and would submit his thesis there. After some hesitation, Wald agreed and so Chernoff's doctoral thesis Asymptotic Studentization in Testing of Hypotheses was supervised by Wald although his official supervisor at Brown University was shown to be James Krumhansl and the degree was awarded by Brown. Chernoff was awarded the degree in 1948 and part of his thesis was published as Asymptotic Studentization in testing of hypotheses in 1949. E L Lehmann writes:-
Methods are developed for obtaining "asymptotically similar" tests, that is, tests the size of which is constant except for an error term of prescribed order in the number of observations.This was in fact Chernoff's third paper since he published A note on the inversion of power series in 1947 which:-
... [treats] the multiplication of power series and their inversion by means of a movable strip of paper on which the coefficients are written.While studying at Brown University, Chernoff met Judy Ullman and they married in September 1947; they had two daughters Ellen and Miriam. After completing the work for his doctorate Chernoff and his wife moved to Chicago where he had an appointment as Research Instructor at the University of Chicago with the Cowles Commission for Research in Economics.
In 1949 Chernoff became an Assistant Professor of Mathematics at the University of Illinois at Urbana. He was promoted to associate Professor of Mathematics there in 1952. He said :-
I liked the University of Illinois very much. being a provincial from the Bronx, the small town college life at Illinois appealed to my personality.Although on the faculty at Illinois until 1952, Chernoff went to Stanford University as a visitor from June 1951 until January 1952, then later that year accepted the position of Associate Professor of Statistics there. He was promoted at Professor of Statistics at Stanford in 1956 and remained there until 1974 :-
I did not care for my introduction to California that well; the social life did not seem to be quite as nice. The intellectual environment in the Statistics department at Stanford, however, did appeal to me.In 1959 Chernoff, in collaboration with Lincoln E Moses, published a classic text Elementary decision theory. D V Lindley provides this summary:-
This is a book about decision making, under uncertainty about the state of nature, for a student with only a background of U.S. high school mathematics. The first six chapters deal with the processing of data (graphical methods, means and variances), probability and random variables, the concept of utility (treated axiomatically following von Neumann) and the comparison of various strategies (Bayes, minimax etc.). The remaining four chapters provide an introduction to classical statistics (the adjective is the authors') and discuss point and interval estimation and tests of hypotheses. There are numerous appendixes in which side-issues or more difficult points are discussed, and a good set of the usual tables which appear at the end of most statistical text books.In 1972 Chernoff published a monograph which incorporated much of his work over many years. This monograph was entitled Sequential analysis and optimal design and is described by P Whittle:-
This 120 page monograph takes the author's own researches as a thread which is unifying rather than dominant. The material of 76 references is knitted into a self-contained and attractive account of the sequential and design-optimal aspects of experimentation. A brief summary of the contents follows. After some elementary preliminaries, Elfving's geometrical treatment of c-optimal designs is introduced and, after some discussion of the large-sample properties of maximum likelihood estimators, the author's own treatment of locally optimal designs for estimation. A list of other topics treated follows: D-optimality and the Kiefer-Wolfowitz equivalence theorem; hypothesis-testing in a treatment which is largely, although not whole-heartedly, decision-theoretical; the large-sample evaluation of risk in terms of the Chernoff bounds (a term not used in the text) and the various information numbers; optimisation of sample size in the case of low-cost experimentation; the sequential probability ratio test, no-overshoot approximations, optimality; the Chernoff "procedure A" for sequential design, and its asymptotic optimality; adjacent hypotheses, and the Schwarz boundaries; testing for the sign of a normal mean, with a general consideration of dynamic programming ideas, and of helpful asymptotics; some discussion of one- and two-armed bandits.In 1974 Chernoff left Stanford University and moved to the Massachusetts Institute of Technology where he was appointed Professor of Applied Mathematics. He held this post until 1985 when he retired and was given the title Professor Emeritus. However, Chernoff did not really retire since he was appointed as Professor of Statistics at Harvard University. He continued in this role until 1997 when in addition to being Professor Emeritus of Applied Mathematics of M.I.T. he became also Professor Emeritus of Statistics of Harvard University.
The authors of  sum up Chernoff's contributions:-
Herman is known to be an inspiring teacher and has greatly influenced a large number of his doctoral students and post-doctoral research associates, who in turn have contributed vastly to the field of statistics.Chernoff has been honoured in many ways for his important contributions. For example he has received honorary doctorates from Ohio State University, The Technion in Israel, La Sapienza in Rome, and the University of Athens. He has been elected to the National Academy of Sciences, the American Academy of Arts and Sciences, and the International Statistical Institute.
Article by: J J O'Connor and E F Robertson