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American Journal of Mathematical and ManagementSciences

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75 Years of Herbert Robbins: The Professional andPersonal Sides

Z. Govindarajulu

To cite this article: Z. Govindarajulu (1991) 75 Years of Herbert Robbins: The Professional andPersonal Sides, American Journal of Mathematical and Management Sciences, 11:1-2, 5-24,DOI: 10.1080/01966324.1991.10737294

To link to this article: http://dx.doi.org/10.1080/01966324.1991.10737294

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AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES CopyrlghtC 1991 by Amerlc.n Sciences Pr ... , Inc.

75 YEARS OF HERBERT ROBBINS: THE PROFESSIONAL AND PERSONAL SIDES

Z. Govindarajulu University of Kentucky Lexington, KY 40506

Herbert Robbins has held many honors, awards, and distinctions, including two Guggenheim Fellowships, the presidency of the Institute of Mathematical Statistics, and membership in the National Academy of Sciences . . Still active in statistical research, a selection of his statistical papers has been published, edited by his former colleagues T.L. Lai and D.O. Siegmund (1985). A conference honoring Robbins was held in 1985 at Brookhaven National Laboratory with the proceedings edited by Van Ryzin (1987). We are familiar with his joint work with Courant (1941), 'What is Mathematics?', and his innovative contributions to stochastic approximation, compound decision theory, empirical Bayes (the latter two were hailed by Jerzy Neyman (1962) as "break­throughs"), sequential adaptive estimation, tests of power one, optimal stopping, and sequential methods for comparative clinical trials form an impressive part of the whole of statistics. His former students include R.R. Bahadur, Gopinath Kallianpur, Sudhish Ghurye, James Hannan, Vernon Johns, David Siegmund, Ester Samuel and many others of distinction.

This article seeks to shed some light on the human and philosophical side of Robbins who has been an inspiration to many, and especially to me in my own research; a major part of my 1987 book Sequential Statistical Analysis is devoted to his ideas. The remainder of this article is based on a conversation I had with him in 1988 in the living room of his house near Princeton; Q stands for questions posed by me, and A for answers by Robbins.

Q: For a special volume of the American Journal of Mathematical and Management Sciences, I am writing an appreciation of Herbert Robbins. Your early education as a

1991, VOL. 11, NOS. 1 & 2, 005-024 0196-6324/91/010005-20 $25.00

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mathematician, your association with Courant, the Robbins-Monro process, empirical Bayes, compound decision theory, power one tests, adaptive sequential estimation, and optimal stopping problems were covered in the paper by Lai and Siegmund (1986). So, I feel I can leave those aspects in the background and concentrate on the human and phi­losophical side, which I hope you will share with me. Would you care to comment on this approach?

A: The trouble is that scientific personalities tend to be rather dull, with a few excep­tions like Richard Feynman and James Watson.

Q: You were trained as a mathematician, and your first publications were in mathemat­ics. How did you get interested in statistics? Was it Harold Hotelling's influence?

A: He called my attention to the fact that little as I knew about mathematics, most sta­tisticians in this country at that time knew even less, so I might have some success in statistics. I took him up on that suggestion and started as a statistician in 1946 in Chapel Hill.

Q: What had been your area of specialization in mathematics? Was it analysis? With whom did you work in mathematics? Was it Courant?

A: Originally I worked in topology. I never worked with Courant except in writing that book. I never studied with him.

Q: Then who influenced your mathematical education?

A: There were a number of mathematicians who were very important to me in my early years. Marston Morse made a big impression on me, although I never took a course with him.

Q: Was he the famous topologist?

A: His field was the calculus of variations. He created the Morse theory, one of the great achievements of American mathematics.

Q: Was he at Harvard?

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A: He was at Harvard during my first two years there, but he left to go to the Institute for Advanced Study while I was still an undergraduate. I took courses with Marshall Stone, George D. Birkhoff, and Hassler Whitney, and wrote my thesis under Whitney's direction. But I didn't really study with any particular mathematician. I didn't know what I wanted to do. I was not very strong in any particular field of mathematics.

Q: Harvard was well-known for mathematics in those days.

A: Princeton, I think, was the leading mathematical center, although Harvard was not far behind. A great change in Harvard and in the American mathematical world occurred during the forties. Many outstanding European mathematicians came to this country, and the character of the Harvard mathematics department changed from being somewhat provincial to being quite cosmopolitan, as the Russians say. The social and intellectual climate at Harvard during the thirties was such that I did not have a very good time there.

Q: How did you happen to write What is Mathematics? Was it your first publication?

A: It was the first major one after my thesis.

Q: What was your thesis on?

A: It was on the homotopy classification of mappings. My first publication was a little note which answered a question raised by G.D. Birkhoff. I got involved in writing the book with Courant because he had some lecture notes for a course on mathematics for the general public that he wanted to expand into a book. He was a very busy man, and he was looking for someone who could take these lecture notes and make a book out of them. I was at the Institute at the time, and he asked Morse to recommend someone for that job, and Morse recommended me. So I got an offer from New York University, at Courant's urging, with the idea that I would help him with the book. What happened, roughly speaking, was that I took his lecture notes and rewrote them and added a lot of other material. The initial conception was Courant's, and at the beginning my main rea­son for working on the book was to make a little extra money. I was supporting my mother and young sister.

Q: Did you join New York University as a faculty member? Could you perhaps give some dates?

A: I got my Ph.D. in '38 and I spent the next year at the Institute for Advanced Study,

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then the next three years at New York University as an instructor, after which I was in the Navy for four years.

Q: How many years of work were involved in the book-writing project?

A: Oh, the better part of 21f2 years. I was 24 when I started working on it. At that time one could still be a general mathematician and be pretty much aware of what was going on in analysis, algebra, topology, and logic. I doubt if that is possible now.

Q: Is it because mathematics has become such a broad field?

A: Well, there were at that time maybe 2,000 members of the American Mathematical Society, and of those maybe 200 actually did any research. Now, I guess, there must be more than ten times that many. Do you have any idea what the membership of the American Mathematical Society is now?

Q: I'm sure it's between 20,000 to 30,000. Now, I want to know how you got

interested in statistics.

A: The first exposure I had to statistics was as a student at Harvard. The department of mathematics there did not concern itself with probability theory, but there was one member of the department, E.V. Huntington, who gave a course on statistics. I went to the first lecture of his course, but I didn't go back to any subsequent lectures. As I recall, he started out by saying something like, ''Statistics is based on the theory of pro­bability, and I have to admit to you that I don't understand what probability is. So I'm not going to usc that concept. Instead we will discuss frequency distributions", and so on. So I didn't take any more of that course. Later on I realized that I didn't understand what probability was either, and I still don't. So I sympathize with Professor Hunting­ton, and wish that I had taken that course, because then I would know a lot more statis­tics than I do. Anyway, that was my first little brush with statistics. The second was when I was at N.Y.U. Willy Feller was supposed to come as a visiting professor and give a course in probability and statistics, but at the last minute he couldn't. So they said "We're advertising this course in probability and statistics. Is there someone who is willing to teach the course so that students can sign up for it?" And there was no one on the faculty who cared about statistics. So they said, "Okay, let's give it to the youngest and most defenseless member of the department. Robbins, this is your job." I'm sure this happened many, many times in America; there was a demand for statistics that nobody could do anything about, so the mathematics department would appoint its youngest and least experienced member to become the expert in statistics. Anyway, I had to prepare to give that course. And I really knew absolutely nothing about statistics.

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It would be embarrassing if I had to tell you what I taught. Anyway, I learned a little about probability and statistics by giving that course. There were no good statistics text­books that I could find.

Q: What text did you use?

A: About the only book on statistics in the N.Y.U. library was by Arne Fisher (not R.A. Fisher). I remember that somebody told me "You'd better look at Fisher's book." So I went to the library and there was Arne Fisher's book. At that time (1940) Uspensky's book on probability had just come out. Anyway, things were in a rather primitive state, and they were not improved by my course, I must say. I had to cover, I think, the whole of probability and statistics in one semester. My real entry into statistics came during World War II. I was in the Navy, and I was not a researcher. I wasn't working on any mathematical problems. But I happened to overhear the conversation of some people concerned with bombing airstrips and military installations. They were having trouble figuring out how much bombing was needed to put an airstrip out of commission. You might drop lots of bombs, but they overlap, and the total area covered is not the sum of the areas of the individual bombs. They wanted to know what the effect of the overlap­ping was so as to determine the number of bombs to be dropped in order to have a high probability of knocking out say 75% of the target. They had not found anybody who could solve that problem mathematically, so they had taken the expedient of simply dropping poker chips on the floor, photographing the patterns, and then measuring the total area covered by the overlapping poker chips. It was what we would now call a simulation or Monte Carlo study, but it was physical Monte Carlo, not mathematical Monte Carlo. Anyway, this got me to thinking, so I wrote a little note and submitted it for publication in the Annals of Mathematical Statistics It came to the attention of Harold Hotelling, and that started my career as a statistician.

Q: Was Hotelling at Columbia at that time?

A: He was there but was about to go to Chapel Hill to start a department of statistics. When I got out of the Navy after the war I did not want to go back to teaching mathematics. So I used some money that had accumulated during my military service to buy a little abandoned farm in Vermont. I lived there for a while and tried to figure out what I wanted to do. While I was up there I got a message from Harold Hotelling stating that he was going to start a department of statistics at Chapel Hill, and asking if I would come there as a mathematician. The mathematics department there was not all that great, in his opinion, and he needed somebody to teach measure theory and proba­bility to graduate students. Well, I needed to make a living, and the salary he offered was twice as much as I had been making before the war. So I said, "Yes, I will."

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Q: How much was it, that first salary?

A: I got my Ph.D. in 1938 and I spent a year at the Institute, where my salary was $1200. Then I took a job at New York University teaching full time, but more than full time because I was teaching at the engineering school in the Bronx and the graduate school in Greenwich Village. Anyway, my salary was $2,500 by 1941, when I joined the Navy. Hotelling offered me $5,000 a year in 1946. In 1946, Harald Cram~r·s book (Mathematical Methods of Statistics) came out, and I tried to read it very quickly before I arrived in Chapel Hill. I spent about two weeks on it, a crash course in probability and statistics for me. When I arrived in Chapel Hill in the fall of 1946 I started to learn statistics by association with statisticians. There were R.C. Bose, S.N. Roy, Wassily Hoeffding, Harold Hotelling and quite a few famous visitors.

Q: Was Chapel Hill the first department of statistics in the country?

A: Hotelling had wanted to start a department when he was at Columbia, and when he left Columbia decided they would start one with Wald as chairman. But how about Neyman? What was his group called? When was the department of statistics at Berke­ley started?

Q: I want to ask you about your first encounter with Neyman.

A: Although there may not have been a formal department in 1946, Neyman had a very strong group at Berkeley. I'd say Columbia, Chapel Hill, and Berkeley were the three main centers at the time. In 1947 Columbia made an offer to Neyman. Berkeley did not want to start a department of statistics because the chairman of the mathematics depart­ment, Professor Evans, opposed it. Neyman gave them an ultimatum. ''Either you start a department, or I'll go to Columbia.'' And so the department of statistics at Berkeley was started in 1947. At least, that is how I remember it.

Q: So, three centers of statistical activity were started just about the same time. How long did you stay at Chapel Hill?

A: 6years.

Q : And is it right that Raj Bahadur was your first student, your first Ph.D?

A: Well, I don't know. I had three Indian Ph.D. students pretty much at the same time:

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Ghurye, Kallianpur, and Bahadur. I think Raj was the first to finish, but I'm not posi­tive.

Q: What about Jim Hannan of Michigan State?

A: I think he was a little bit later than Bahadur, but they were all pretty much contem­poraneous.

Q: How did you come upon the Robbins-Mom-a process?

A: Sutton Monro was a graduate student at Chapel Hill. He had been working, I think, at Bell Labs as an engineer in the field of quality control, and he decided to get a degree in statistics. He asked me if he could work with me. Suppose you want to design an experiment for testing the sensitivity level of something like an explosive, for example. You want to know, for example, from what height it will be safe to drop it. Suppose you drop a shell on the ground. If it drops from a large height it will explode. From a small height it won't explode. What is the height at which 50% of the shells will explode if you drop them? People said, "Well, what we'll do is drop some of them from 1 foot, some of them from 2 feet, some of them from 3 feet, etc.'' - the usual bioassay type of thing. But we felt that this was not very efficient, that there should be some method of designing the experiment while it was going on rather than a priori . The so-called probit analysis was rather arbitrary. Working together, Monro and I got some results on what is now known as the Robbins-Monro process. I said, "What you should do now is extend this to the case of finding the maximum. Instead of finding the LD 50, let 's find the maximum of an unknown regression function." And Monro started working on that, but Kiefer and Wolfowitz, to whom I had communicated the problem, got results before Monro. He was very upset and he never went on with his Ph.D.

Q: But the amazing thing was that the Robbins-Monro process was a nonparametric procedure.

A: Well, I've always liked nonparametric methods.

Q: Were you familiar with nonparametric methods before then?

A: I had written a little note on tolerance limits, which is nonparametric.

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Q: Yes, that was nonparametric. I am aware of it. You showed that distribution-free tolerance limits that are symmetric in the observations are necessarily order statistics.

A: Yes.

Q: Why did you leave Chapel Hill?

A: I had been at Chapel Hill for six years, and I had a sabbatical coming, so I applied for a Guggenheim Fellowship to spend a year at the Institute for Advanced Study again. While we were at Princeton my wife told me that she wanted a divorce. She went to New York City, where her family was living, and settled down there with our two daughters, who were seven and four years old then. At this time I was offered a position at Columbia, and I thought I should accept it, at least temporarily, in ordcrto be with the children.

Q: But it seems that you did not stay at Columbia very long.

A: No, that's not true. I joined the faculty of Columbia in 1953, and I stayed there with only sabbatical leaves until 1966. That was 13 years. And then I resigned at Columbia because I remarried in 1967, and my wife didn't want to live in New York. I spent half a year at Minnesota, another half at Purdue, a year at Berkeley as a visiting Miller Pro­fessor (in electrical engineering, of all things, not statistics), and then a year at Michi­gan, which I thought would be a permanent thing. But while I was away, the Columbia department was deteriorating and Columbia asked me to come back to rebuild it. Vari­ous people had left, T.W. Anderson among others. So I went back to Columbia and stayed there for almost 20 years more. I've been at Columbia from 1953 to 1985, with a three year gap.

Q : Are you sure that you stayed only for a year at Michigan?

A: Quite sure. When I went to Michigan, statistics was part of the mathematics depart­ment. I got an agreement on the part of the administration to start a department of statistics, which started at the same time that I left, so I was never a member of it.

Q : When was your first encounter with Neyman?

A: I think it was at some Institute of Mathematical Statistics meeting in the late forties. Of course, I knew him by reputation long before I met him. We didn't get along

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particularly well. We were both rather difficult people, I suppose. I don't recall any disputes we had, but we were not close friends. And then, much to my surprise, Ney­man asked me to participate in the Second Berkeley Symposium in 1950, and I wrote a paper, but was unable to go because of illness in the family. So Wassily Hoeffding gave my paper (on compound decision theory). I got absolutely no comments on it, so when the next Berkeley symposium came along I decided to write on the same subject from a slightly different point of view. I wrote the 1955 paper on empirical Bayes, although to me the two things, compound and empirical Bayes, were essentially the same. These two papers made quite an impression on Neyman, which was surprising because it's unusual for an older man to look at something that a younger person has done and say "Hey, this is great stuff! ". Moreover, what I was doing was opposed to Neyman's own non-Bayesian approach, and I considered my empirical Bayes to be more Bayesian than non-Bayesian.

Q: At that time as a graduate student I read of the great controversies between Baye­sians and non-Bayesians. So Neyman saw you as a middle-of-the road person?

A: When I came into statistics in 1946, it seemed to me that there was a lot in the Baye­sian point of view that parameters were not just constants about which nothing is known. On the other hand, I didn't understand how one could be a subjective Bayesian and assume that the state of your mind in some sense dictates the state of nature. I said to myself, ·'There is indeed an a priori distribution of parameters, but it's not a subjec­tive distribution. It 's something which is a part of the state of nature itself. So let's see what happens if we try to find out what it really is." And, of course, everybody opposed that. The Bayesians opposed it because they were in no doubt as to what the a priori distribution was. They had only to ask themselves, and that was it. And the non­Bayesians, except for Neyman, who were not used to thinking in any kind of Bayesian terms, were equally opposed to my way of thinking. It was not until Stein's multivariate means paper of 1955 that people started taking seriously the idea that the compound approach could possibly be of some value because of the fact that for sample sizes as small as two or three, the standard estimator is inadmissible. Now, I never felt myself that admissibility was of such great importance. An estimator may be good if it's admissible, it may be good if it's inadmissible. Anyway, the general acceptance of empirical Bayes and compound decision theory came, I think, from the effect of Stein's work on Efron and Morris who adopted what I call the "restricted" or "linear approach". Well, I don't want to characterize their philosophy for them. I always ask myself, "If there is a prior distribution and you don't know what it is, can you figure out what it is approximately, and therefore act as though you knew it?" Many statisticians just ask, "If the prior distribution is known to be a conjugate prior, what would you do?", which I don't much like because I don't see why we should consider only conju­gate priors. Stein's original result had no philosophical background. He simply showed that the multivariate mean is inadmissible with respect to squared error loss. It doesn't

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answer the question, "Well, what do you do about it?". Only somewhat later did Efron and Morris and some others say "Not merely does Stein's worlc show that the usual multivariate estimator is inadmissible, but you can actually improve on it, and this new thing really could be, should be used." Aside from that, I don't really understand the philosophical basis, if there is one, of empirical Bayes in the minds of some people. Is it that they want to approximate the best linear function of the observations, or do they want to approximate the Bayes estimator with respect to the true prior, irrespective of whether it's conjugate or not?

Q: I heard you comment about it at the Annual IMS Meetings in Cincinnati.

A: I sometimes want to write an essay on the philosophical basis of statistics, but I have resisted the desire so far. Most of the time the response to empirical Bayes during my lifetime has been rather negative. An example of that is a paper by J.B. Copas [J.R.S.S., Ser. B, Vol. 31 (1969), pp. 397-425) with comments by the Royal Statistical Society. What he said was, "Well, these are interesting ideas, and I guess empirical Bayes may be good, but compound decision theory is bad.'' The commentators were mostly nega­tive about the whole thing. When things like that happen, I don't feel like arguing. I'd rather just go and do something else. Now everybody seems to be an empirical Baye­sian to some extent, but what that means is not clear. Anyway, I feel that empirical Bayes and compound decision theory are great stuff and that they have changed the whole horizon of what a statistician can do, but it would take a long time to give you the psychological origin of how I came to it. Someday I will, because it has to do with my whole history, my whole pre-statistical background. It's something which should have been done a long time ago. Why did it take so long?

Q: There is always a first one. It happened to be you.

A: Nobody was trying to exploit the idea that there is a knowable prior distribution, not based on subjective grounds, but in the real world.

Q: What are your feelings on admissibility? Is it really that imponant?

A: I think it's a useful concept, but it's not of great imponance, because you can work with a non-admissible estimator provided it has useful properties. Let's put it this way. First of all, something may be inadmissible, but nevenheless so close to being admissi­ble that it doesn't really maner. The other thing is, something may be inadmissible, but only because there is a theoretically superior thing which you can't even write down in practice. The imponant point is not so much "What is admissible?", but, "What is the proper problem? What is the proper domain of this investigation?'' Stein's paper was

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on the multivariate nonnal mean. Now, to me that's not the way it should be put. It's on the univariate mean when you have a lot of different univariate problems. There is really no multivariate structure in Stein's problem. I would say that what Stein was really doing was the single nonnal mean in a compound setting, whereas he thought of it as a multivariate nonnal mean.

Q: Yes, I think I have an idea of what you mean.

A: In classical statistics you are repeatedly observing a random variable with the same unknown distribution, because most of classical statistics deals with random samples and asymptotic properties. If I want to take someone's blood pressure, I realize that it varies a great deal from hour to hour and from day to day with the psychological state of the person, etc. So I say, well, if I want to find your true blood pressure, I will ask you to come in 5000 times and I will measure your blood pressure 5000 times and I will add them all and divide by 5000, and that will be my estimate of your blood pressure. This is what I call "capitalist statistics" -many observations, one parameter- because someone as rich as a Rockefeller has billions of dollars, is very concerned about his health, and is willing to pay for a lot of work to estimate his blood pressure very accu­rately. The other is what I call "communist statistics" . Here the problem is that we have a lot of people and we have measured their blood pressure just once each. We also have 5000 observations, but there are 5000 different parameters. So the question is "How can I best estimate each of these 5000 blood pressures?", not "How can I best estimate one person's blood pressure by 5000 measurements?". It seemed to me quite obvious that true blood pressures are distributed around some population mean and part of your observed blood pressure is also due to chance variation, so I can get a better esti­mate of your true blood pressure by looking at everybody else's as well as yours. It doesn't mean that I can get a very good estimate because I have only one observation. But by looking at the measured blood pressure oflots of people I can get some idea of the distribution of the parameter itself. And once I have that, I can act like a Bayesian. This aspect, I think, is absent from Stein's paper. He was not concerned with the distri­bution of the parameter. I don 't think he had that idea in mind. Let me say one other thing. What I am alluding to now is the psychological background of statistical ideas, because statisticians, so far, are not robots. They do not function in an abstract environ­ment. They are human beings. Most of what you read doesn't get down to the ambi­tion, desires. and motives of the people who write it. Only rarely do these things come to the surface. You won't find them in the "literature".

Q: Can you touch upon some things which we never get from your papers?

A: Well, I'm not prepared to do a background analysis of my papers at the moment.

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Q: No, no, we do not need that here.

A: I have always intended to discuss these things some day. To be a mathematical scientist is a very doubtful way of life, because you are so restricted in your communica­tion with the rest of the universe. You don't do anything that normal people would regard as reasonable, and you can't tell them what it is that you do. At this stage in my life, at my age, I feel that I can, so to speak, relax. I don't have to prove anything to any­

body.

Q: You have proved it already.

A: Well, yes and no. In particular, my two sons, I don't think there is any chance that they will go into the mathematical profession. Is that right, David?

DAVID (Robbins' second son): I have no idea.

Q: Nowadays, young people have lots of pressures and competition. I do not blame them if they are not sure of what they want to do later on.

A: At their age I wasn't either. I didn't decide to go into mathematics until I was half­way through college. Now it's as though I were a tennis player at the age of 78, still try­ing to win the championship at Wimbeldon, and everybody says, "Oh, isn't that nice that he's still in there trying to play tennis even though he can hardly walk."

Q: Well there's no real age restriction for a statistician.

A: But practically speaking, . ..

Q: You're not standing in anybody's way. Ideas are ideas. Anybody is welcome to come up with new ideas.

A: Well , I don't know that I have any new ideas. You don't know what it's like, and you won't until you get to be my age. The number of new roads has decreased, so that you can still do things, but it takes such a long time compared to the way it used to be. It's like walking up the stairs when you get older. It just takes longer, and longer, and longer, until finally you say, "Oh, to hell with it! I'll just stay where I am" . I'm still try­ing to climb the stairs, but it takes me a long time.

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Q: So, what about the biostatistical unit you are building up at Rutgers?

A: I'm just serving as an entrepreneur. I don't presume to be a biostatistician myself; I'm just trying to use what influence I have with the authorities to establish a good pro­gram. John Van Ryzin, who was a friend of mine, was going to do it at Rutgers, and it needs to be done, and somebody has to do it, and so I thought I would see if I could do some of the things he wanted to do in the way of getting the program going.

Q: Speaking for myself, you've touched upon many areas of statistics, and made contri­butions to several areas, and many people have pursued your ideas.

A: I've done maybe three or four main things, and a lot of little things. None of them required great mathematical power. The only thing that they have in common is that they are all very simple things which nobody thought of doing before. So if I have any claim to attention it's that I touched upon these ideas for the first time.

Q: Some people think power one tests are catching on now. Do you agree?

A: Power one tests have certainly not caught on yet. Whether they will find practical use, I don't know. I think so, but maybe not in an obvious manner. I can think of appli­cations which nobody has bothered to imagine. Maybe I should make them myself. In a way I'm rather pleased with them because I'm an ex-mathematician. I take a certain amount of pleasure in the fact that, motivated by statistical considerations, I was able to do some things which mathematicians had never bothered with, for example, iterated logarithm inequalities. They were all doing limit theory, and I did inequalities. Well, limit theorems are of great philosophical value in statistics, but inequalities are what really count. You have to know the actual probability, and not just what it is asymptoti­cally. Some of the ideas involved in power one tests have, I think, useful applications in designing clinical trials. Right now, I'm working with a couple of young people on comparative clinical trials of two or more treatments with the object of minimizing the number of trials of the inferior treatments in order to cut down on the total human cost of the experiments, and some ideas involved in the tests of power one definitely come in here. Well, who knows? Anyway, it's interesting.

Q: What do you think of the future of statistics?

A: I'm pessimistic, because of my age. The general situation for people who have passed their prime is not to be able to communicate very well with younger people, and not to really understand what's going on. I think that happened to Einstein after he came

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to this country, after he had done his great work in Europe. In my case, I look at the situation in statistics now, and I'm no longer as excited about it as I was 30-40 years ago when I thought I saw a lot of things that I could do that nobody had done, and I tried to do them. I don't know what the young people see now when they look at statistics, but I hope that the things that they want to do now seem as interesting as the things that I wanted to do 40 years ago. I don't know anything about computing, so when people tell me about statistical techniques that exploit high-speed computing, I am out in the cold. What really worries me is that young people (I mean, under 60) have a very different cultural background from what I was familiar with. A lot of people nowadays are exiled from their countries. Well, everybody who gets old is exiled too, to a different time, and it's very difficult for me to communicate with young people because of the time differ­ence. I fear that the people who are now studying science do not have the same cultural interests that people of my generation did. To put things very simply, I don't think that the graduate students I now encounter in the sciences read books. They read textbooks. But they don't read poetry, drama, history, because it's not relevant to their jobs, and it's not required, and it's not fun for them. I remember in my day meeting colleagues occasionally who would read popular fiction or things which I regarded as semi-literate, and I looked at them with great surprise. ''What are they doing in a university if that's the level at which they function?" But now I don't think they even read science fiction stories. I don't think they read anything. I suppose they watch television. I don't know what they do.

Q : Perhaps due to job pressures, young people today are not as much concerned about current social issues as we were in our youth.

A: As far as I can see in this country, they're concerned with personal and family com­fort. I don't see much ambition in a general sense. They certainly are not interested in political and social movements. In my generation God was dead. Now, Communism is dead. Everything now is, I think, every man for himself. The question that concerns my children is "What kind of new car are we going to get? How will it compare with our neighbor's new car?". This is a part of my life that has never been developed. So I find

it depressing, particularly because many of my graduate students are of foreign origin, and I used to think that foreigners would come here with their own cultural background and learn whatever they wanted to learn here, but also leave with us some of their cul­tural heritage. And I see them coming now without any culture at all and acquiring none from us. I am depressed by this. They seem to be just as lacking in cultural interests as our American students. (Added in proof, with apologies: the students in Tiananmen Square can certainly not be accused of a lack of interest in political or social movements, nor with an undue concern for personal or family comfort.)

Q : Do you think that the current graduate students are receiving good training in

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75 YEARS OF HERBERT ROBBINS 19

statistics in our universities? Do they have adequate mathematical training?

A: Well, I never studied statistics. I studied mathematics.

Q: But essentially you are a self-made statistician.

A: A rather preposterous figure. I know it's a mystery, maybe, what graduate students are doing now, and what they want. And yet, why are they doing it at all? Why are they going into statistics instead of computer science or medicine or law where they could make more money? Of course, making money is a part of the game-getting contracts, summer pay, extra money as consultants, and so on. It plays a large role. It played no role at all when I was young. There weren't any NSFs and NIHs, but this is the standard complaint of the older generation- things are going to hell, and young people are no damned good. Of course, it's true. It has always been true. My family was not an academic one, so I didn't have these problems. The academic world was a fascinating one. What was the alternative? Selling shoes? Reading books was great! But what do kids nowadays think about life? I don't think they want to read books. What do they want? Do you know?

Q: No, I wish I did. What do you think of statistics as a discipline? For example, when I first entered graduate school, statistics was one discipline, everything in one. Now there are so many shades of statistics, business statistics, medical statistics, etc. Is statistics a discipline or a multi-disciplinary activity?

A: Some people think that statistics departments should not exist.

Q: That's the import of my question.

A: Some people think that there is no central discipline which needs to be studied, that you should learn whatever statistical methods you need to be an economist, or a psychologist, or an engineer, but primarily you should have a specific subject matter, rather than be a statistician in general. Of course, I could say that of mathematics as well. But most mathematics is a highly specialized subject. If you are going to work, say for General Motors or IBM or what have you, you don't need all of mathematics. You just need certain parts of it from time to time. I don't believe that is the correct way to educate people. I still think that statistics is a legitimate subject for study, but I am not very happy with the statistics departments that I see. There are very few places where a person can get a good education by concentrating on statistics. One thinks of Berkeley and Stanford as the outstanding centers of statistics in this country, but,

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although they may be the top, I don't think that either of them is doing such a great job as far as the education of future scientists goes. My age makes me look back on the good old days.

Q : At international meetings, I meet some Russian counterparts. For them, apparently, there is no difference between probability and statistics. What do you think of their atti­tude towards probability and statistics?

A: As far as I can see, there is no statistics in Russia. So in a sense there is no difference between probability and statistics because the Russians do not do statistics, period. And the reason they don't do statistics is that statistics in Russia is a dangerous subject. The Russian state, the Communist apparatus, regards information as a state monopoly, and the knowledge of even such things as infant mortality or life expectancy is restricted by the state. Recently there were some Russian delegates at a meeting of the AAAS here, and someone interviewed them on questions of computing science, and the Russians' attitude was "Our computer scientists are better than American computer scientists. Unfortunately, we don't have any computers. " Well, it's the same way in statistics. It may be that Russian statistical theory is better than ours, but they don't have any statis­tics to work with. In fact they don't have any statisticians either. You see, if you work in probability theory you're a mathematician, and the Russians have liked probability theory very much for the last 150 years, because it had some vague applications to artil­lery fire- how to direct artillery fire, and the circular probable error. They can see some practical use of probability theory in the military field . But what are some of the things we use statistics for in this country? I'll give you three examples.

1. Quality control: the Russians have no quality control, and they don't have any use for quality control, because they don't make anything to sell. They don't com­pete in the international market.

2. Clinical trials: the Russians have no use for statistics in clinical trials. They don't have any clinical trials, because they import drugs from the west; they don't develop them themselves.

3. Public opinion surveys: it is needless to say anything about this.

Q : Do you think that they are not interested in public opinion?

A: Right. Why should a Russian study statistics? There is no useful application for statistics, because there aren't any statistics to work with. So of course, a Russian would go into pure probability theory. For example, recently one of the leading Russian statis­ticians was finally allowed to leave the Soviet Union and will come to this country, I guess. He is the nearest thing they had to a mathematical statistician, but the statistical problems he worked on were purely theoretical. They're essentially mathematical prob­lems that arise in statistics, but certainly do not come out of any practical need. By the way, what I'm saying about Russians, to a lesser extent is true of Europe in general.

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75 YEARS OF HERBERT ROBBINS 21

There seems to be nothing going on in the field of statistics in France, which ranks very high in probability theory. We may be in a period when statistics, as such, is rather quiet-nothing much going on. This may change overnight. In fact, it may have already changed. I was lucky that when I went into statistics, I went into a very stimulating atmosphere. Hotelling had assembled a very powerful group of people, both permanent and visitors, which I don't think you have anymore, who talked to one another, were in and out of each others' offices, would go to each others' lectures, and so on. Within a few months I was exposed to so much. I have a feeling that most of the centers now are not really cohesive, collegial; they are just isolated people coming from their homes and then going back again. I don't know what the environment is for a student, whether he feels excitement and fellowship in statistics the way he would, say, in a biological laboratory where you walk along the aisles and see this one working on gene splicing, and that one working on an antibiotic, and you have the feeling that you are part of a liv­ing organism. I don't see that very much anymore. When Neyman was at Berkeley, I think he was the catalyst that brought everyone together. Then the younger people revolted against his domination, but they have nothing to replace it with. So I think the spirit has gone out of the flesh, which is staggering around on inertia rather than inter­nally generated power.

Q: When I first entered the statistical field, I saw essentially three schools of thought. One was the British school of thought. Another one was the Neyman school of thought. And the third one was the Indian school of thought. Do you see any differences between the three schools of thought?

A: Certainly the British influence has been dominant in the evolution of statistical theory. I'm not talking about probability-I'm talking about statistics. Especially, I would say, it started with Francis Galton, Karl Pearson and R.A. Fisher. Without those three people, we wouldn't have anything at all. It didn't have a very profound mathematical content. Now, in England, I would say, nothing much of theoretical importance is going on with the conspicuous exception of D.R. Cox. For one thing, the British educational system is increasingly starved. There are very few positions avail­able, and very few graduate students. There is no active school of statistics that I can see anywhere in England comparable to Berkeley or Stanford.

Q: Someone should have the spark.

A: You have to cohere around occasional super-good people, and they're not very fre­quent, and where they will tum up, God only knows. Certainly, England was one place. Wald didn't live long enough; he was cut down early in his career. Hotelling became senile rather young. Neyman was the only one who retained much intellectual power through his later years. So he has always been for me the ideal of a scientist who not only does important work himself, but inspires it in others, and he organized a school and continued to work until his last days. When you look at the statistical scene in this country now, I'm sure a lot is going on that I don't see. But what I see is not very inspiring. Nothing is going on at MIT. If you '11 pardon my saying so, nothing is going on at Harvard, either, although some people might differ there. Columbia is on its last legs. Princeton went out of business a couple of years ago. Chapel Hill is on the decline. The whole eastern seaboard in the United States is very dim. Chicago,

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likewise. I would say that Seattle, Berkeley, and Palo Alto are the only places where anything is going on. They are 3,000 miles away from here, so I don't have a first-hand impression of how active they are. But I don't think they have the same feeling of excitement that was present in the '40s and '50s. I hope I'm wrong.

Q: So do you think we have reached a plateau in statistics?

A: The plateau is probably human rather than intrinsic. I don't think we've come any­where near to the limits of statistics itself. I think the human energy that is going into it now is at a much lower level. Something of the nature of another war might be the answer. Unfortunately, another war would kill everybody. If you could have another old-fashioned war, it might be good for statistics.

Q: Some people believe that testing hypotheses is somewhat artificial, that 100 years from now, it will not be a pan of scientific development. What is your opinion on that?

A: I agree. Hypothesis testing is something I have never had much use for. I've never been able to understand the point of testing hypotheses. But this is of course largely a linguistic matter. People don't really test hypotheses. They just call it that.

Q: If Wald were alive, what kind of sequential estimation procedures would he have come up with?

A: Well, Wald was very proud of his work on statistical decision theory.

Q : He has a book on decision theory.

A: He has two books, Sequential Analysis and Statistical Decision Theory . The latter subject has some value, but, really, who needs it? The ideas of minimax, admissibility, and so on, are really rather feeble concepts. Something much stronger is needed. In Sequential Analysis, simple versus simple is pretty useless. Wald made a great begin­ning, but it's only a begirming. So it's hard to say what the lasting residue of Wald's career will be. He needed another twenty years, I think. It's certainly true that no one of his order of magnitude has come along since he died. So, it's been 35 years, and you can say pretty clearly, nobody of that order of magnitude exists. And that's in the whole world, with billions of people. But then, how long has it been since a Mozart came along? These things are at the moment not under human control. I have a feeling that biostatistics is a field that requires more attention than it has had. And it will have to come from various fronts, not only theoretical, but practical. Attention has to be paid to biostatistics in Washington as well as in Harvard, because people are doing theoretical work that is not used by NIH, FDA, EPA and others. Anyway, I'm glad the field of statistics has yet to make its real impact. I feel, for example, that more lives are going to be saved by statisticians than by doctors in the immediate future.

Q : That's very interesting. Are you serious?

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75 YEARS OF HERBERT ROBBINS 23

A: Well, it's an exaggeration, but take the case of two causes of death: cancer and heart disease. Essentially nothing has been done about the cure of cancer or heart disease. The emphasis is on prevention . And prevention means finding, by statistical analysis, those aspects of the environment and lifestyle that will reduce the incidence of these diseases. Right now we are attempting to abolish smoking because it is a clear cause of death, but what is the evidence that smoking is a cause of death? It's not medical, in the sense that nobody in the laboratory has yet found a direct link. It is a purely statistical fact that people started looking at death rates and making crude analyses, so crude, in fact, that Fisher himself, during his last years, testified as an expert witness for the tobacco industry, that there was no demonstrable effect of smoking on cancer. Did you know that?

Q : No!

A: Yes. He testified for the tobacco companies sometime during the 40's or 50's when there were some of the first cases brought by people who were suing the tobacco com­panies on the grounds that their husbands or fathers had died of lung cancer caused by smoking. Fisher testified that there was no statistical evidence that cancer was caused by smoking, because it could well be that people who have an inborn tendency to cancer also have an inborn tendency to smoke. And therefore, since there had been no double­blind prospective study, there was no real evidence. Now that is just a footnote. Now the questions are of exercise, health, or at what level should cholesterol be. Does aspirin help you? For example, you have two studies published in the early part of 1988. One study says aspirin is good for preventing heart attacks, and another says it isn't. Did you read those reports?

Q : Yes I did.

A: Okay. Now, this is obviously what the statistician has got to decide, and not just the doctors. I think it is a scandal the way some studies are conducted essentially by doctors who have gotten advice from statisticians to a greater or lesser extent, but in such a way that studies have no meaning after they are done. It's a waste of money. It's going to be the statisticians who have to design and interpret these things, not people hurriedly bringing out a study. For example, one study was terminated long before it was sup­posed to be, because the doctors thought, "Gee, it's unethical for us to continue this study when it is so obvious that aspirin is good.'' Even though, as it seems, more peo­ple were dying in the aspirin group, but they were dying of strokes, not heart attacks. This is called " sequential analysis" . So, sequential analysis seems to mean that you design a study to spend a given amount of money over a given period of time, and then if you get worried, you stop it. That's the sequential part. If the monitoring group feels that maybe you should stop it, then you stop it. In other words, there is no sequential design at all, it's just a panic button, and when anybody wants to push the panic button, then that 's the sequential stopping point.

Q : So what advice do you give to young people who are aspiring to be biostatisticians?

A: Well, I was talking to a biostatistician in Washington who said that he would not

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recommend to his students to go into biostatistics because the control of statistical methods at NIH is so much in the hands of doctors. The statisticians are regarded as computer technicians, rather than being respected by the medical profession as fellow scientists. Of course, the doctors are the authorities in matters of health. They know what's good for you. But we have to help them more, whether they like it or not.

ACKNOWLEDGEMENT

I thank Brian Moses for transforming tape-recorded informal conversations into the present manuscript.

REFERENCES

Courant, R. and Robbins, H.E. (1941). What is mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, London and New York.

Govindarajulu, Z. (1987). Sequential Statistical Analysis of Hypothesis-Testing, Point and Interval Estimation and Decision Theory. American Sciences Press, P.O. Box 21161, Columbus, Ohio, ix + 680 pp.

Lai, T.L. and Siegmund, D.O. (1985). Herbert Robbins: Selected Papers. Springer­Verlag, New York.

Lai, T.L. and Siegmund, D.O. (1986). The contributions of Herbert Robbins to mathematical statistics. Statistical Science, 1, 276-284.

Neyman, J. (1962). Two breakthroughs in the theory of statistical decision-making. Rev. Intern. Statist. Inst., 30, 11-27.

Van Ryzin, J. (1987). Adaptive Statistical Procedures and Related Topics. IMS Lec­ture Notes-Monograph Series, IMS, Haywood, California, Vol., 8, 476 pp.

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