Professor C.R. Rao Interview
I am aware that you received your M.A. in Mathematics from Andhra University in India, your M.A. in Statistics from Calcutta University and your Ph.D. from Cambridge University. Could you tell us when did your interest in Statistics originally start? Have any family members, even distant ones, followed your statistical interests?
I received the M.A. degree in Mathematics from Andhra University in 1940 and, after six months of frustration of applying for jobs and not getting any response, I joined the Indian Statistical Institute (ISI) in 1941 to undergo what was called a one-year training course in statistics to improve my prospects of getting a job. The course was elementary, but it gave me an opportunity to start doing research in
collaboration with K.R. Nair, one of the mathematicians recruited by P.C. Mahalanobis, the founder of the ISI, as that was the best way of learning statistics in the absence of books and advanced courses in statistics at the universities at that time. When I was half way through the training program at the ISI, Calcutta University started a Masterâ€™s course in Statistics, which I took and earned an M.A. degree in Statistics by the end of 1943, with a first class, first rank and record number of marks, which remains unbeaten till today. Mahalanobis offered me the job of Technical Apprentice at the ISI on Rs. 75 a month in 1943, which was the beginning of my career as a statistician. During the period 1941-46, I published 33 research papers; one of which published in 1945 is the most quoted one in literature and generated the technical terms, Cramer-Rao Inequality, Rao-Blackwell Theorem, Fisher-Rao metric and Rao distance in statistical inference. Another paper published in 1946 contained a combinatorial arrangement called orthogonal arrays, which is widely used in industrial experimentation to design new products. Mahalanobis doubled my salary to Rs. 150 per month. This encouraged me to pursue my career as a statistician! In 1946, I got an
offer from Cambridge University, UK, to do statistical analysis on some skeletal measurements, based on methods developed in India. I was glad to accept the offer considering
underdeveloped country to an advanced country. I arrived in Cambridge in the fall of 1946 and started working at the University’s Museum of Archeology and Ethnology as a Research Fellow on £20 a month and also received admission in King’s College (of which I am now one of the 11 Life Fellows) and registered myself as a research scholar under the supervision of R.A. Fisher for a Ph.D. degree. I completed the project by the middle of 1948 and also wrote a thesis for my Ph.D. degree that was approved by the examiners. I returned to India in the fall of 1948 to continue my research work at the ISI.
Mahalanobis doubled my salary to Rs. 300 a month, perhaps as a reward for my additional qualification of a Ph.D. degree from Cambridge University and a few additional papers I published during my stay in Cambridge. The stage was set for me to continue to work at the ISI for
the next 32 years until I took mandatory retirement at the age of 60.
I chose to study statistics out of necessity to improve my job prospects and found it an interesting area to pursue as a career. I started to learn statistics at a time when it was a new field that provided job opportunities. But now, several new fields have opened up, especially in information technology, with better job opportunities. None of my family members, even distant ones, chose statistics as a career.
Who were the three people who influenced your career the most, and why?
The three people who had an early influence on my career in statistics are P.C. Mahalanobis, R.A. Fisher and R.C. Bose. Mahalanobis encouraged me to pursue research in statistics and put me in a responsible position at the ISI as the Head of the Research and Training School for the development of statistical education and research in India. R.A. Fisher was a frequent visitor to the ISI and he was an
inspiring guide when I was working at Cambridge. R.C. Bose made remarkable contributions to combinatorial mathematics, the most famous of which is disproof of Eulerâ€™s Conjecture on the nonexistence of orthogonal Latin squares of certain orders, and my work on orthogonal arrays was inspired by his contributions. Of all your accomplishments, and there are countless, which one in particular are you most proud of achieving?
I have some intellectual satisfaction for the esteem I earned from the peers in my profession, who introduced some technical terms in statistical inference, attaching my name to them. The most widely quoted term in the literature on statistics, engineering and lately in quantum physics is the Cramer-Rao inequality. Perhaps, my greater contribution is the encouragement and guidance I provided to my Ph.D. students (fifty to date), some of whom have made outstanding contributions to statistics and who, in turn, produced about 300 Ph.D.â€™s (according to the information on the genealogy website). This is a matter of great pride for me. I understand that you have received numerous honorary degrees from universities and institutions around the
world. How do you feel about all that you have achieved so far?
Up to date, I have 29 honorary degrees from universities in 17 countries. This statistic may or may not be meaningful, but what I value most is my Sc.D. degree from Cambridge University which, I am told, is based on a peer evaluation of published research work and its contribution to natural knowledge. If you were to start up a statistics department at a new university, what advice would you give to the new Department Head?
Statistics is not a discipline like physics, chemistry or biology where we study a subject to solve problems in the same subject. We study statistics with the main aim of solving problems in other disciplines. So, the teaching of statistics must be different from that of other disciplines. Of course, curriculum of a statistics course should include the established statistical methods in common use, and also selected areas of mathematics and probability necessary to develop new statistical methodology. But emphasis should be given to the application of various tools using real data for demonstration.
The statistics department should also take the responsibility of developing special courses in statistics for students studying other subjects like psychology, sociology, biology, etc., with greater emphasis on applications than on theory. Finally, I would encourage the statistics department to run consultation services to help research workers in other departments in designing experiments, collecting necessary data and drawing inferences. The consulting division of the department would also be useful in providing hands-on experience for students taking regular courses in statistics.
Can you identify any gaps in statistical methodology that are most deserving of the professionâ€™s attention? What new statistical challenges are awaiting you in the forthcoming years?
The methods of statistical analysis are changing with the advent of computers, availability of large data sets and refined measurement techniques. Model based techniques developed for analyzing small data sets using hand driven computers are being replaced by algorithmic and computer intensive methods without model assumptions. A new brand
of statistics is coming up in the name of data mining. There are different schools of statisticians postulating different approaches to statistical inference and it is not unusual for different statisticians working on the same data arriving at different conclusions. It is not so much the gaps in statistical methodology that we have to worry about. Judging from the current trends of drop in the demand for courses in statistical theory, static or diminishing number of members of statistical societies and assumed advantage of each subject matter department developing its own courses in statistics with special applications to the concerned subject, there is an apprehension that autonomous university statistics departments may cease to exist. There are some serious
development of statistics in the 21st century. We have to think of the future of statistics as a separate discipline with a well-defined philosophy and methodology, and the role of autonomous statistics departments in educating statisticians and developing research in statistics. One of the problems that many universities (and the International Statistical Institute) are currently faced with is the decreasing number of young students. What can you
say to anyone who is considering a career in the statistical profession? What would your advice be to young academic statisticians today?
Because of availability of attractive jobs in the emerging areas of information science and technology, young students are attracted to courses in computer and information sciences to qualify themselves for the new jobs. Perhaps, courses in statistics with specialization in some field of application would be more attractive to young students rather than a course confined to what is known as mathematical statistics dealing with rigorous derivations of statistical
theorems. I believe there is more to be done by way of research in developing a coherent and practically oriented theory for statistics. I would encourage some academically bent students to pursue a research career in the foundations of statistics. You have held professorships and taught in different countries such as India and the United States of America.
Were there differences in the academic philosophy in the institutions you attended that you found to be significant?
The courses in statistics offered at the Indian universities are somewhat rigid. All students have to take a prescribed set of courses covering theoretical and applied statistics. The syllabus followed in all the Indian universities are more or less the same. In the USA, the students can exercise their choices depending on their interests. However, in the USA there is greater emphasis on mathematical statistics, whereas some areas of statistics like design of experiments and sample surveys are not covered in many universities. In India, courses seem to be well-balanced between theory and practice. But the academic philosophy is the same, since the courses are taught using the same set of text books. How many ISI Sessions have you attended? Do you have any suggestions as to how the ISI Session concept can be improved? As an ISI Honorary member, what new roles do you think that the ISI can play in the international statistical community in the future?
I have attended quite a number of ISI Sessions. I am glad to find that the scientific program of these Sessions is
becoming more and more broad based to meet the increasing demand for statistical methodology and statistical thinking
The programs of the special Sections of the ISI, the Bernoulli Society, International Associations for Official Statistics, Education, Statistical Computing and Survey Statisticians, are also well-balanced and reflect the modern trends of theoretical research and applications in these areas. There is probably a need to create special sections in other areas such as bio-informatics (including mathematical genetics) and environmental statistics, where there is much activity all over the world.
You have published a large volume of work. Have you ever stepped back and tried to identify the genesis of the creative process in yourself? For example, some people find that they are more creative in certain environments, or at certain times in the day. Are there any â€œcommon denominatorsâ€? in your own creative process?
When I was working in India during the period 1941-1978, there were only a few people to consult or collaborate with in research work. About 76% of my published papers are authored by me and 19% with one joint author and 5% with more than one joint author. The source of problems on which I worked arose from the applied work I was involved in or questions raised or not fully answered by authors in statistical journals. Under these circumstances, there is plenty of scope for doing creative work. You have to think for yourself. My 1945 paper, where the Cramer-Rao Inequality is derived, actually arose out of a question put to me by a student when I was teaching estimation theory. There is no particular time or environment conducive to do research work. I generally do research work in the early hours of the morning, a habit which was forced on me by my mother who used to wake me up from my bed at 4AM in the
morning, light up the lamp and made me study. Some researchers prefer to work late in the night and wake up late in the morning. I do have some experience of going to bed thinking of some difficult problem and finding an easy way of arriving at a solution on getting up in the morning. Perhaps the brain keeps working even if you are not in a fully conscious state!
mathematics departments have a large number of faculty members and visitors, I found opportunities for collaborative work. About 80% of my published papers, while working in the USA during 1979-2004, are jointly with others and 20% by me, which is a reverse of the corresponding figures in India. The average number of papers published per year increased by 3. It is difficult to evaluate the difference in quality of my papers published while working in India and the USA. Einstein has said: â€œCreative work depends more on oneâ€™s imagination rather than on other inputs through discussion with others, knowledge from published works, and particular work environments.â€?
Of the twelve books and hundreds of research papers you have (co-) authored, which one(s) was (were) the most challenging/satisfying? Which do you find to be your most defining work?
It is difficult to answer this question as every piece of work, a research paper or book, needs some imagination and hard work, and is never completed to the authorâ€™s satisfaction. Usually, the author is not aware how important his work is until he comes to know how well it is received. Some of my books and papers are mathematically oriented and written for advanced students. The most challenging and satisfying book I have written is Statistics and Truth: Putting Chance to Work (published by the World Scientific Press, Singapore and translated into 6 languages), which is not a traditional book summarizing available knowledge, but discusses philosophical and practical aspects of knowledge and the role of statistics in acquiring knowledge on which we can act upon. Two of my papers, which generated some technical terms with my name attached and are reproduced in the book on Breakthroughs in Statistics (1890-1990), will probably be part of statistical literature for some time.
There are a number of theorems credited with your work and name. Which one did you find to have the most interesting process prior to becoming a theorem?
I have already mentioned four of my papers published during 1945-1949 provided some technical terms named after me. There are a few other papers published after 1949 that also generated technical terms bearing my name. Some results catch the attention of the researchers immediately after publication, a few others take time and many go unnoticed and unreferred to by other writers. My result on the lower bound to the variance of an unbiased estimator published in 1945 was named as Cramer-Rao Inequality by Neyman and Scott in 1948. It is a simple result obtained by using the Cauchy-Schwarz Inequality and it generated considerable research. It is frequently quoted in papers on statistics, engineering and has begun to appear in papers on quantum physics. I introduced a new asymptotic test criterion called the Score Test in a paper published in 1948. It took almost 40 years for it to be recognized as a useful criterion and find a place in text books as Raoâ€™s Score Test. I mention these two as they are the most quoted results. My paper on orthogonal arrays developed during 1945-1949, on
which there is a full-length book, led to considerable research in combinatorial mathematics and applications in industry,
Another result of mine is generalized inverse of a matrix introduced in 1965, which has been accepted as a useful contribution in the discussion of linear models and multivariate distributions with a singular dispersion matrix. In my 1945 paper, I introduced differential geometric methods in statistical inference which led to the keywords, Rao Metric, Rao Distance, Rao Measure and Cramer-Rao Functional, and which is a topic of current research in statistics. In my recent reseach, I introduced the concept of quadratic entropy and cross entropy, which are finding applications in statistical inference and some areas of physics. I learned that there was only a limited amount of literature available on statistical theory and practice when you were a student of statistics, in particular, Sir R.A. Fisherâ€™s Statistical Methods for Research Workers. How greatly has R.A. Fisher influenced you and your work? Are there any other individuals you feel have done the same?
As I mentioned before, R.A. Fisher is the statistician who influenced my work in statistics. Long before I met him, I learnt all my statistics from his book, Statistical Methods for Research Workers. I think it is a classic that served as a guide to research workers over a long period of time; especially,
to those working in
assumption of normality of measurements holds at least approximately, and the number of observations is limited. There are, however, some controversial issues, which are inevitable when someone is creating a new branch of knowledge. He was involved in bitter controversies over some of them. However, the three methodological problems, specification,
enunciated by him as relevant to statistical analysis of data, constitute the framework for all statistical theory and research. If you could relive any part of your career, which part would it be? If you would prefer to omit any part of your career, which part would it be? If you were to start again and choose a profession other than statistics, what would you want to be?
If I could relive any part of my career, it would be my stay in India, the country to which I belong, where I had the opportunity to guide students in new areas of research.
The choice of oneâ€™s profession, however, depends much on what is attractive in terms of job satisfaction and remunerative to meet the needs of the family. Statistics provided such opportunities when I completed my university education. Under the present circumstances, I might have chosen some areas of information science and technology, which offers greater challenges. You will be eighty-five this September. What plans do you have to mark this special occasion?
There are several unfinished tasks. I only hope that I will be able to complete some. The Government of Andhra Pradesh (one of the states in India) has announced plans to develop an Institute for basic research. They have named the Institute as C.R. Rao Advanced Institute for Mathematics, Statistics and Computer Science, the three basic sciences that have a fundamental role in the improvement of natural knowledge. Perhaps, I will spend some time in developing this Institute