Calyampudi Radhakrishna Rao

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10 September 1920
Hoovinna Hadagali, Karnataka State, India
22 August 2023
Buffalo, New York, USA

C R Rao is an Indian-born mathematician and statistician who has worked most of his life in America. His statistical work has been applied in many fields.


Calyampudi Radhakrishna Rao's parents were C Daraiswamy Naidu (1879-1940), a police inspector, and A Laxmikanthamma. He was the eighth of his parents' ten children, two of whom died as infants. We should explain that the name Rao is part of his given name - in fact all the male children in his family were named Rao. His other given name Radhakrishna comes from the god Krishna (who was the eighth of his parents children and, for that reason, the custom was to name the eighth child after Krishna). Only the boys in the family were given a school education, as was the tradition at the time, but this was rather disrupted by frequent moves that the family made as their father was transferred from one place to another. Calyampudi Radhakrishna completed his first two years schooling in Gudur, his next two in Nuzvid, and then grades 6 and 7 in Nandigama, all towns in the state of Andhra Pradesh. During these years CR, as we will call Calyampudi Radhakrishna Rao throughout this biography, saw little of his father who was totally absorbed in his work but his mother was a major influence [11]:-
... in my younger days, [my mother] woke me up every day at four in the morning and lit the oil lamp for me to study in the quiet hours of the morning when the mind is fresh.
In 1931 CR's father retired and the family settled down in Visakhapatnam, on the coast of Andhra Pradesh. The family chose this city because of the excellent educational facilities that were available there for their children. CR studied there for ten years, first at high school, then mathematics, physics, and chemistry at the intermediate Mrs A V N College before attending Andhra University. At the intermediate college he won the Chandrasekara Iyer Scholarship in each of his two years. The School Magazine for 1935 published his picture with the caption:-
C Radhakrishna Rao who has won the Chandrasekara Scholarship this year. He has had the unique distinction of knocking off the most coveted prizes throughout his school career.
CR spoke in the interview [5] about the influence of his father:-
When I was 11, I could do complicated arithmetical problems without paper and pencil. My father appreciated my interest in mathematics and my good performance in school, and he thought that I should eventually get a degree in mathematics and proceed to do research to get a doctorate degree. He presented me with a book called 'Problems for Leelavathi', a collection of problems set by a mathematician for his daughter Leelavathi to solve. He asked me to work out 5 to 10 problems in the book every day. I enjoyed solving these problems, which aroused further interest in me to pursue mathematics. Thus, my entry into mathematics resulted from the encouragement I received from my father and my own interest in solving mathematical problems.
He graduated M.A. with First Class Honours in Mathematics from Andhra University in 1940. He applied for a research scholarship from Andhra University but his application was rejected on the grounds that it had been received after the deadline. At this stage, encouraged by his family, he decided to sit the competitive Indian Civil Service examinations but, being only twenty years old, he had to wait eighteen months before being allowed to take the examinations. He applied for job as a mathematician in an army survey unit to fill out the time before taking the Civil Service examinations. He was called to Calcutta for an interview but failed to get the job. However, this was a turning point for CR, for he stayed in the South Indian Hotel before his interview and there he met a young man who was being trained in statistics at the Indian Statistical Institute. CR had taken a course on probability while studying for his Master's degree at Andhra University but he had never heard of the Indian Statistical Institute. The young man took CR to visit the Institute, at that time located in the Physics Department of Presidency College. It seemed to provide both a job and a chance to test whether he would like research so CR applied for the one-year training course in statistics.

The family were in some financial difficulties by this time since CR's father had died in the previous year. However, one of his brothers and his mother managed to finance him through the year at the Institute. The training course was rather a disappointment, taught by people with little understanding of statistical theory. However, there was the head of the Institute P C Mahalanobis, as well as other top researchers working at the Institute such as K Raghavan Nair, Samarendra Nath Roy and Raj Chandra Bose. CR began undertaking research with Nair and they published a joint paper Confounded designs for asymmetrical factorial experiments (1941). In the following year he published six papers, four of them joint publications with Nair, for example A general class of quasi-factorial designs leading to confounded designs for factorial experiments and A note on partially balanced incomplete block designs.

A few months after he began training at the Indian Statistical Institute, Calcutta University announced a new Master's degree in statistics. Lecturers at the University were the same statisticians working at the Institute and he took courses from K Raghavan Nair, Samarendra Nath Roy and Raj Chandra Bose. He was awarded the degree in 1943 with First Class and the gold medal. His thesis was a major piece of work in four areas: the design of experiments, linear models, multivariate analysis, and the characterization of probability distributions. These would be the topics he continued to study throughout his career. As well as research in statistics, CR began to look at combinatorial problems with R C Bose and number theory problems with Sarvadaman Chowla.

CR was appointed as a Technical Apprentice at the Indian Statistical Institute beginning in November 1943 and, a few months later, in June 1944, he also worked as a part-time lecturer at Calcutta University. By the end of 1946 he had over thirty papers in print [5]:-
I continued my research on combinatorics with reference to design of experiments and wrote a number of papers, some jointly with R C Bose and S D Chowla. I developed a general theory of least squares without any assumptions on the concomitant variables. I found a test for redundancy of a specified set of variables in multivariate analysis.
The most significant result CR obtained during this period is now called the Cramér-Rao inequality and gives a bound for the variance of an unbiased estimate of a parameter. It appears in his paper Information and accuracy attainable in the estimation of statistical parameters (1945). The significance of this paper can be seen from the fact that it was republished in S Kotz and N Johnson (eds.), Breakthroughs in Statistics: 1889-1990 (Springer Verlag, New York, Berlin, 1991). In August 1946 CR boarded a ship sailing from Calcutta to England. He had been offered a position undertaking statistical work at the Anthropological Museum in Cambridge and he also registered as a research student at Cambridge University. His Ph.D. studies at King's College, Cambridge, were supervised by R A Fisher. CR worked at the Anthropological Museum every day and every evening he spent a few hours in Fisher's genetics laboratory mapping the chromosomes of thousands of live mice. While at Cambridge CR wrote his influential paper Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation which was published in the Proceedings of the Cambridge Philosophical Society in 1948. He was awarded his doctorate in 1948 for his thesis Statistical Problems of Biological Classification which was examined by John Wishart.

He returned to Calcutta in August to his post at the Indian Statistical Institute. Soon after arriving, on 9 September 1948, he married Smt Bhargavi, a girl he had known from childhood; they had a daughter Tejaswini and a son Veerendra. CR was appointed as a Superintending Statistician on his return but a major grant to the Institute from the Indian government allowed the Institute to set up a Research and Training School and appoint professors, assistant professors and other academic grades. Those senior to CR, R C Bose and A Bhattacharya, left the Institute around this time. S N Roy was appointed to a professorship and CR to an assistant professorship. However, Roy left for the United States soon after his appointment and CR became a professor in July 1949 at the young age of 28. The Institute was given authority to award its own degrees in 1959 and started up its own undergraduate programme. Being understaffed CR found himself doing quite a bit of undergraduate teaching but found that a rewarding experience. In 1964 he became the director of the Research and Training School, and then in 1972 he was appointed Director-Secretary of the Indian Statistical Institute. He was named Jawaharlal Nehru Professor in 1976.

Let us now look at some of the highly influential books CR has published. The first was Advanced statistical methods in biometric research (1952) which he began writing while working for his doctorate at Cambridge. His aim, stated in the Preface, is:-
... to present a number of statistical techniques, keeping in view the requirements of both the student who questions the basis of a particular method employed and the practical worker who seeks a recipe for the reduction of his data.
His next book Linear statistical inference and its applications (1965) was more mathematical, and designed to be used for mathematical statistics courses in universities. He published Computers and the Future of Human Society in 1970, and in the following year, jointly with Sujit Kumar Mitra, he published Generalized inverse of matrices and its applications. R J Plemmons explains that they:-
... present a general unified treatment of the concept of inversability of singular, and in general rectangular, matrices over the complex field. ... the material should make a good text for a one term course in matrix algebra or mathematical statistics.
Further books followed such as Characterization Problems of Mathematical Statistics (Russian edition 1972, English translation 1973) with A Kagan and Yu V Linnik.

In 1979 CR left the Indian Statistical Institute, just before reaching the mandatory retirement age of sixty, and went to the United States where he was appointed to a University Professorship at the University of Pittsburgh. In 1988 he moved to Pennsylvania State University in Pittsburgh where he was named Eberly Family Professor in Statistics. After retiring he became Emeritus Eberly Professor.

Books he published after moving to the United States include: (with J Kleffe) Estimation of variance components and applications (1988), Statistics and truth. Putting chance to work (1989), (with D N Shanbhag) Choquet-Deny type functional equations with applications to stochastic models (1994), (with H Toutenburg) Linear models. Least squares and alternatives (1995), and (with M B Rao) Matrix algebra and its applications to statistics and econometrics (1998).

C R Rao has received so many honours that it would be quite impossible to list them all. Here is a selection to give an impression of the high esteem that he is held in throughout the world. First we list some of the prizes and awards he has received: the S S Bhatnagar Prize of Council of Scientific and Industrial Research (1963), the Guy Medal in Silver of the Royal Statistical Society (1965), awarded the title of Padma Bhushan by the Indian Government (1968), received the Megnadh Saha Medal of the Indian National Science Academy (1969), the Jagdish Chandra Bose Gold Medal of the Bose Institute (1979), the Silver Plate of the Andhra Pradesh Academy of Sciences (1984), The Times of India listed Rao as one of the top 10 Indian scientists of all time (1988), awarded the Samuel S Wilks Memorial Award of the American Statistical Association (1989):-
For major contributions to the theory of multivariate statistics and applications of that theory to problems of biometry; for world wide activities as advisor to national and international organizations; for long time conscientious as a teacher, editor, author and founder of academic institutions; and for the great influence he has had on the applications of statistical thinking in different scientific disciplines, embodying over a career of more than 40 years the spirit and ideals of Samuel S Wilks.
Other awards include: the Mahalanobis Birth Centenary Gold Medal by the Indian Science Congress (1996), the Distinguished Achievement Medal from the American Statistical Association (1997):-
... for outstanding contributions to the development of methods, issues, concepts, and applications of environmental statistics ...
the Padma Vibhushan by the Government of India (2001), the National Medal of Science (2002), presented by President George W Bush on 12 June at a ceremony in the White House:-
... for his pioneering contributions to the foundations of statistical theory and multivariate statistical methodology and their applications, enriching the physical biological, mathematical, economic and engineering sciences ...
the Srinivasa Ramanujan Medal of the Indian National Science Academy (2003), the International Mahalanobis Prize of the International Statistical Institute (2003), and the Guy Medal in Gold of the Royal Statistical Society (2011):-
In the 115-year history of Royal Statistical Society, he is the 34th recipient of the award. He is the first Asian, first non-European and first non-American to receive the award.
He has been elected to the Royal Society of London (1967), the National Academy of Sciences, USA (1995), the American Academy of Arts and Science, the Indian National Science Academy, the Lithuanian Academy of Sciences, and the Third World Academy of Sciences. He was made an Honorary Member of the International Statistical Institute (1983), the International Biometric Society (1986), the Royal Statistical Society (1969), the Finnish Statistical Society (1990), the Portuguese Statistical Society, the Institute of Combinatorics and Applications, and the World Innovation Foundation.

He has been awarded thirty-three honorary degrees by universities in eighteen countries, in six continents, including: Andhra University, India (1967), Leningrad University, USSR (1970), Delhi University, India (1973), University of Athens, Greece (1976), Osmania University, Hyderabad, India (1977), Ohio State University, Columbus, USA (1979), Universidad Nacional de San Marcos, Lima, Peru (1982), University of the Philippines, Manila (1983), University of Tampere, Finland (1985), Indian Statistical Institute, India (1989), Université de Neuchâtel, Switzerland (1989), Colorado State University, Fort Collins, USA (1990), University of Hyderabad, India (1991), Agricultural University of Poznań, Poland (1991), Slovak Academy of Sciences, Bratislava, Slovakia (1994), University of Barcelona, Spain (1995), University of Munich, Germany (1995), Sri Venkateswara University, Tirupati, India (1996), University of Guelph, Canada (1996), University of Calcutta, India (2003), University of Pretoria, South Africa (2004), University of Rhode Island, Kingston, USA (2007), and Jawaharlal Nehru Technological University, Kakinada, India (2011).

The Journal of Quantitative Economics published a special issue in Rao's honour in 1991. The preface gives the following tribute:-
Dr Rao is a very distinguished scientist and a highly eminent statistician of our time. His contributions to statistical theory and applications are well known, and many of his results, which bear his name, are included in the curriculum of courses in statistics at bachelor's and master's level all over the world. He is an inspiring teacher and has guided the research work of numerous students in all areas of statistics. His early work had greatly influenced the course of statistical research during the last four decades. One of the purposes of this special issue is to recognize Dr Rao's own contributions to econometrics and acknowledge his major role in the development of econometric research in India.
C R Rao has served as president of five statistical societies: the Indian Econometric Society (1971-1976); the International Biometric Society (1973-1975); the Institute of Mathematical Statistics (1976-1977); the International Statistical Institute (1977-1979), and the Forum for Interdisciplinary Mathematics (1982-1984).

Finally we record that CR has many hobbies such as gardening, photography, cooking, and Indian classical dance. In Calcutta he played soccer and badminton with staff and students in the evenings. In the United Sates his relaxation is walking.

References (show)

  1. S Das Gupta, S K Mitra, P S S N V P Rao, J K Ghosh, A C Mukhopadhyay and Y R Sarma (eds.), C R Rao, Selected papers of C R Rao, Volume 1 (John Wiley & Sons, Inc., New York, 1994).
  2. S Das Gupta, S K Mitra, P S S N V P Rao, J K Ghosh, A C Mukhopadhyay and Y R Sarma (eds.), C R Rao, Selected papers of C R Rao, Volume 2 (John Wiley & Sons, Inc., New York, 1994).
  3. S Das Gupta, S K Mitra, B L S Prakasa Rao, S B Rao, J K Ghosh, A C Mukhopadhyay, P S S N V P Rao and Y R Sarma (eds.), C R Rao, Selected papers of C R Rao, Volume 3 (John Wiley & Sons, Inc., New York, 1995).
  4. C R Rao, Selected papers of C R Rao, Volume I (Indian Statistical Institute, Calcutta, 1989).
  5. A K Bera, The ET interview: Professor C R Rao, Econometric Theory 19 (2) (2003), 331-400.
  6. C M Cuadras, C R Rao: a life devoted to statistics (Catalan), Questiio (2) 19 (1-3) (1995), 7-12.
  7. M H DeGroot, A conversation with C R Rao, Statist. Sci. 2 (1) (1987), 53-67.
  8. Y Dodge, C Radhakrishna Rao's contributions to statistics, in Statistical data analysis and inference, Neuchâtel, 1989 (North-Holland, Amsterdam, 1989), 3-18.
  9. G P Patil, Banquet address in honor of C R Rao at the International Conference on Statistical Research in the 21st Century, Montreal, Canada, J. Statist. Res. 35 (1) (2001), 1-8.
  10. Publications of C R Rao, in Statistics and probability: essays in honor of C R Rao (North-Holland, Amsterdam-New York, 1982), 713-722.
  11. S Puntanen and G P H Styan, A brief biography and appreciation of Calyampudi Radhakrishna Rao [b. 1920], with a bibliography of his books and papers, Linear Algebra Appl. 237/238 (1996), 1-40.
  12. C R Rao National Medal of Science Laureate, Bull. Inst. Combin. Appl. 36 (2002), 20-22.
  13. Short biographical sketch of C R Rao, J. Statist. Res. 34 (2) (2000), iv-vi.
  14. K Stavseth, MT4501 Project: C R Rao, 1920-Present, University of St Andrews (2007).

Additional Resources (show)

Honours (show)

Honours awarded to Calyampudi Radhakrishna Rao

  1. RSS Guy Medal in Gold 2011
  2. Popular biographies list Number 11

Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update March 2011