Dr CR Rao: The Pride of India

Dr Calyampudi Radhakrishna Rao, the India-born American mathematician and statistician has been awarded the 2023 International Prize in Statistics— equivalent of the Nobel Prize—for the monumental work done by him 75 years ago that revolutionized statistical thinking. This award shall be presented to the 102-year-old Dr Rao at the International Statistical Institute World Statistics Congress in Ottawa, Canada, in July 2023. 

Dr Rao reported this remarkable work in a paper, “Information and Accuracy Attainable in the Estimation of Statistical Parameters”, published in The Bulletin of the Calcutta Mathematical Society, Vol. 37, pp. 81-91, 1945. The origin of this paper is equally interesting: As Dr Rao, while teaching a course on estimation to the senior students of MA statistics class at Calcutta University in 1944, mentioned without proof Fisher’s information inequality for the asymptotic variance of a consistent estimate, a bright student in the class, VM Dandekar raised a question: “Whether such an inequality exists for the exact variance of an estimate in small samples?” Dr Rao replied: “I would try and let you know”. That night he read Fisher’s papers but could not get any clue, as Fisher did not give satisfactory proof of his inequality. So, he began to think about the problem independently. Trying for a couple of hours with an assumption that an “estimate T is unbiased for a parametric function ф (θ) … he could arrive at the desired inequality where I (θ) is information defined by Fisher”. In the next lecture, he presented this proof in the class.

That was how this seminal 1945 paper emanated demonstrating three fundamental results: First, the Cramer-Rao lower bound that provides a means for knowing when a method is as good as any method can be; the second, Rao-Blackwell Theorem that provides a means for transforming an estimate into a better—in fact, an optimal—estimate; and the third provides insights that pioneered a new interdisciplinary field, which has become popular as “information geometry” that has been recently used for the optimization of Higgs boson measurements at the Large Hadron Collider, the world’s largest particle accelerator and also recently in research on radars, antennas, in the advancements in AI, data science, signal processing, shape classification, and image segregation. This paper was included in Breakthroughs in Statistics, Vol. I.

That being the significance of the paper of his twenties—to be precise written at the age of 25— which opened new areas of research and generated a number of further technical terms such as Quantum Cramer-Rao Bound providing sharper versions of Heisenberg’s Principle of Uncertainty in Quantum Physics, it is not surprising that Guy Nason, Chair of the International Prize in Statistics Foundation, said: “In awarding this prize, we celebrate the monumental work by CR Rao that not only revolutionized statistical thinking in its time but also continues to exert enormous influence on human understanding of science across a wide spectrum of disciplines.”

Dr Rao has indeed started working on the design of experiments and some problems in multivariate analysis on his own right from 1941 and published papers. For his MA statistics degree of Calcutta University, he submitted thesis in 1943 on “Researches in the Theory of the Design of Experiments and Distribution Problems Connected with Bivariate and Multivariate Populations” in lieu of two practical papers. This thesis was indeed a reflection of his “early interests in four areas of statistics: the design of experiments, linear methods, multivariate analysis, and the characterization of probability distributions”, which he says “kept him engaged for the next 60 years”. He received his MA degree in statistics in 1943 from Calcutta University with a first class first and the marks that he scored remained an unbroken record to date.

Working as TA at Indian Statistical Institute from 1944 to 1946, he had an eventful research career: researched on combinatorics with reference to design of experiments and wrote a number of papers jointly with RC Bose and SD Chowla; developed a general theory of least squares without any assumptions on the concomitant variables; and found a test for redundancy of a specified set of variables in multivariate analysis. On an inquiry of his colleague, “Whether the Neyman-Pearson theory could be used to test a hypothesis about a parameter when the alternatives are one-sided”, he, saying “yes”, gave him an immediate solution. Later this was published as a note, an extension of which to the multiparameter case led to his much-celebrated score test paper, as an alternative to the likelihood ratio and Wald tests—“Large Sample Tests of Statistical Hypotheses Concerning Several Parameters with Applications to Problems of Estimation”—which was included in Breakthroughs in Statistics, Vol. III of 1948. 

Books authored by Dr CR Rao

1.         Linear Statistical Inference and Its Applications 2.         Advanced Statistical Methods in Biometric Research 3.         With S K Mitra. Generalized Inverse of Matrices and Its Applications 4.         With A Kagan and Yu V Linnik. Characterization Problems of Mathematical Statistics 5.         Computers and the Future of Human Society 6.         With R K Mukerjee and J C Trevor. Ancient Inhabitants of Jebel Moya 7.         With P C Mahalanobis and D N Majumdar. Anthropometric Survey of the United Provinces 8.        With D N Majumdar. Race Elements of Bengal: A Quantitative Study 9.         With A Matthai and S K Mitra. Formulae and Tables of Statistical Work 10.     With J Kleffe. Estimation of Variance Components and Its Applications 11.      Statistics and Truth: Putting Chance to Work 12.      With D N Shanbhag. Choquet-Deny Type Functional Equations with Applications to Stochastic Models 13.      With H Toutenburg. Linear Models: Least Squares and Alternatives 14.      With MB Rao. Matrix Algebra and Its Applications to Statistics and Econometrics

At the request of the Department of Anthropology, Cambridge University to send a person to analyze measurements made on human skeletons by the University Museum of Archeology and Anthropology to trace the origin of the people of Jebel Maya using the method of Mahalanobis D-square statistic, Prof Mahalanobis sent Rao to Cambridge in 1946. Working at the University Museum during 1946-48 as a visiting scholar, Rao developed new methods of analysis of multiple measurements and used them to analyze the data. These results were published in the book, Ancient Inhabitants of Jebel Maya.

Simultaneously, he worked for PhD on “Statistical Problems of Biological Classification” under the guidance of Ronald F Fisher and submitted his dissertation to Cambridge University in 1948. Based on his dissertation, he published three papers: “Tests with Discriminant Functions in Multivariate Analyses”, “Utilization of Multiple Measurements in Problems of Biological Classification” and “Tests of Significance in Multivariate Analysis”, laying the foundation for the modern theory of multivariate methodology. He worked in this area all through his career and made significant contributions.

On returning to India in July 1949, he was appointed as professor at the young age of 28 “in recognition of his creativity”. During his long tenure at ISI (1941-78), he worked in various capacities: in 1960 became professor and head of the Research and Training School; in 1964 became the director of RTS; and finally in 1972, on the death of Prof Mahalanobis, he took over his designation: became Director of ISI. In 1976 he, accepting the Jawaharlal Nehru professorship, continued to work at ISI till he retired in 1984.

During his tenure of 40 years at ISI, Dr Rao made huge contributions towards its growth:   published 201 research papers spreading across varied fields of statistics. In a series of papers Dr Rao developed a theory behind a set of combinatorial arrangements called orthogonal arrays. These are widely used in industrial experimentation to determine the optimum mix of factors for maximizing output. Notable among them is “the foundation blocks of what is now quite famously known as Taguchi methodology for applying statistics to improve the quality of manufactured goods”. 

He also made significant contributions to results on the characterization of probability distributions. This led to the emergence of technical terms such as Rao’s damage model; Rao-Rubin Theorem; Kagan, Linnik and Rao Theorems.

In 1955, using the idea of canonical correlations to estimate dominant factors that explain the correlation between measurements, Rao published a paper. This method is called Rao’s canonical factor analysis. 

He launched a variety of courses to train statisticians to work in different applied areas, established research units to work on special projects in subjects such as economics, sociology, psychology, genetics, anthropology, geology, etc., and developed a four-year program for B Stat and a two-year M Stat program at ISI. Immediately after returning from Cambridge, he launched the PhD program in theoretical statistics and probability with D Basu as his first PhD student. During this period, he also won many awards: Bhatnagar Award in 1963, FRS in 1967, and Padmabhushan in 1968.

Dr Rao is also an accomplished teacher. His former students said that he made difficult mathematical concepts so simple, using good humour and interesting anecdotes—for instance, he used to give “tailor’s measurements” as an example of a “vector”. Listening to his lecture on multivariate analysis, a student of Pen State said: “Watching Professor Rao lecture is like watching a skilled artist at work, with every statistical function and procedure at his command”. He has also mentored many scholars. According to Mathematics Genealogy Project, he has nearly 650 academic descendants. This noble teacher “followed the policy of not associating my [his] name with papers arising out of their [research scholars guided by him] thesis, even when I [he] had a large input”.

After his retirement, Dr Rao, being desirous of having a job to continue his research work with no administrative burden, accepted positions of distinguished professorships offered by American universities. He worked for 8 years as a University Professor at the University of Pittsburgh and as Eberly Chair Professor of Statistics at The Pennsylvania University for 13 years continuing his research in diverse areas of statistics. Later, working as the Director of the Center for Multivariate Analysis at Penn State till 2003, he continued his research. Currently, he is Professor Emeritus at Pennsylvania University and Research Professor at the University at Buffalo.

Working from the US on wide fields in statistical theory and practice, Dr Rao has published 274 papers. In the 80s, Rao introduced a series of measures that quantify information and variation in data. In collaboration with Burbea, he introduced one such series of measures based on information-theoretic notions of entropy. He also developed analysis of diversity (ANODIV), which generalized the idea of analysis of variance (ANOVA). He also introduced a general measure of variance known as Rao’s Quadratic Entropy, which is being used by ecologists. He also introduced the concept of cross-entropy in a paper written jointly with Nayak.

Continuing his research on the characterization of probability distributions in the US in collaboration with Khatri and Shanbhag he summarized the results in the book, Choquet-Deny Functional Equations with Applications to Stochastic Models written jointly with Shanbhag. 

Along with Ka-Sing Lau, Dr Rao published a paper in 1984 introducing a new equation in the area of functional equations in mathematics called the Cauchy functional equation offering a general technique for characterizing probability measures and solving problems of stochastic modelling of data for statistical analysis.

He made a notable contribution to the theory of matrices by introducing the concept of a generalized inverse of a matrix, both for singular and non-singular matrices, and through it offering a general technique for characterizing probability measures and solving problems of stochastic modelling of data for statistical analysis. Later he came up with a unified theory of least squares estimation. Generalizing Kantorovich inequalities on matrices, he enabled their use in statistics that had thrown open a new area of research in matrix algebra.

Dr Rao has also made significant contributions to econometrics. It started in 1947 with his answering a foundational econometric problem raised by Ragnar Frisch to J Neyman (Econometrica). He founded the Indian Econometric Society in 1971 and nurtured it for long. Much of his work such as the Cramer-Rao efficiency bound, the Rao-Blackwellization, the score test, MINQUE theory, the F-test, quadratic entropy, Rao’s distance measure, g-inverse etc., had a tremendous impact on the practice of econometrics. Suffice it to say that Dr Rao made fundamental contributions to the four stages of modelling in economics viz., specification, identification, estimation, and testing of hypotheses.  

Dr Rao, however, has not confined himself just to statistics. Interestingly, he has intense love for music and dance. Acting as president of Andhra Association in Calcutta he organized several cultural events. While in Delhi, he acted as the president of the Kuchipudi Dance Academy. He pursues his hobbies of photography and gardening. Some of the photos taken by him appeared in newspapers and photographic journals.

This living legend, whose work in statistics has far-reaching implications for varied fields such as economics, genetics, anthropology, geology, national planning, demography, biometry, medicine, quantum physics, data analytics, AI, etc., was born on September 10, 1920 in a Telugu family of Smt Laxmikanthamma and Sri Doraswamy Naidu in Hadagali village, Karnataka. His father was a police inspector and owing to his job their family was to frequently move from one place to the other. He thus completed his schooling in Gudur, Nuzividu, Nandigama, and Visakhapatnam of Andhra Pradesh. This however did not affect his studies, for his parents fostered children’s “innate abilities with proper guidance”, in “an environment conducive to study”.

Right from his earliest years, Dr Rao had an interest in mathematics. “Appreciating his interest in mathematics and hoping he would eventually …get a doctorate degree”, his father presented him with a book, Problems for Leelavathi and advised him to work out 5-10 problems in the book every day. This, he says aroused further interest in him to pursue mathematics. His mother, being a stern disciplinarian, had set up a daily routine for the boys to follow: play from 4 to 6 p.m.; then study for 2-3 hours and go to sleep; then wake up at 4 a.m. and study in the quiet hours of morning when the mind is fresh.

After completing high school Dr Rao joined AVN College, Visakhapatnam, for Intermediate course. Though the Chandrasekara Iyer scholarship in physics that he won for two consecutive years at AVN College induced in him an aptitude for physics, he finally decided to stick to mathematics, which was his father’s wish. Later joining Andhra University, he obtained BA (H0ns) degree in mathematics with first class first at the age of 19. Wanting to pursue a research career in mathematics, he applied for a research scholarship at Andhra University, but due to certain administrative formalities, it was turned down.

He then applied for a job of mathematician to work in an army survey unit. He was called for an interview in Calcutta. He attended the interview but did not get the job due to his under-age. He, however, found something that kept him engaged for the rest of his life: A chance encounter with one, Mr Subramanyam that he had while staying in a South Indian Hotel in Calcutta, made him aware of ISI and the one-year training program that it offered in statistics and the ample scope for getting a job with a certificate of training from the ISI. Thinking that by joining the training program he could achieve his “twin objectives of getting a job and also testing my [his] abilities to do research”, he, “with a letter of recommendation to Prof PC Mahalanobis from VS Krishna, Vice Chancellor of Andhra University”, joined the training program at ISI.

Having thus “stumbled into Statistics by chance” and “as a last resort”, he never looked back. He has published 475 research papers, authored 14 books, edited 42 volumes of the Handbook of Statistics, guided 51 scholars for PhD, and received 38 honorary doctorates from universities in 19 countries spanning six continents. In 1974, Cambridge University awarded him Sc.D. degree based on a peer review of his publications and he was made an Honorary Life Fellow of King’s College, Cambridge.

In recognition of his contributions to the theory of multivariate statistics and its application to problems of biometry, he was awarded the Samuel S Wilks Memorial Medal in 1989. He was honored by President George Bush at the White House with the National Medal of Science on June 13, 2002 “… 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…”

And now the Nobel equivalent prize. On this happy occasion, I join the nation in saluting him for his eight decades of pioneering research in statistics—an inspiring and lasting legacy—and placing ISI on a global pedestal and praying for his many more happy and functional years of life.

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