# CR Rao, a dean of statistics, turns 100

CR Rao shaped the dramatic growth of mathematical statistics in the 20th century, refining and restructuring it from its somewhat ad hoc origins

Stephen Stigler, a historian of statistics, proposed the law that no scientific discovery bears the name of its original discoverer. Although there is the slightest exaggeration in Stigler’s law of eponymy, Calyampudi Radhakrishna Rao might agree with his colleague’s judgment. The widely used statistical test in econometrics, widely known as the Lagrange multiplier test, was developed by Professor Rao in 1947. This is not even recognized by most Indian econometricians. It is also not widely recognized by the general public in our country as it deserves.

At the dawn of an era of big data, data analytics, machine learning, bioinformatics and artificial intelligence, we have more reasons to honor and celebrate Professor Rao’s contributions and achievements. Along with a handful of others like Egon Pearson, Ronald Fisher, Andrey Kolmogorov, and Jerzy Neyman, he shaped the dramatic growth of mathematical statistics in the twentieth century, refining and restructuring it from its origins somewhat ad hoc, making this new era possible. . And it is remarkable that it is almost entirely a “product of India” – as the website of the CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, affiliated with the University of Hyderabad, puts it. It turns 100 on September 10, and continues to guide and inspire students and practitioners of statistics as it has over the past eight decades.

Calyampudi Radhakrishna Rao was born as the eighth child of CD Naidu and A. Laxmikanthamma in the small town of Huvina Hadagali in present-day Karnataka (in the former province of Madras). He was an exceptionally good student and obtained a BA (Hons.) In Mathematics from Andhra University with distinction and first rank. He hoped to get a scholarship for graduate studies in mathematics, but was unable to achieve this goal. World War II had started by this time, and he decided to apply for a job in the Army’s North African Investigative Unit. He visited Calcutta (as it was then called) for this purpose, but he was not qualified for the post. Destiny, in the form of a friend of his, took him to the nascent Indian Institute of Statistics; the legendary Prasanta Chandra Mahalanobis had created the institute ten years earlier. The young CR Rao joined a training program offered by ISI.

In 1943, Professor Rao completed his Masters in Statistics from the University of Calcutta, still with a first rank. By the age of 22, he had already published seven technical papers in mathematical statistics. Mahalanobis recognized the potential of this young man and CR Rao was delegated to the University of Cambridge in 1946 to assist in the analysis of certain anthropological data. While at Cambridge, he enrolled in a doctorate under the supervision of renowned professor Ronald A. Fisher. He returned to ISI in 1948 after obtaining his doctorate. and was appointed professor there the following year.

**ISI Veteran**

Professor Rao has spent more than three and a half decades at ISI, and he has served the institution in a variety of capacities: as Director and Director of the ISI Research and Training School, as Professor Jawaharlal Nehru of the Institute and soon. In 1972, he succeeded Mahalanobis as director. Professor Rao has thus played a key role in building the ISI’s worldwide reputation. In a speech at the institute, Professor Fisher once pointed out that more than half of the qualified statisticians working in the world were Indians, for some time. Most of them were students of Professor Rao.

Upon retirement from ISI in 1979, he joined the University of Pittsburgh as a professor and then joined Pennsylvania State University as the Eberly Family Chair Professor of Statistics. He was director of the Center for Multivariate Analysis at Penn State until 2008. One statistic about this statistician is simply astounding: he has published over 270 journal articles since his retirement from the Indian Statistical Institute.

It is hardly possible to describe the outlines of Professor Rao’s research in an article like this. We will discuss some outstanding results of his work. In 1922, Professor Fisher had defined a metric that conveys the amount of information a sample can provide about the value of an unknown population parameter. Professor Rao (and Harald Cramér, independently) proved that any unbiased estimator of the parameter must have a variance greater than the reciprocal of Fisher’s metric. This is the well-known Cramér-Rao lower bound, one of Professor Rao’s most renowned contributions. This result, and the famous Rao-Blackwell theorem (independently discovered by DH Blackwell), appeared in a famous article that Professor Rao published in 1945 – when he was only 24 years old. As mentioned earlier, in a 1947 article, Professor Rao presented the Rao Test score; this test has some practical advantages over the other two tests commonly used for the same purpose – the Wald test and the likelihood ratio test.

Asked by Frank Nielsen of Ecole Polytechnique (France) on his three major contributions, Professor Rao listed orthogonal matrices (OAS) and quadratic entropy (QE) other than score statistic as the main ones. The OAS was then used by Genichi Taguchi of Japan to develop Taguchi’s well-known methods for improving the quality of manufactured products. QE is a perfect diversity measure that can be used to perform diversity analysis (ANODIV) of any order.

As India began to build its institutional infrastructure after independence, Professor Rao was instrumental in establishing our national statistical framework. During the 1960s, he was Chairman of the Statistics Committee of the Government of India. He also chaired the Mathematics Committee of the Atomic Energy Commission and was a member of the Science and Technology Committee. He was the driving force behind ISI’s International Center for Statistical Education (ISEC), where students and government statisticians from developing countries could learn statistical techniques and methods of establishing their national statistical offices. Statistics, as we all know today, are an indispensable tool in fields as diverse as economics, psychology, management and epidemiology. Professor Rao anticipated this development and created units at ISI to train statisticians in these various applied disciplines.

**Exemplary teacher**

He had the reputation of being an exemplary teacher. When asked about the greatest achievement of his life, he was quick to respond that it was his students’ outstanding contribution to statistical theory and practice. He supervised 50 students in their doctoral research. He has been the mentor, and in some cases also the colleague, of many outstanding statisticians and mathematicians. The late Debabrata Basu, who was a professor at ISI and later at Florida State University, is one of them. Another is SR Srinivasa Varadhan (aka Raghu Varadhan), professor at the Courant Institute of Mathematical Sciences in New York City and winner of the Abel Prize in 2007. Others like R. Ranga Rao, KR Parthasarathy and the late VS Varadarajan are among the many distinguished academics mentored by Professor Rao.

**Prizes and distinctions**

He has received countless awards and honors. He received the Padma Vibhushan in 2001. The then President of the United States, George W. Bush, awarded him the National Medal of Science in 2002. He received the Shanti Swarup Bhatnagar Award from Prime Minister Nehru in 1963. The American Statistical Association awarded him the Samuel S Wilks Memorial Award in 1989. In 2011, he became the first non-European and non-American to receive the Guy Gold Medal from the Royal Statistical Society of the United Kingdom. Professor Rao has received 38 honorary doctorates from universities in 19 different countries.

The extent to which a company recognizes and celebrates its true heroes is an indicator of the cultural and institutional climate it has built to produce, retain and nurture its talent. In our recent history, India has produced very few scientists whose output can compete with that of Professor Rao. It deserves to be known, appreciated and celebrated by all the citizens of the country. Barely a week after celebrating Teachers’ Day, on his 100th birthday, we offer our humble tribute to this outstanding teacher, scholar and institution builder. Professor Rao currently lives in the city of Buffalo, New York, with the family of his daughter Tejaswini. The inspiration he provides should and will live on in the hearts of every Indian.

*(R. Hemant Kumar is Principal of Ettumanoorappan College, Kottayam, and Emmanuel Thomas is Assistant Professor at St. Thomas’ College, Thrissur)*