C. R. Rao, a giant in statistical theory who blazed the field over an 80-year research career died on August 22, 2023, two weeks short of his 103rd birthday.
Calyampudi Radhakrishna Rao was born in Karnataka, India, on September 10, 1920. Rao was known to be a very good student in mathematics among his juniors and classmates since his high school days in India (Efron B. et al, 2020, “C.R. Rao’s century,” Significance). His twelfth research article published in 1945 from India when Rao was just 25 years old broke the ground in the field of statistical sciences (Rao, C.R., Bull Calcutta Math Soc, 1945). That article created several newer theories such as the Rao–Blackwell Theorem, Cramér–Rao inequality, and Rao distances, and made statistics a more formidable force in science. Prior to this 1945 article, Rao published articles in the field of number theory independently and also by collaborating with S. Chowla and R.C. Bose—for example, Rao. C.R., 1942; Rao, C.R., 1944; Bose, R.C., Chowla, S., Rao, C.R.. 1945a,1945b. He served as president of the IMS, the International Biometrics Society and ISI, Netherlands. His passion for Statistics and applications continued till the end. Refer to IMS Bulletin’s special tribute, C.R. Rao at 100, for his education background and list of key honors and awards.
The IEEE, honoring Rao as an honorary life member in 2022, and the International Prize in Statistics 2023 citation mentioned how Rao’s path-breaking 1945 article shaped the field of statistics. The director of the Indian Statistical Institute recently wrote that C.R. Rao “will continue to be remembered as an influential figure who enriched the discipline of Statistics and who helped shape the Indian Statistical Institute. For that, we remain grateful to the entire [C.R.] Rao family.”
By 1960, when C.R. Rao was 40, he was already an established name in the world of statistics. The Cramér–Rao Theorem and Rao–Blackwellization had essentially become parts of most standard mathematical statistics textbooks and courses. Numerous highly prestigious awards and recognitions subsequently came his way during the 1960s, including the Indian government’s Padma Shri, Padma Bhushan, S.S. Bhatnagar Award for Science and Technology, Fellow of the Royal Society and a DSc from Cambridge University. But the best of Rao was yet to be seen. In fact, 1960–1980 was the prime period of Rao. This is the time when under his leadership as head of research and training, the Indian Statistical Institute became an institute of national importance and began to award degrees. Many of Rao’s PhD students from that period have subsequently established themselves as leading statisticians, probabilists and mathematicians. This was the era when Rao played an instrumental role in shaping the theories and introducing the methodologies for multivariate analyses, linear models, discriminant analysis, variance component estimation, selection and screening problems, experimental designs, matrix theory with special emphasis on generalized inverses, asymptotic theory and characterization of probability distributions.
What could not escape the notice of any mathematical statistician of that period was the publication of his book Linear Statistical Inference and Its Applications, about which Bradley Efron, who was then a graduate student, says, “When the book came out in 1965, it was a real eye-opener for me.” In fact, this was perhaps the first unified account of the entire field of mathematical statistics along with a solid foundation where the latest results known up to that point had been incorporated. This major book essentially defined how statistics was taught all over world in the ’70s and ’80s. He published more than 14 textbooks and about 40 edited books in the Elsevier Handbook of Statistics series. Many of his textbooks were translated in several languages in the world.
M-estimation, introduced by P. J. Huber, is an important extension of likelihood techniques and robust statistics. C.R. Rao had become very interested and, together with the students under his guidance, had made significant contributions to this area. In a paper published in Statistica Sinica, they established the CLT (central limit theorem) under very general conditions when the discrepancy function is convex. Later, they extended their results to the case where the discrepancy function is a difference of two convex functions, which covers almost all currently known results of M-estimation in linear models (see Bai, Rao, Wu, 1997).
In multivariate analysis, many statistics can be written as a function of sample means. Edgeworth expansions of the testing statistics can be used to calculate various error probabilities of multivariate tests and hence it is important here. A significant result on it was obtained by Bhattacharya and Ghosh (1978). Rao worked with his coauthor and established the Edgeworth expansion under the so-called “partial Cramér condition” (terminology coined by C.R. Rao), that is, the absolute value of the conditional Fourier transformation of one component given all other components has an upper limit less than one when the transformation variable tends to infinity.
It is well known that the CT scanner is an important medical diagnostic instrument. Based on projections in all directions, the inner structures of the human body can be precisely imaged. However, it does not work with moving parts, such as the left ventricle of the heart. As it can only take two orthogonal pictures at one time, the reconstruction can only rely on these two pictures. The cardiologist P.S. Reddy (Pittsburgh Medical School), jointly with C.R. Rao, P. Krishnaiah and others in the Center for Multivariate Analysis, applied a huge grant from Siemens to solve this problem. More details on this can be found in a tribute to C.R. Rao (Andrews et al, Notices of the AMS, 2022, https://www.ams.org/notices/202206/rnoti-p982.pdf).
Since Akaike proposed the AIC for model selection criterion in 1973, various information criteria were proposed in the literature. The most famous ones are AIC and BIC. It is easy to show that AIC is usually overestimating the true model, more precisely, it is not consistent in the sense that there is a positive probability that the true model is over estimated when the number of unknown parameters is bounded. In contrast, the BIC is frequently underestimated the true model. Due to this, C.R. Rao and collaborators proposed a GIC (general information criterion). See Rao, Wu, 1989; Review paper by Rao, Wu, 2001, Handbook of Statistics. Although GIC is more flexible, the penalty parameter needs to be decided. To this end, C.R. Rao, Bai and Wu (Rao’s PhD student) proposed the data-driven information criterion in Bai, Rao, Wu, 1999. C.R. Rao also spent substantial time during the 1980s in density estimation for directional data.
The intent of C.R. Rao’s book Statistics and Truth: Putting Chance to Work (2nd edn., 1997) was to explain the essence of statistics with real examples, for a broad readership. He introduced the following logical equation that attracted the scientific community:
Uncertain knowledge + Knowledge of the extent of uncertainty in it
= Usable knowledge.
This is a new way of thinking, and this logical equation is exceptionally useful for both understanding and explaining statistics.
Further, in the book, C. R. Rao express the main principle of data analysis in the form of a fundamental equation:
Data analysis = Answering specific questions
+ Providing information for new lines of research.
Rao distances and information geometry principles due to C.R. Rao were now extended to complex plane manifolds, for example Rao (Arni), 2022; Rao (Arni) and Steven, 2020: Steven and Rao (Arni), 2022; etc. These complex manifolds were shown to have implications in virtual tourism and universe climate analysis.
What is C.R. Rao’s legacy? Is it the simplicity of Cramér–Rao inequality? Is it the enigmatic Rao–Blackwellization? Is it the complexity of information in the data? Is it the industriousness of orthogonal arrays? Is it the ubiquity of the score statistic? Is it the fashionable characterizations? Is it the discerning discriminant analysis? Is it the enveloping book, Linear Statistical Inference? Is it the mettle-testing book, Advanced Statistical Methods in Biometric Research? We give up. C.R. Rao might have left this world. He will remain with us forever.
C.R. Rao, who despite being renowned as one of the world’s foremost mathematicians and statisticians, believed that the real value of numbers lay not in their ability to predict, but in their capacity to inspire and illuminate. Through the precision of digits and the elegance of probability, C.R. Rao taught us to seek truth in its purest form, to cut through the noise and find the signal that guides our understanding of the world. Rao, the genius, was not confined to equations. Behind the intense gaze that focused on complex algorithms was a heart that pulsed with a passion for humanity. C.R. knew how to bring out the best in others. To him, every individual was like a unique data point, possessing potential waiting to be unlocked. With unyielding patience and an open heart, C.R. Rao mentored, guided, and inspired countless individuals, ensuring that their potential was not only recognized but also realized. In a world often caught up in tempests of distraction, C.R. was able to see and share beauty. Whether it was the graceful arc of tennis volley, the subtle expression of a dancer, the giggle of child, or the delicate bloom of a flower, C.R. Rao celebrated the aesthetic marvels of our universe. As we bid farewell to this luminary, we are reminded not just of the legacy he leaves in awards, degrees, articles, textbooks, and lectures, but also, and more importantly, in the life lessons shared. Rao’s life serves as a testament to the belief that in seeking truth, sharing beauty, and uplifting others, we find the true essence of existence. The numbers have lost their greatest champion, but the world has gained from his immeasurable wisdom.
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Written by Ravindra Khattree, Zhidong Bai, Paula Caligiuri, Yasunori Fijikoshi, Yuehua Wu, Arni S.R. Srinivasa Rao, and corresponding author Marepalli B. Rao.