The work that made CR Rao a legend in statistics

Aug 23, 2023 05:51 PM IST

CR Rao made his reputation at the age of 24 in 1945 when he wrote a seminal paper in the Bulletin of the Calcutta Mathematical Society

NEW DELHI: Pioneering statistician C Radhakrishna Rao died in the United States on Wednesday, statisticians in India said. Rao, 102, was professor emeritus at Pennsylvania State University.

Penn State professor emeritus and preeminent statistician C.R. Rao (
Penn State professor emeritus and preeminent statistician C.R. Rao (

Rao, born in Bellary in 1920 (he would have turned 103 on September 10), made his reputation at the age of 24 in 1945, when he wrote a seminal paper in the Bulletin of the Calcutta Mathematical Society. He was then with the Indian Statistical Institute (ISI).

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“That one paper has changed the complete statistics scenario. There are three fundamental problems he has addressed, which have impacts on every field of science: social sciences engineering sciences, biological sciences, everywhere,” said Rao’s former PhD student Debasis Kundu, now a professor of statistics at IIT Kanpur.

Rao went on to make many more significant contributions over his long career, but the three results he described in the 1945 paper are recognised as his definitive achievements. In April this year, when he was chosen for the International Prize in Statistics, the announcement dwelt on those three results: the Cramér-Rao lower bound, the Rao-Blackwell Theorem, and insights that pioneered a field known as “information geometry”.

Quality of an inference

In statistics, one often needs to make estimates and inferences from data collected. Two of Rao’s 1945 results, the Cramér-Rao lower bound and the Rao-Blackwell Theorem, relate to the quality of such inferences.

In April, Probal Chaudhuri, a professor at ISI Kolkata, described these two concepts to HT. “These essentially give you an idea about how much accuracy you will get if you use a certain amount of data to draw a certain inference or get an estimate. Especially, the Cramér-Rao lower bound gives you the amount of uncertainty you have when you draw your inference,” he had said.

If the Cramér-Rao lower bound allows a statistician to assess how accurate an estimate is, the Rao-Blackwell Theorem is a procedure to improve that estimate. As described by the International Prize in Statistics Foundation, the theorem “provides a means for transforming an estimate into a better — in fact, an optimal — estimate”.

Independently, Swedish mathematician Harold Cramér and American statistician David Blackwell each established one of these results, which is why they are called the Cramér-Rao lower bound and the Rao-Blackwell theorem. Cramér would describe the lower bound in his book Mathematical Methods of Statistics in 1946; Rao was not aware of Cramér’s independent result when he arrived at it himself, according to a 2021 article in the International Statistical Review, co-authored by Nandini Kannan (Indo-US Science and Technology Forum) and Kundu (Rao’s former student).

The way Rao reached this breakthrough is the stuff of legend. Back in 1944, one of the existing methods for assessing the accuracy of estimates was known as the Fisher Information, named after the British statistician and geneticist RA Fisher. This, however, works only for large data samples, while Rao’s results work for samples of any size.

Rao was teaching Fisher’s results in class when a student asked what would happen if the sample size was not large. Rao is said to have gone back, worked on it and established his famous result, all in 24 hours.

The impact

The Cramér-Rao lower bound and the Rao-Blackwell theorem have applications in practically every field where data is analysed and interpreted. The former has been used in signal processing, spectroscopy, radar systems, multiple image radiography, risk analysis, and quantum physics, while the Rao-Blackwell process has been applied to stereology, particle filtering, and computational econometrics, among others, the International Prize in Statistics Foundation noted.

Rao’s paper was also one of the earliest to approach probability models with differential geometry (the study of certain shapes and spaces using algebra and calculus). This was the third of the celebrated results. It pioneered the field called information geometry, which is the study of probability theory using differential geometry.

Information geometry has been used to aid the understanding of Higgs boson measurements at the Large Hadron Collider, in recent research on radars and antennas, and in advancements in artificial intelligence and signal processing.

Signal processing, in which both the Cramér-Rao lower bound and information geometry has applications, is a field of study that involves the analysis of signals such as sound, images, or potential fields. It was on statistical signal processing that Kundu worked under Rao’s supervision; he said Rao was possibly the first statistician to recognise the importance of statistics in signal processing.

“I don’t think too many people know about Rao’s contribution to the field of signal processing, which is an area of electrical engineering. Rao made some significant contributions to statistical signal processing in the 1990s,” Kundu said. “He always tried to think in a different way rather than the traditional way. That made him different.”

His legacy

Technical terms bearing Rao’s name appear in textbooks on statistics and other subjects in which his work has applications. Apart from Cramér-Rao lower bound and Rao-Blackwell Theorem, other such terms include Fisher-Rao Theorem, Rao Distance, and Rao’s Orthogonal Arrays.

Rao’s work has earned him the Padma Bhushan (1968) and the Padma Vibhushan (2001), the Shanti Swarup Bhatnagar Award (1963) and the India Science Award (2009), as well as the National Medal of Science (2002) in the US, besides the International Prize in Statistics this year.

The Indian government has instituted a biennial ‘The Professor C R Rao’ Award in statistics. while the CR Rao Advanced Institute of Mathematics, Statistics and Computer Science and Prof C R Rao Road in Hyderabad are named after him. Pennsylvania State University has instituted a C R and Bhargavi Rao Prize in Statistics.

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    Puzzles Editor Kabir Firaque is the author of the weekly column Problematics. A journalist for three decades, he also writes about science and mathematics.

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