The Indian-American statistician Calyampudi Radhakrishna Rao has been awarded the 2023 International Prize in Statistics, which is statistics’ equivalent of the Nobel Prize. It was established in 2016 and is awarded once every two years to an individual or team “for major achievements using statistics to advance science, technology and human welfare.”
Prof. Rao, who is now 102 years old, is a ‘living legend’ whose work has influenced, in the words of the American Statistical Association, “not just statistics” but also “economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine”. The citation for his new award reads: “C.R. Rao, a professor whose work more than 75 years ago continues to exert a profound influence on science, has been awarded the 2023 International Prize in Statistics.”
What was Rao’s 1945 paper about?
Rao’s groundbreaking paper, ‘Information and accuracy attainable in the estimation of statistical parameters’, was published in 1945 in the Bulletin of the Calcutta Mathematical Society, a journal that is otherwise not well known to the statistics community. The paper was subsequently included in the book Breakthroughs in Statistics, 1890-1990.
This was an impressive achievement given Rao was only 25 at the time and had just completed his master’s degree in statistics two years prior.
He would go on to do his PhD in 1946-1948 at King’s College, Cambridge University, under the supervision of Ronald A. Fisher, widely regarded as the father of modern statistics.
The Cramér-Rao inequality is the first of the three results of the 1945 paper. When we are estimating the unknown value of a parameter, we must be aware of the estimator’s margin of error. Rao’s work provided a lower limit on the variance of an unbiased estimate for a finite sample. The result has since become a cornerstone of mathematical statistics; researchers have extended it in many different ways, with applications even in quantum physics, signal processing, spectroscopy, radar systems, multiple-image radiography, risk analysis, and probability theory, among other fields.
In an article published in the journal Statistical Science in 1987, the American statistician Morris H. DeGroot set out an intriguing story (corroborated by Rao’s own account) of how Rao arrived at the lower limit. Prof. Fisher had already established an asymptotic (i.e. when the sample size is very large) version of the inequality, and it seems a student had asked Rao, “Why don’t you prove it for finite samples?” in 1944. A then-24-year-old Rao did so in under 24 hours!
The second outcome of the 1945 paper was the Rao-Blackwell Theorem, which offers a method to improve an estimate to an optimal estimate. The Rao-Blackwell theorem and the Cramér-Rao inequality are both related to the quality of estimators.
A new interdisciplinary area called ‘information geometry’ was born as a result of the paper’s third finding. This field integrated principles from differential geometry into statistics, including the concepts of metric, distance, and measure. Erich L. Lehmann, a renowned statistician, said in 2008 that “this work [of Rao’s] was before its time and came into its own only in the 1980s”.
So overall, Rao’s 1945 paper made an outstanding contribution, boosting the development of modern statistics and its widespread application in modern research. In a 2008 book, Reminiscences of a Statistician: The Company I Kept, Lehmann also discussed the generative nature of the paper – i.e. the goldmine of insights that it was – and acknowledged that “several of my early papers grew out of Rao’s paper of 1945”.
How did Rao enter the field of statistics?
The Australian statistician Terry Speed claimed that the “1940s were ungrudgingly C.R. Rao’s. His 1945 paper … will guarantee that, even had he done nothing else – but there was much else.”
Indeed, one of Rao’s papers in 1948 offered a novel generic approach to testing hypotheses, now widely known as the “Rao score test”. In fact, the three test procedures – the likelihood ratio test of Jerzy Neyman and E.S. Pearson (1928), the Wald test (1943) of Abraham Wald, and the Rao score test (1948) – are sometimes called “the holy trinity” of this branch of statistics.
Rao also contributed to orthogonal arrays, a concept in combinatorics that is used to design experiments whose results are qualitatively good, as early as 1949. A 1969 Forbes article described it as “a new mantra” in industrial establishments.
Given the magnitude and relevance of his contributions, it might seem surprising that Rao entered the field of statistics by chance.
Despite scoring first in mathematics at Andhra University, a 19-year-old Rao didn’t secure a scholarship there for administrative reasons. He was also rejected for a mathematician’s job at an army survey unit because he was judged to be too young.
When he was staying at a hotel in Calcutta, he met a man who was employed in Bombay and had been sent to Calcutta to be trained at the Indian Statistical Institute. He asked Rao to apply to the institute as well. Rao did so, for a year-long training programme in statistics, hoping the additional qualification would help him land a job.
P.C. Mahalanobis, then director of the institute, replied promptly and Rao was enrolled. That marked the beginning of a four-decade-long stay at the institute. Rao retired in 1979 and afterwards settled in the U.S.
The first half of the 20th century was the golden period of statistical theory in general, and Rao is undoubtedly one of the reasons for this being the case, thanks to his mathematical ingenuity. In the words of the late mathematician Samuel Karlin, Rao’s contributions to statistical theory have “earned him a place in the history of statistics”.
Indian statisticians also owe Prof. Rao gratitude for his enormous contributions to the growth of statistics in the country, notably at the Indian Statistical Institute (where this author works). As Lehmann wrote, Rao was “the person who did the most to continue Mahalanobis’s work as a leader of statistics in India.”
Atanu Biswas is professor of statistics, Indian Statistical Institute, Kolkata.