C. R. Rao Explained

C. R. Rao
Image Upright:	1.2
Birth Name:	Calyampudi Radhakrishna Rao
Honorific Suffix:	FRS
Birth Date:	10 September 1920
Birth Place:	Hoovina Hadagali, Madras Presidency, British India (present day Vijayanagara, Karnataka, India)^[1]
Citizenship:	Indian (until 1995) American (1995–2023)^[2]
Death Place:	Buffalo, New York, U.S.
Field:	Mathematics and Statistics
Education:	Andhra University (MA) University of Calcutta (MA) King's College, Cambridge (PhD, DSc)
Doctoral Advisor:	Ronald Fisher
Known For:	Cramér–Rao bound Rao–Blackwell theorem Orthogonal arrays Score test
Thesis Title:	Statistical Problems of Biological Classifications
Thesis Year:	1948
Prizes:	Padma Vibhushan National Medal of Science (2001) S. S. Bhatnagar Prize Guy Medal International Prize in Statistics (2023)

Calyampudi Radhakrishna Rao (10 September 1920 – 22 August 2023) was an Indian-American mathematician and statistician.^[3] He was professor emeritus at Pennsylvania State University and research professor at the University at Buffalo. Rao was honoured by numerous colloquia, honorary degrees, and festschrifts and was awarded the US National Medal of Science in 2002.^[4] The American Statistical Association has described him as "a living legend" whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine."^[4] The Times of India listed Rao as one of the top 10 Indian scientists of all time.^[5]

In 2023, Rao was awarded the International Prize in Statistics,^[6] an award often touted as the "statistics' equivalent of the Nobel Prize".^[7] Rao was also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on raising South Asian cardiovascular disease awareness.^[8]

Early life

C. R. Rao was the eighth of the ten children born to a Telugu family in Hoovina Hadagali, Bellary, Madras Presidency, Britain ruled India (now in Vijayanagara, Karnataka, India). His schooling was completed in Gudur, Nuzvid, Nandigama, and Visakhapatnam, all in the present state of Andhra Pradesh.^[1] He received an MSc in mathematics from Andhra University and an MA in statistics from Calcutta University in 1943.^[4] He obtained a PhD degree at King's College, Cambridge, under R. A. Fisher in 1948, to which he added a DSc degree, also from Cambridge, in 1965.

Career

Rao first worked at the Indian Statistical Institute and the Anthropological Museum in Cambridge. Later he held several important positions, as the Director of the Indian Statistical Institute, Jawaharlal Nehru Professor and National Professor in India, University Professor at the University of Pittsburgh and Eberly Professor and Chair of Statistics and Director of the Center for Multivariate Analysis at Pennsylvania State University.^[9]

As Head and later Director of the Research and Training School at the Indian Statistical Institute for a period of over 40 years, Rao developed research and training programs and produced several leaders in the field of Mathematics. On the basis of Rao's recommendation, the Asian Statistical Institute (ASI), now known as the Statistical Institute for Asia and Pacific, was established in Tokyo to provide training to statisticians working in government and industrial organisations.^[10]

Among his best-known discoveries are the Cramér–Rao bound and the Rao–Blackwell theorem both related to the quality of estimators.^[11] Other areas he worked in include multivariate analysis, estimation theory, and differential geometry. His other contributions include the Fisher–Rao theorem, Rao distance, and orthogonal arrays. He was the author of 15 books^[9] and authored over 400 journal publications.

Rao received 38 honorary doctoral degrees from universities in 19 countries around the world and numerous awards and medals for his contributions to statistics and science. He was a member of eight National Academies in India, the United Kingdom, the United States, and Italy. Rao was awarded the United States National Medal of Science, that nation's highest award for lifetime achievement in fields of scientific research, in June 2002. He was given the India Science Award in 2010, the highest honour conferred by the government of India in a scientific domain. He was most recently honoured with his 38th honorary doctorate by the Indian Institute of Technology, Kharagpur, on 26 July 2014 for "his contributions to the foundations of modern statistics through the introduction of concepts such as Cramér–Rao inequality, Rao–Blackwellization, Rao distance, Rao measure, and for introducing the idea of orthogonal arrays for the industry to design high-quality products."

Rao was the President of the International Statistical Institute, Institute of Mathematical Statistics (USA), and the International Biometric Society. He was inducted into the Hall of Fame of India's National Institution for Quality and Reliability (Chennai Branch) for his contribution to industrial statistics and the promotion of quality control programs in industries.

The Journal of Quantitative Economics published a special issue in Rao's honour in 1991: "Dr. Rao is a very distinguished scientist and a highly eminent statistician of our time. His contributions to statistical theory and applications are well known, and many of his results, which bear his name, are included in the curriculum of courses in statistics at bachelor's and master's level all over the world. He was an inspiring teacher and has guided the research work of numerous students in all areas of statistics. His early work had greatly influenced the course of statistical research during the last four decades. One of the purposes of this special issue is to recognise Dr. Rao's own contributions to econometrics and acknowledge his major role in thedevelopment of econometric research in India."

Areas of research contributions

• Estimation theory
• Statistical inference and linear models
• Multivariate analysis
• Combinatorial design
• Orthogonal arrays
• Biometry
• Statistical genetics
• Generalized matrix inverses
• Functional equations

Death

Rao died in Buffalo, New York on 22 August 2023, aged 102.^{[12] [13] [14] [15]}

Obituaries describing his professional contributions and impact on the society have been published in the IMS Bulletin https://imstat.org/2023/09/30/obituary-c-r-rao-1920-2023/ and Nature https://www.nature.com/articles/d41586-023-03200-5.

Awards and medals

• Guy Medal in Gold (2011) of the Royal Statistical Society
• India Science Award 2010 (the highest award in a scientific field presented by government of India)^[16]
• International Mahalanobis Prize (2003) of the International Statistical Institute^[17]
• Srinivasa Ramanujan Medal (2003) of the Indian National Science Academy
• Sardar Patel Lifetime Achievement International Award (Sardar Ratna) (2014) by Sardar Patel International Foundation, India^[18]
• President George W. Bush, on 12 June 2002, honoured him with the National Medal of Science, the highest award in US in the scientific field, as a "prophet of new age" with the citation, "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."
• Padma Vibhushan (2001) by the Government of India^[19]
• Mahalanobis Centenary Gold Medal (1993?) of the Indian Science Congress
• Wilks Memorial Award (1989) of the American Statistical Association
• Padma Bhushan (1968)
• Megnadh Saha Medal (1969) of the Indian National Science Academy
• Guy Medal in Silver (1965) of the Royal Statistical Society
• S. S. Bhatnagar Award (1963) of Council of Scientific and Industrial Research
• JC Bose Gold Medal of the Bose Institute
• Gold Medal of the University of Calcutta
• He was also awarded an honorary Doctor of Science by the University of Calcutta in 2003.^[20] Also honorary doctorates from a number of universities and institutes around the world.

• 2023 International Prize in Statistics

In his honour

• The Pennsylvania State University has established C. R. and Bhargavi Rao Prize in Statistics
• CR Rao Advanced Institute of Mathematics, Statistics and Computer Science
• National Award in Statistics established by Ministry of Statistics and Programme Implementation (MoSPI), Government of India.
• The road from IIIT Hyderabad passing along Central University of Hyderabad crossroads to Alind Factory, Lingampally is named as "Prof. C.R. Rao Road".^[21]

Selected publications

Books

• 2020. PEREIRA, B. de B.; RAO, Calyampudi Radhakrishna.; OLIVEIRA, F. B.;Statistical Learning Using Neural Networks: A Guide for Statisticians and Data Scientists with Python, CRC Press, London.
• 2017. Book Review: Multivariate Statistical Methods, A Primer
• 2016. (with Lovric, M.) Testing Point Null Hypothesis of a Normal Mean and the Truth: 21st Century Perspective
• 2009. (with PEREIRA, B. de B.) Data Mining Using Neural Networks: A Guide for Statisticians. State College, Pennsylvania, 2009. 186 p.
• 1999. (with Helge Toutenburg, Andreas Fieger et al.). Linear Models: Least Squares and Alternatives, 2ed, Springer Series in Statistics. Springer
• 1998. (with M. Bhaskara Rao). Matrix Algebra & Its Applications to Statistics & Econometrics. World Scientific
• 1997. Statistics And Truth: Putting Chance To Work, 2ed. Wspc
• 1996. Principal Component and Factor Analyses. PN
• 1996. Extensions of a Characterization of an Exponential Distribution Based on a Censored Ordered Sample.
• 1996. Bootstrap by Sequential Resampling. PN
• 1993. Applications of Multivariate Analysis. PN
• 1992. Signal Estimation, Multitarget Tracking and Related Areas.
• 1989. Multivariate Analysis and Its Applications. PN
• 1988. Linear Transformations, Projection Operators and Generalized Inverses; A Geometric Approach. PN
• 1984. Recent Results on Characterization of Probability Distributions: A unified Approach through Extensions of Deny's Theorem. PN
• 1973. Linear Statistical Inference and Its Applications, 2nd Edition. Wiley-Interscience
• 1973. On a Unified Theory of Estimation in Linear Models. PN
• 1973. (with A.M. Kagan et al.). Characterization Problems in Mathematical Statistics, Wiley Series in Probability & Mathematical Statistics.
• 1972. (with Sujit Kumar Mitra). Generalized Inverse of Matrices and Its Applications, Probability & Mathematical Statistics. John Wiley & Sons
• 1963. Essays on econometrics and planning. Statistical Pub. Society

Edited volumes

• 2015. (Ed. with Marepalli B. Rao). Handbook of Statistics 32: Computational Statistics with. Elsevier
• 2013. (Ed. with Venu Govindaraju). Handbook of Statistics 31: Machine Learning: Theory and Applications. North Holland
• 2012. (Ed. with Tata Subba Rao and Suhasini Subba Rao). Handbook of Statistics 30: Time Series Analysis: Methods and Applications . North Holland
• 2012. (Ed. with Ranajit Chakraborty and Pranab K. Sen). Handbook of Statistics 28: Bioinformatics in Human Health and Heredity. North Holland
• 2011. (Ed. with Dipak K. Dey). Essential Bayesian Models. North Holland
• 2011. (Ed. with Danny Pfeffermann). Essential Methods for Design Based Sample Surveys. North Holland
• 2009. (Ed. with Danny Pfeffermann). Handbook of Statistics 29A: Sample Surveys: Design, Methods and Applications. North Holland
• 2009. (Ed. with Danny Pfeffermann). Handbook of statistics 29B Sample Surveys: Inference and Analysis. North Holland
• 2007. (Ed. with J. Philip Miller and D.C. Rao). Handbook of Statistics 27: Epidemiology and Medical Statistics. North Holland
• 2006. (Ed. with Sandip Sinharay). Handbook of Statistics, Volume 26: Psychometrics. North Holland
• 2005. (Ed. with Dipak K. Dey). Handbook of Statistics 25: Bayesian Thinking, Modeling and Computation. North Holland
• 2005. (Ed.). Handbook of Statistics 24: Data Mining and Data Visualization. North Holland
• 2004. (Ed. with N. Balakrishnan). Handbook of Statistics 23: Advances in Survival Analysis. North Holland
• 2003. (Ed. with Ravindra Khattree). Handbook of Statistics 22: Statistics in Industry. North Holland
• 2001. (Ed. with N. Balakrishnan). Handbook of Statistics 20: Advances in Reliability, North-Holland Mathematics Studies. Elsevier
• 2001. (Ed. with D. N. Shanbhag). Handbook of Statistics 19: Stochastic Processes: Theory and Methods. North-Holland
• 2000. (Ed. with Pranab Kumar Sen). Handbook of Statistics 18: Bioenvironmental and Public Health Statistics. North-Holland
• 1998. (Ed. with N. Balakrishnan). Handbook of Statistics 17: Order Statistics: Applications. North-Holland
• 1998. (Ed. with N. Balakrishnan). Handbook of Statistics 16: Order Statistics: Theory & Methods. North-Holland
• 1997. (Ed. with G.S. Maddala). Handbook of Statistics 15: Robust Inference. Elsevier
• 1996. (Ed. with G.S. Maddala). Handbook of Statistics 14: Statistical Methods in Finance. Elsevier
• 1996. (Ed. with S. Ghosh). Handbook of Statistics 13: Design and Analysis of Experiments. North Holland
• 1994. (Ed. with G.P. Patil). Handbook of Statistics 12: Environmental Statistics. Elsevier
• 1994. (Ed. with G.P. Patil). Multivariate Environmental Statistics., Volume 6 in North-Holland Series in Statistics and Probability. North Holland
• 1993. (Ed. with G. S. Maddala and H. D. Vinod). Handbook of Statistics 11: Econometrics, Techniques and Instrumentation in Analytical Chemistry. Elsevier
• 1993. (Ed. with N. K. Bose). Handbook of Statistics 10: Signal Processing and its Applications, Elsevier
• 1993. (Ed.). Handbook of Statistics 9: Computational Statistics. North-Holland
• 1991. (Ed. with R. Chakraborty). Handbook of Statistics 8: Statistical Methods in Biological and Medical Sciences. North-Holland.
• 1988. (Ed. with Krishnaiah P.R.). Handbook of Statistics 7: Quality Control and Reliability.
• 1988. (Ed. with P. R. Krishnaiah). Handbook of Statistics 6: Sampling. North-Holland.
• 1969. (Ed. with R. C. Bose, I. M. Chakravarti et al.). Essays in Probability and Statistics, Monograph Series in Probability and Statistics. The University of North Carolina Press

Textbooks

• 1970. Advanced Statistical Methods in Biometric Research. Macmillan

Collected works

• 1996. Selected Papers of C.R. Rao 3. (Ed). S. Das Gupta et al. Wiley-Interscience
• 1995. Advances in Econometrics and Quantitative Economics, Essays in Honor of Professor C.R. Rao. (Eds.) G. S. Maddala, T. N. Srinivasan, and Peter C. B. Phillips. Wiley-Blackwell
• 1994. Selected Papers of C.R. Rao 2. (Ed). S. Das Gupta et al. Wiley-Interscience
• 1994. Selected Papers of C.R. Rao 1. (Ed). S. Das Gupta et al. Wiley-Interscience
• 1982. Statistics and Probability: Essays in Honor of C. R. Rao. (Ed.). Kallinapur. Elsevier

Further reading

• Nalini Krishnankutty, 1996. Putting Chance to Work: A Life in Statistics: A Biography of C. R. Rao.
• किशोर खनाल, 2080. "विश्वविख्यात प्रोफेसर सीआर रावको सम्झनामा". विश्वविख्यात प्रोफेसर सीआर रावको सम्झनामा
• Kishor Khanal and Shankar Prasad Khanal, 2023. " An interview with Professor C.R. Rao on the occasion of his 100th birthday". https://journals.ku.edu.np/kuset/article/view/136

Sources

• Calyampudi Radhakrishnan Rao. University of Minnesota Morris
• Statisticians in History: Calyampudi R. Rao" . American Statistical Association
• "Prof. C R Rao : An Eminent Statistician" By Bibhuprasad Mohapatra . Govt. of Odisha.

External links

• C. R. Rao's profile at the Advanced Institute of Mathematics, India
• C. R. Rao's profile at Penn State
• C. R. Rao's profile at the University at Buffalo
• ET Interviews: C. R. Rao on the Econometric Theory page.
• Royal Society citation 1967
• Rao was awarded the Samuel S. Wilks Award in 1989
  • June 2022 . C. R. Rao (1920–) Celebrates His 101st Birthday . . 69 . 6 . 982–997 . 10.1090/noti2492 . free . Andrews . George E. . Babu . G. J. . Bai . Zhidong . Banks . David . Chernoff . Herman . Frieden . Roy . Fujikoshi . Yasunori . Mardia . Kanti V. . De Braganca Pereira . Basilio . Pingle . Urmila . Rao . M. B. . Srinivasa Rao . Arni S. R. . Rosenberger . James . Varadhan . S R.S. . Wu . Yuehua.

For the Cramér–Rao inequality and the Rao–Blackwell theorem see the relevant entries on

• Earliest Known Uses of Some of the Words of Mathematics

For Rao contribution to information geometry

• Cramer-Rao Lower Bound and Information Geometry

Photograph of Rao with Harald Cramér in 1978

• C. R. Rao from the PORTRAITS OF STATISTICIANS
• President George Bush awarding Rao the National Medal of Science

Notes and References

1. Web site: C. R. Rao: A life in statistics. Prakasa Rao. B. L. S.. 10 September 2014. Current Science. 17 February 2020.
2. News: The Numberdars. The Times of India. 1 October 2011. dead. https://web.archive.org/web/20160323041225/http://www.timescrest.com/coverstory/the-numberdars-6380. 23 March 2016.
3. Web site: The Numberdars. 1 October 2001. Times Crest. 14 March 2016. https://web.archive.org/web/20160323041225/http://www.timescrest.com/coverstory/the-numberdars-6380. 23 March 2016.
4. Web site: Statisticians in History: Calyampudi R. Rao. 30 November 2016. American Statistical Association. 4 October 2009. 3 March 2016. https://web.archive.org/web/20160303195003/http://www.amstat.org/about/statisticiansinhistory/index.cfm?fuseaction=biosinfo&BioID=13. dead.
5. Web site: C.R.Rao in News. C.R.Rao Advanced Institute of Mathematics, Statistics and Computer Science.
6. Web site: International Prize in Statistics . 10 April 2023 . www.statprize.org . en.
7. News: Biswas . Atanu . 8 April 2023 . C.R. Rao wins top statistics award – a look back at his pioneering work . en-IN . The Hindu . 10 April 2023 . 0971-751X.
8. Web site: Indian Heart Association. Indian Heart Association Webpage. 27 April 2015.
9. DasGupta . Anirban . 2024-02-20 . C.R. Rao: Paramount statistical scientist (1920 to 2023) . Proceedings of the National Academy of Sciences of the United States of America . 121 . 9 . e2321318121 . 10.1073/PNAS.2321318121 . 38377193. 10907269 .
10. Web site: Profile. C.R.Rao Advanced Institute of Mathematics, Statistics and Computer Science.
11. Web site: Telugu professors do homeland proud. 3 July 2010. The Times of India. 22 October 2016.
12. News: Indian American Mathematician CR Rao No More . 23 August 2023 . Deccan Chronicle . 23 August 2023.
13. News: Renowned Indian Mathematician C.R Rao passes away in America . 23 August 2023 . The Hans India . 23 August 2023.
14. Web site: Legendary statistical scientist CR Rao dies . 23 August 2023 .
15. Web site: C.R. Rao, Amherst resident and winner of statistics' 'Nobel Prize,' dies at 102 . 23 August 2023 .
16. Web site: C.R. Rao Receives the India Science Award. 19 October 2010. Eberly College of Science, Penn State University. 28 December 2013. https://web.archive.org/web/20150926001137/http://science.psu.edu/news-and-events/2010-news/Rao10-2010. 26 September 2015. dead.
17. "The previous winners of the Award are Professor C.R. Rao (India) in 2003..." Web site: Archived copy . 1 January 2014 . dead . https://web.archive.org/web/20140101160834/http://www.isi-web.org/component/content/article/43-about/about/588-2013-mahalanobis-international-award-in-statistics-announcement . 1 January 2014.
18. Web site: NRI Award, Sardar Patel Award, Sardar Ratna, Pravasi Bharatiya Award, International Award. www.sardarpatelaward.org.
19. Web site: Padma Awards . Ministry of Home Affairs, Government of India . 2015 . 21 July 2015 . dead . https://web.archive.org/web/20151015193758/http://mha.nic.in/sites/upload_files/mha/files/LST-PDAWD-2013.pdf . 15 October 2015.
20. Web site: Recipients of Honorary Degrees. University of Calcutta. dead. https://web.archive.org/web/20131230234707/http://www.caluniv.ac.in/convocation/hony_degrees.htm. 30 December 2013.
21. News: Road to be named after Prof. C.R. Rao. https://web.archive.org/web/20090913102243/http://www.hindu.com/2009/09/10/stories/2009091059940300.htm. dead. 13 September 2009. 6 May 2012. The Hindu. 10 September 2009.