C.R.rao: One hundred years of statistics

Author:High Energy Institute of the C Time:2022.06.16

The following article comes from WeChat public account: Teacher Guo statistics small classroom, author Guo Xu compiled

The well -known statistical journal "International Statistics" published the "C. Radhakrishna Rao: A Century in Statistics" written by NANDINI KANNANNNNNNNNNN and Debasis Kundu. Speaking of C.R.rao, you should have learned the statistics of C-R and Rao-Blackwell theorem. In the Chinese world, his "all knowledge in the ultimate analysis is history; in the abstract sense, all science is mathematics; in a rational world, all judgments are statistical" words " It is also very famous. Today we read the statistics experienced by C.R.rao with this article for a hundred years.

This article summarizes the legendary life of C. R. Rao, a centenary old man. His life has experienced Pearson, and Fisher has also experienced big data and artificial intelligence. His life is a hundred years of statistics. RAO is still alive, he will continue to witness the second hundred years of statistics!

1. Early time of RAO

Rao was born on September 10, 1920 and is the eighth child in the family. His father was a patrol police who quickly realized RAO's talent in mathematics and encouraged him to engage in research. His mother called him to get up at four o'clock every day when Rao was a kid. This has a profound impact on his extraordinary career. At the age of 20, RAO received a master's degree in mathematics. Due to the war, it is difficult to find a mathematical work, and RAO applies for a position of the military investigation bureau. In the process, he went to Calcutta, and this trip completely changed his career.

By chance, Rao and a young man who had attended the statistical training program in Calcutta had a conversation. After some researchers who knew the Indian Institute of Statistics (ISI) learned some of the projects, Rao was convinced that this statistical training program could not only improve his employment prospects, but also provides an opportunity to continue research. In this way, RAO took the Monthly 30 rupees to ISI on January 1, 1941.

Legendary scientist, mathematician and statistician Prasanta Chandra Mahalanobis had a very important and far -reaching impact on RAO's life. Mahalanobis is specifically referred to as "Professor" in ISI. When reading physics in Cambridge, Mahalanobis accidentally saw the first copy of Biometrika (Biometrika was created by Karl Pearson, Francis Galton and Raphael Weldon. It is still one of the top four journals in the field of statistics.) At that time, it was gradually growing into a new discipline, and Mahalanobis soon realized that the potential impact of statistics in many applications. Mahalanobis founded ISI in 1931 after returning to India.

Rao met with ISI very outstanding researchers at the time, such as R. C. Bose, S. N. Roy, K. R. NAIR, and A. Bhattacharya. Just a few months after joining ISI, Rao and K.R.NAIR published papers on confusion design! During this period, RAO obtained a master's degree in statistics from the University of Calcutta with a maximum scoring record of 87%.

For a 23 -year -old young man, no matter from which angle, RAO's master's dissertation is an extraordinary job. There are three parts of this master's dissertation. The first part is about 119 pages of the experimental design. The second part of the second part is about diversified tests, and the last 42 pages are about binary distribution. In fact, his master's dissertation has reached the level of doctoral dissertation. His papers have a lot of original contributions, such as a solution to Norwegian economist Ragnar Frisch. And this part of the result was published in Econometrica in 1947!

2. 1945-Create legendary breakthrough articles!

The article published by RAO in 1945 "Information and Accuracy Attainable in the Estimation of Statistics of Statistics One stroke. This article has only 10 pages of papers and has a beautiful demonstration, but it shows the two basic results of statistical inferring and breeding the field of information geometry.

2.1 Cramer-Rao Unqualified Model-Information Industry

In the early 1920s, R.A.Fisher introduced the concept of Fisher information. Fisher information can be used to measure the amount of information that contains unknown parameters in a random sample. It measures the expectations of the number of like -minded functions; the greater the degree of bending of the number of numbers, the data can provide more information about parameters. Fisher proves that there is a lower bound to the approximate difference in the great estimation of great estimates. In an article in 1945, RAO proves that for any unbiased estimation, its variance is greater than the countdown to the Fisher information. This result was cleverly used by cauchy-schwarz. During the same period, Harold Cramer, a Swedish mathematician, also independently established the information discomfort and included the result in the book "Statistical Mathematics Method" published in 1946.

Cramer-Rao's lower bound (CRLB) has important applications in almost all science and engineering fields. There are many promotion in the literature, such as quantum CRLB and Bayesian CRLB. Dembo, COVER, and Thomas (1991) summarize the various inequality in the information theory and its unlimited connections in the fields of mathematics and physics. They further showed that Weyl-Heisenberg uncertainty criteria can be obtained from the Cramer-Rao.

2.2 Rao-Blackwell theorem

The second basic results established in the article in 1945 involved the use of sufficient statistics to improve the efficiency of estimation. In the article, RAO pointed out that "if there is a sufficient statistics and an unbiased estimation for parameters, the optimal and unbiased estimation must be a function of sufficient statistics." Strictly, this result shows that if it is an estimate of an unknown parameter, it will be expected to have a smaller equalizer error on sufficient statistical measurement conditions. In an article in 1947, David Blackwell proves the same result. This result is called Rao-Blackwell theorem, and a process that is estimated to be estimated to be replaced with the corresponding condition is called Rao-Blackwell. This classic results also have many modern applications. DOUCET et al. (2000) Use Rao-Blackwell to improve the efficiency of different dynamic Bayesian network particles filtering. Robert and Roberts (2021) discusses the application of Rao-Blackwell's sampling and more general MCMC.

2.3 Licule geometry and statistics: The beginning of information geometry

RAO's article in 1945 proposes the concept of the distance or scattering of different probability distribution. This is the one in the early few articles to study the probability model using micro -division geometry and create a new field of information geometry. RAO regards the parameter as Riemann as the same as the Riemann quantity of the Fisher information matrix. He proposed the measuring the difference between the two probability distribution differential distribution of the floor line distance induced by Riemann. Riemann's linear volume is also called Fisher-Rao distance.

EFRON (1975) research on statistical curvature has made micro -split geometric methods began to play an increasingly important role in mainstream statistical research. In recent years, the role of information geometry in the fields of optimization, machine learning and deep learning has become increasingly greater.

During this period, a anthropologist at the University of Cambridge J. C TREVOR asked Mahalanobis if he could assign an ISI expert to the UK for some anthropological data analysis. Another turning point of this fate will open a new chapter in RAO.

3. Cambridge -the years with Fisher

RAO arrived in England in 1946 and entered the King's College of Cambridge University, following Ronald Aylmer Fisher, the founder of modern statistics. Fisher's theory of statistics in 1925 "The theory of Statistics" establishes the cornerstone of statistical inference. This article introduces a large number of basic concepts such as compatibility, adequateness, effectiveness, Fisher information, great granting and optimality. EFron (2003) In his article, the statistics "written in the century": "I regard 1925 as a year of statistics from the clever technology to become a coherent discipline."

By the 1940s, Fisher had worked at Cambridge and was named Balfour lecturer professor named genetic. Fisher agrees to be the instructor of Dr. RAO's thesis, but the premise is that RAO needs to work in the genetic laboratory of Fisher. So Rao was full of day: to analyze bone data at the Museum of Archaeological and Ethnicology in the morning, stay in the genetic laboratory at night, and at other times to study for a doctoral degree.

3.1 1948: Scoring test -The second breakthrough article!

During the work of Fisher's genetic laboratory, Rao realized the necessity of homogeneous testing, and he also introduced a new score -based and information matrix statistics.

RAO recalled the scene at the time: he showed Fisher to see the manuscript of the score score, but Fisher insisted that RAO must show the performance of statistical quantities on actual data. According to Fisher's suggestion, RAO applies score statistics to the related data of newspaper Chunhua and submits the article to the journal Heredity founded by Fisher. Another theoretical basis and nature of the development score scores were published in PROceedings of the Cambridge Philosophical Society in 1948. The scoring statistics mentioned by RAO have now become a tool for every statistician and actual workers. NEYMAN and Pearson's grant ratio, WALD statistics and RAO's scores are often called the trinity of statistics.

RAO has a very fulfilling life in Cambridge. Not only does it follow Fisher to learn genetics, but he also studies the random process from Bartlett, studying the "game theory and economic behavior" of von Numermart and Morgan Stan, and also participated in the debate.

Although Fisher is less directly participating in RAO, his impact on the RAO business is second only to Mahalanobis. Like Mahalanobis, Fisher realizes the importance of data and practical applications in the development of statistical methods. EFron (2003) once wrote: "No one can do better than Fisher's statistical connotation (mathematical foundation) and extension (statistical method) and their fusion. His theoretical frame The application is connected. In RAO's 100th birthday, EFron wrote that "RAO is indeed a student student in continuing Fisher's statistical tradition."

Both Mahalanobis and Fisher are extraordinary figures, very enthusiastic about work, but often unknown compromise. They firmly believe that statistical research should be driven by practical applications. They understand how applications have spawned the development of new theories, and they are aware of the importance of data quality. Rao believes that they are the guidance for his life.

RAO returned to ISI in August 1948 and was awarded a professor position in July 1949. He was less than 29 years old!

4. 1950-1978: RAO and Indian Institute

In 1947, India faced many economic and social challenges after Britain's independence. Mahalanobis believes that data and statistical methods are the key to solving these challenges and can help the government make more reasonable decisions. India's first Prime Minister is also a friend of Mahalanobis. Nehru also thought and appointed Mahalanobis as a statistical consultant of the Indian government. Nicru's support is very important for the Indian Institute of Statistics ISI to become an important national institute.

Under the leadership of Mahalanobis, ISI has gradually become a world -class organization. He can identify people and promote them to work in II. At the same time, he clarified the value of international cooperation and often invited well -known experts to visit ISI. After R. C. Bose and S. N. ROY went to the United States, RAO has become the second figure in ISI.

The first job after RAO returned to China was to complete his book "Advanced Statistical Methods in Biological Research". This is an entry book about multiple statistical methods and its application. At the same time, RAO began to guide doctoral students. His first doctoral student was BASU. BASU studied PhD since 1950 and received a doctorate in 1953. In the later years, BASU will make basic contributions to statistics, including the well -known BASU theorem.

During the ISI period, Rao basically maintained 5-6 doctoral students at the same time, and involved completely different areas. These include sampling surveys (A.C. DAA, Des Raj), multiple analysis (J. ROY), quality control (A. MATHAI), feature (R. G. Laha), experimental design (I. M. Chakraborti), directional data (J.S. RAO), Combination and diagram theory (U.S. R. Murthy, S. B. Rao). Based on the knowledge and insights he learned in Cambridge, he also instructed students to engage in statistical genetics (D. C. Rao, K. Dronamraju, Ranajit Chakraborty), probability theory and random process (S. R. S. Varadhan, K. R. PARTHASARATHY), and, and Gaming theory (T. PARTHASARATHY, T.E.S. Raghavan). In a relatively new field for him, he will let students read related literature carefully and write a review. He is very good at establishing intuitive connections between different fields and providing unique solutions.

In India statistics centered on ISI, the period of 1930-1960 was often accused of statistics "golden age". ISI invited a series of well -known scholars to visit, including Norbert Weiner, Ragnar Frisch, John Kenneth Galbraith, J. B. S. Haldane, Walter Shewhart, A. N. Kolmogorov, and J. L. Doob. Rao encourages the teachers and doctoral students of the Institute to seize these rare opportunities and take the initiative to understand the new fields of statistics and related disciplines. The academic atmosphere of this period can be said to be very active. In order to meet the needs of the government and industry, RAO has updated the ISI training project and added courses on statistical quality control and industrial statistics. In 1953, ISI added a new branch of statistical quality control to provide consulting services to the industry. A visiting person in this period was a Japanese engineer and statistician Genichi Taguchi. After World War II, he and Edward Deming helped Japanese companies adopt the principle of quality control. RAO's work on orthogonal tables has played a very critical role in the development of steady design and the formulation of industrial test standards.

RAO also dominates the training program of the International Statistics Center. He also participated in the design of the national sampling survey and the formulation of regulations.

In order to promote the teaching and research of measuring economics and quantitative economics, the Indian Economic Association was established in 1960. RAO is one of the founders. Ragnar Frisch is Norwegian economist. He shared the first Nobel economist with Jan Tinbergen in 1969. At the same time, he was also one of the founders when the Metrology Economics was founded in the 1930s. The article published by Rao published in Econometrica proposed a solution to a featured problem proposed by FRISCH on the structural equation. Many of RAO's work have greatly affected the development of economics.

In 1953, Rao went to University of Illinois at Urbana-Champaign (UIUC) to make a one-year visit to research professor. During this period, RAO participated in the programming course. After returning ISI, RAO began to promote the use of computer in India. However, many people are worried that automation can cause unemployment and greatly delay the use of computer. Many years later, Rao said, "We have lost a good opportunity to develop statistical methods based on complex computing."

Despite the heavy administrative affairs, RAO still adheres to scientific research. He started to write a new book of statistics. This book "Linear Statistical Inference and Application" published in 1965 has now become a must -read book for statistical majors. At the same time, Rao was the editor -in -chief of the SANKHYA journal after Mahalanobis.

After Mahalanobis died in 1972, RAO began to be the leader of ISI. Rao was appointed as "Professor Nehru". Another opportunity in Pittsburgh will change his career again.

5. The years in the United States

In 1978, Rao was a Mellon professor at the University of Pittsburgh, and later served as a visiting professor at Ohio State University. Both universities provide him with permanent positions. RAO was close to 60 years old, and most agencies in India were the age of compulsory retirement. Accepting invitations from American universities can focus on research and write a new chapter in his career.

Coincidentally, Rao's son Verendra decided to transfer to the University of Pittsburgh to complete his bachelor's degree in engineering. So Rao was invited by the University of Pittsburgh and became a professor at the Department of Mathematics and Statistics in the fall of 1979.

Although the department had many top mathematicians at the time, there were few teachers in statistics in the 1970s. In order to enhance the strength of teaching and scientific research in statistics, the department hired P. R. Krishnaih in 1976. Krishnaih is the authority in multiple statistical analysis. He has worked as a mathematical statisticians at the Rongte Patson Air Force Base for many years. He is the editor -in -chief of Journal of Multivar association Analysis (JMVA). He is the editor -in -chief of the statistical manual series, and at the same time organized multiple international conferences.

Under the support of the US naval and air force, the Multi -Statistical Analysis Center was established in 1982. Krishnaiah and Rao can get many large -scale research funds from the US Air Force to fund the foundation and application research in the fields of multiple analysis, model recognition and signal processing. He attracted a large number of scholars at the University of Pittsburgh to go to Pittsburgh for exchanges.

Krishnaiah died of cancer in 1987. After Krishnaih's death, Rao began to host the center's affairs and JMVA's editing work. The Department of Statistics at the State University of Pennsylvania persuaded Rao as the first Eberly lecture professor to join the Department of Statistics. Rao went to the Department of Statistics at Pennsylvania in 1988. RAO has been a lecture professor of the Department of Statistics from 1988 until his retirement in 2001, and in 2009, he served as Professor Rongxiu. After retirement, Rao served as a research professor at the Department of Biology Statistics at Buffalo University.

6. Rao research contribution

RAO's pioneering contribution involves almost all areas of theory and application statistics, as well as many disciplines including economics, electronic engineering, anthropology and genetics. Overview some of RAO's work here.

6.1 portrayal of probability distribution

One day, he accidentally read H. V. Allen to write an article, and RAO began to become interested in characterized issues. This article gives a solution to the following problems raised by Ragnar Frisch. This question is as follows: If the two random variables are satisfied

Under what conditions, the right regression is linear. This is an independent variable but an unknown constant. The results of RAO have established permission, and the sufficient necessary conditions for the infinite variance. The result of this part is part of the RAO master's paper, and published in Econometrica in 1947. His research in this area gives important portrayal of Poisson, Gamma, one yuan diverse normal, infinitely stable distribution of infinite variance.

6.2 Guangxian inverse

RAO considers the general theory of the smallest two -by -multharma when the matrix strange matrix is strange in the regular equation. In an article in 1955, Rao proposed the concept of pseudo -reversal of the strange matrix. In the same year, Roger Penrose published the "broad sense of the parallel" in the processings of the Cambridge Philosophical Society. Roger Penrose won the Nobel Prize in Physics in 2020. Although the pseudo-pseudo-counterfeit of RAO is not satisfied with the limitation of the MOORE-PENROSE, RAO's pseudo-reverse provides an answer to the minimum secondary multiplication in general situations.

6.3 Multiple statistical analysis

In the calculation of the distance of Mahalanobis, the inverse of the coordinated matrix is involved. When the matrix is large, it was not easy to solve. Rao proposed a method of simplifying solution. While following Fisher's doctoral degree, Rao promoted Fisher's work on classification issues.

In 1948, RAO analyzed the one -dollar variance analysis to a variety of situations. RAO develops an approximate method for the commonly used Wilk in the analysis of multiple variance analysis.

When analyzing some binary data, RAO found that it was not significant whether it was based on Mahalanobis's distance or Hotelling statistics, but the marginal test was significant. Healy calls this phenomenon a RAO paradox. This also reflects the cursing of dimensions.

RAO has promoted the analysis of the variance and proposed a variety of analysis, which has many applications in ecology, sociology and economics.

RAO also studied how to determine the number of factor.

In addition, RAO also contributes many basic contributions to growth curve analysis, main component analysis, and variance component estimation. Due to the contribution of diversified statistical analysis, Rao was awarded the Samuel S. Wilks Memorial Award in 1989.

6.4 Statistical signal processing

While working at the University of Pisburg, RAO and Krishnaih received many projects from the US Air Force to engage in theoretical and methods of signal processing. The US Air Force is interested in how to propose effective statistical methods to analyze the noise -containing signal from multiple sensors. RAO led his team to develop corresponding parameters, assumptions inspection, model selection, and effective algorithms.

Rao's vision: statistics in the era of big data

During the 80 -year -old career, RAO not only witnessed the gradual development of statistics into an independent science discipline, but also one of the few founders of statistical foundation. He understands that for statistics, in order to maintain an influential discipline, statistical science must adapt to the changing world.

In the first half of the twentieth century, the basis of statistical science was developed by using powerful mathematics and probability theory. Efron called this period as the golden age of statistical theory in the "Statistics Century" of his article. In this article, EFron also wrote that "" Fisher, Neyman, Pearson, Hotelling, WALD, Cramer, and Rao's intelligent people lead statistical theory to mature ".

In the work with Mahalanobis and Fisher, Rao realized the importance of research driven by actual application.By the 1950s, Rao began to experience the key role of computers and computing methods in the development of statistical methods.As an extremely predictive scientist, RAO foresees the big data revolution and a cross discipline composed of mathematics, statistics, and computer science.In 2007, he established C.R. RAO Mathematics, Statistics and Computer Science Store Institute (AIMSCS).RAO continues to expand statistical boundaries to solve the challenges in the era of big data and artificial intelligence.This article is authorized to reprint from WeChat public account: Teacher Guo's statistical small classroom author: Guo Xu