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Breakthroughs in Statistics pp 227–234Cite as

Introduction to Rao (1945) Information and the Accuracy Attainable in the Estimation of Statistical Parameters

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Abstract

The object of this introduction is to present a brief account of a paper that remains an unbroken link in the continuing evolution of modern statistics. In it are contained some of the fundamental paradigms of the foundations of statistics, e.g., unbiased minimum variance estimation, the Cramér-Rao inequality, the Rao-Blackwellization, and the Rao space of probability distributions furnished with a Riemannian quadratic differential metric and the associated geodesic distance (called the Rao distance). The paper furnishes some of the basic theoretical tools for the discussion of a variety of problems in statistical inference.

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References

  • Aitken, A.C. (1941). On the estimation of statistical parameters, Proc. Roy. Soc. Edin., 61, 56–62.

    Google Scholar 

  • Amari, S. (1985). Differential-Geometric Methods in Statistics. Springer-Verlag, New York.

    Book  Google Scholar 

  • Bahadur, R.R. (1957). On unbiased estimates of uniformly minimum variance, Sankhyā, 18, 211–224.

    MathSciNet  MATH  Google Scholar 

  • Barankin, E.W. (1950). Extension of a theorem of Blackwell, Ann. Math. Statist. 21, 280–284.

    Article  MathSciNet  MATH  Google Scholar 

  • Basu, D. (1958). On sampling with and without replacement, Sankhyā, 20, 287–294.

    MATH  Google Scholar 

  • Bhattacharya, A. (1946). On some analogues to the amount of information and their uses in statistical estimation, Sankhyā, 8, 1–14, 201-208.

    Google Scholar 

  • Black well, D. (1947). Conditional expectation and unbiased sequential estimation, Ann. Math. Statist., 18, 105–110.

    Article  Google Scholar 

  • Blyth, C.R. (1974). Necessary and sufficient conditions for inequalities of Cramér-Rao type, Ann. Statist., 2, 464–473.

    Article  MathSciNet  MATH  Google Scholar 

  • Blyth, C.R., and Roberts, D.M. (1972). On inequalities of Cramér-Rao type and admissibility proofs, in Proceedings of 6th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Vol. 1, pp. 17–30.

    MathSciNet  Google Scholar 

  • Brown, L.D. (1982). A proof of the central limit theorem motivated by the Cramér Rao inequality, in Statistics and Probability: Essays in Honor of CR. Rao (Kallianpur et al., eds.). North-Holland, Amsterdam, pp. 141–148.

    Google Scholar 

  • Burbea, J., and Oller, J.M. (1989). On Rao distance asymptotic distribution. Mathematics Reprint Series, No. 67, University of Barcelona, Spain.

    Google Scholar 

  • Cedarquist, J., Robinson, S.R., and Kryskowski, D. (1986). Cramér-Rao lower bound on wavefront sensor error, Opt. Eng., 25, 586–592.

    Google Scholar 

  • Chapman, D.C., and Robbins, H. (1951). Minimum variance estimation without regularity assumptions, Ann. Math. Statist., 22, 581–586.

    Article  MathSciNet  MATH  Google Scholar 

  • Cramér, H. (1945). Mathematical Methods of Statistics, Almqvist and Wiksell, Uppsala, Sweden.

    Google Scholar 

  • Cramér, H. (1946). Contributions to the theory of statistical estimation, Skand. Aktuarietidsk, 29, 85–94.

    MATH  Google Scholar 

  • Doob, J.L. (1953). Stochastic Processes. Wiley, New York.

    MATH  Google Scholar 

  • Efron, B. (1975). Defining the curvature of a statistical problem (with applications to second order efficiency), Ann. Statist., 3, 1189–1242.

    Article  MathSciNet  MATH  Google Scholar 

  • Fabian, V., and Hannan, J. (1977). On the Cramér-Rao inequality, Ann. Statist., 5, 197–205.

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher, R.A. (1922). On the mathematical foundations of theoretical statistics, Philos. Trans. Roy. Soc. London, Ser. A, 222, 309–368.

    Article  MATH  Google Scholar 

  • Fossi, M., Giuli, D., and Gherardelli, M. (1989). Cramér-Rao bounds and maximum-likelihood estimation of Doppler frequency of signals received by a Polarimetric radar, IEEE Proc, Part F, Radar and Signal Processing, 136, 175–184.

    Article  Google Scholar 

  • Hall, W.J., and Mathiason, D.J. (1990). On large-sample estimation and testing in parametric models, Int. Statist. Rev., 58, 77–97.

    Article  MATH  Google Scholar 

  • Hammersley, J.M. (1950). On estimating restricted parameters, J. Roy. Statist. Soc., Ser. B, 12, 192–240.

    MathSciNet  MATH  Google Scholar 

  • Hodges, J.L., Jr., and Lehmann, E.L. (1950). Some problems in minimax point estimation, Ann. Math. Statist., 21, 182–197.

    Article  MathSciNet  MATH  Google Scholar 

  • Hodges, J.L., Jr., and Lehmann, E.L. (1951). Some applications of the Cramér-Rao inequality, in Proceedings of 2nd Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, pp. 13–22.

    Google Scholar 

  • Kallianpur, G., and Rao, C.R. (1955). On Fisher’s lower bound to asymptotic variance of a consistent estimate, Sankhyä, 17, 105–114.

    MathSciNet  Google Scholar 

  • Kolmogorov, A.N. (1950). Unbiased estimates, Izvestia Akad. Nauk SSSR, Ser. Math., 14, 303–326 (American Mathematical Society Translation No. 98).

    MathSciNet  MATH  Google Scholar 

  • Kremers, W.K. (1986). Completeness and unbiased estimation for sum-quota sampling, J. Amer. Statist. Assoc., 81, 1070–1073.

    MathSciNet  MATH  Google Scholar 

  • Lehmann, E.L., and Scheffé, H. (1950, 1955, 1956). Completeness, similar regions, and unbiased estimation, Sankhyā, 10, 305–340; 15, 219-236; 17, 250.

    Google Scholar 

  • Mayer-Wolf, E. (1988). The Cramér-Rao functional and limit laws. Mimeo Series, No. 1773, Department of Statistics, University of North Carolina.

    Google Scholar 

  • Mitra, S.K., and Pathak, P.K. (1984). The nature of simple random sampling, Ann. Statist., 12, 1536–1542.

    Article  MathSciNet  MATH  Google Scholar 

  • Pathak, P.K. (1964a). On sampling schemes providing unbiased ratio estimators, Ann. Math. Statist., 35, 222–231.

    Article  MathSciNet  MATH  Google Scholar 

  • Pathak, P.K. (1964b). On inverse sampling with unequal probabilities, Biometrika, 51, 185–193.

    MathSciNet  MATH  Google Scholar 

  • Pathak, P.K. (1965). Estimating population parameters from conditional sampling schemes. 35th Session of the International Statistical Institute, Belgrade, Yugoslavia.

    Google Scholar 

  • Pathak, P.K. (1975). An extension of a theorem of Hoeffding, Studia Sci. Math. Hung., 10, 73–74.

    Google Scholar 

  • Pathak, P.K. (1976). Unbiased estimation in a fixed cost sequential sampling scheme, Ann. Statist., 4, 1012–1017.

    Article  MathSciNet  MATH  Google Scholar 

  • Rao, C.R. (1946). Minimum variance and the estimation of several parameters, Proc. Cambridge Philos. Soc., 43, 280–283.

    Article  Google Scholar 

  • Rao, C.R. (1948). Sufficient statistics and minimum variance estimation, Proc. Cambridge Philos. Soc., 45, 215–218.

    Google Scholar 

  • Rao, C.R. (1961). Asymptotic efficiency and limiting information, in Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1. University of California Press, pp. 531–546.

    Google Scholar 

  • Rao, CR. (1963). Criteria of estimation in large samples, Sankhyā, 25, 189–206.

    MATH  Google Scholar 

  • Simons, G. (1980). Sequential estimators and the Cramér-Rao lower bound, J. Statist. Planning and Inf., 4, 67–74.

    Article  MathSciNet  MATH  Google Scholar 

  • Simons, G., and Woodroofe, M. (1983). The Cramér-Rao inequality holds almost everywhere, in Recent Advances in Statistics (papers in honor of Herman Chernoff on his 60th birthday) (Rizvi et al., eds.). Academic Press, New York, pp. 69–93.

    Google Scholar 

  • Torgersen, E. (1988). On Bahadur’s converse of the Rao-Blackwell theorem. Extension to majorized experiments, Scand. J. Statist., 15, 273–280.

    MathSciNet  MATH  Google Scholar 

  • Wolfowitz, J. (1947). The efficiency of sequential estimates and Wald’s equation for sequential processes, Ann. Math. Statist., 18, 215–230.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. University of New Mexico, USA

    P. K. Pathak

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  1. P. K. Pathak
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Editor information

Editors and Affiliations

  1. College of Business and Management, University of Maryland at College Park, College Park, MD, 20742, USA

    Samuel Kotz

  2. Department of Statistics Phillips Hall, The University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA

    Norman L. Johnson

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© 1992 Springer Science+Business Media New York

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Pathak, P.K. (1992). Introduction to Rao (1945) Information and the Accuracy Attainable in the Estimation of Statistical Parameters. In: Kotz, S., Johnson, N.L. (eds) Breakthroughs in Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0919-5_15

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  • DOI: https://doi.org/10.1007/978-1-4612-0919-5_15

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94037-3

  • Online ISBN: 978-1-4612-0919-5

  • eBook Packages: Springer Book Archive

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