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Mathematical Statistics: Basic Concepts and Theoretical Tools for Finite Sample Analysis

  • Takanori Sugiyama
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

In this chapter, we introduce some fundamentals of statistical estimation theory. There are three purposes to this chapter. The first is to set down mathematical preliminaries in order to treat several quantum estimation problems. The second purpose is to explain how to evaluate the estimation errors and the effects of statistical errors on the estimation errors. The third is to explain known results in statistical parameter estimation which have been used in quantum estimation theory or will be used in chapters that follow. In Sect. 3.1, we explain fundamental concepts and technical terms in probability theory and statistical parameter estimation. In Sect. 3.2, we explain some known results in the asymptotic theory. In Sect. 3.3, we explain known results for expected loss and error probability involving finite samples.

Keywords

Figures of merit in statistical estimation Tools for finite data analysis 

References

  1. 1.
    E.L. Lehmann, Theory of Point Estimation. Springer Texts in Statistics (Springer, 1998)Google Scholar
  2. 2.
    M. Akahira, K. Takeuchi, Non-Regular Statistical Estimation. Lecture Notes in Statistics (Springer, 1995)Google Scholar
  3. 3.
    C.R. Rao, Linear Statistical Inference and Its Applications, 2nd edn. Wiley Series in Probability and Statistics (Wiley, New York, 2002). (originally published in 1973)Google Scholar
  4. 4.
    R.R. Bahadur, J.C. Gupta, S.L. Zabell, Large Deviations, Tests, and Estiamtes. Asymptotic Theory of Statistical Tests and Estimates (Academic Press, New York, 1980), p. 33Google Scholar
  5. 5.
    R.R. Bahadur, Sankhyā 22, 229 (1960)MATHMathSciNetGoogle Scholar
  6. 6.
    R.R. Bahadur, Ann. Math. Statist. 38, 303 (1967). doi: 10.1214/aoms/1177698949
  7. 7.
    M. Hayashi, K. Matsumoto, IEICE Trans. A 83, 629 (2000)Google Scholar
  8. 8.
    T. Sugiyama, P.S. Turner, M. Murao, Phys. Rev. A 83, 012105 (2011). doi: 10.1103/PhysRevA.83.012105 Google Scholar
  9. 9.
    W. Hoeffding, J. Am. Stat. Assoc. 58, 13 (1963). doi: 10.2307/2282952 Google Scholar
  10. 10.
    I.N. Sanov, (English translation from Mat. Sb. (42)) in. Selected Translations in Mathematical Statistics and Probability 1(1961), (1957)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Department of Physics Graduate School of ScienceThe University of TokyoTokyoJapan

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