Skip to main content
SpringerLink
Log in

A Uniformly Asymptotically Efficient Estimator of a Location Parameter

  • Chapter
  • First Online:
Contributions on Theory of Mathematical Statistics

  • 798 Accesses

Abstract

Suppose that a sample of size n from a continuous and symmetric population with an unknown location parameter is given. We consider a fictitious random subsample of size k drawn from the original sample and construct the best linear estimator based on the subsample. Applying the Rao–Blackwell-type argument, we get an estimator which is supposed to be uniformly efficient for a wide class of distributions. Monte Carlo experiments established that this estimator is highly efficient for small samples of size 10 or 20.

This chapter was first published in Takeuchi (1971) Journal of the American Statistical Association, 66, 292–301.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Adichie, J.N.: Estimates of regression parameters based on rank test. Ann. Math. Statist. 38, 894–904 (1967)

    Article  MathSciNet  Google Scholar 

  • Bennett, C.A.: Asymptotic properties of ideal linear estimators. Ph.D. Dissertation, Unversity of Michigan (1952)

    Google Scholar 

  • Bhattacharya, P.K.: Efficient estimation of a shift parameter from grouped data. Ann. Math. Statist. 38, 1770–1787 (1967)

    Article  MathSciNet  Google Scholar 

  • Bickel, P.J.: On some robust estimatesof location. Ann. Math. Statist. 36, 847–858 (1965)

    Article  MathSciNet  Google Scholar 

  • Birnbaum, A., Laska, E.: Optimal robustness: a general method with applications to linear estimators of location. J. Amer. Statist. Assoc. 62, 1230–1240 (1967)

    Article  MathSciNet  Google Scholar 

  • Birnbaum, A., Meisner, M.: Optimally robust linear estimators of location. (mimeographed) (1968)

    Google Scholar 

  • Blom, G.: Statistical Estimates and Transformed Beta-Variables. Wiley, New York (1958)

    MATH  Google Scholar 

  • Chernoff, H., Gastwirth, J.L., Johns Jr., M.V.: Asmptotic distributions of functions of order statistics with applications to estimation. Ann. Math. Statist. 38, 52–72 (1967)

    Article  MathSciNet  Google Scholar 

  • CramĂ©r, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton, N. J. (1946)

    MATH  Google Scholar 

  • Chow, E.L., Siddiqui, M.: Robust estimation of location. J. Amer. Statist. Assoc. 62, 358–389 (1967)

    MathSciNet  Google Scholar 

  • Filliben, J.J.: Simple and robust linear estimator of the location parameter of a symmetric distribution. Ph.D. Thesis, Princeton Unversity (1969)

    Google Scholar 

  • Gastwirth, J.L.: On robust procedures. J. Amer. Statist. Assoc. 61, 929–948 (1966)

    Article  MathSciNet  Google Scholar 

  • Gastwirth, J.L., Cohen, M.S.: The small sample behavior of some robust linear estimators of location. Department of Statistics, Johns Hopkins Univ. Tech. Report. 91, Baltimore, Md. (1968)

    Google Scholar 

  • Gastwirth, J.L., Rubin, H.: On robust linear estimators. Ann. Math. Statist. 40, 24–39 (1969)

    Article  MathSciNet  Google Scholar 

  • Hájek, J.: Asymptotically most powerful rank order tests. Ann. Math. Statist. 33, 1124–1147 (1962)

    Article  MathSciNet  Google Scholar 

  • Hodges Jr., J.L., Lehmann, E.L.: Estimates of location based on rank tests. Ann. Math. Statist. 27, 324–335 (1963)

    Article  MathSciNet  Google Scholar 

  • Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325 (1948)

    Article  MathSciNet  Google Scholar 

  • Hogg, R.V.: Some observations on robust estimation. J. Amer. Statist. Assoc. 62, 1179–1186 (1967)

    Article  MathSciNet  Google Scholar 

  • Huber, P.: Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101 (1964)

    Article  MathSciNet  Google Scholar 

  • Jung, J.: On linear estimates defined by a continuous weight function. Ark. Mat. B. 3, 199–209 (1955)

    Article  MathSciNet  Google Scholar 

  • Lloyd, E.H.: Least squares estimation of location and scale parameters using order statistics. Biometrika 39, 88–95 (1952)

    Article  MathSciNet  Google Scholar 

  • MikĂ©, V.: Contributions to robust estimation. Courant Institute of Mathematical Sciences, Tech. Rep. ONR 042-206 (1967)

    Google Scholar 

  • Stein, C.: Efficient non-parametric testing and estimation. Pro. Third Berk. Sympo. Math. Statist. Prob. 1, 187–195 (1956)

    Google Scholar 

  • Takeuchi, K.: A uniformly asymptotically efficient estimator of a location parameter. J. Amer. Statist. Assoc. 66, 292–301 (1971)

    Article  Google Scholar 

  • Takeuchi, K.: A test for the shape of a symmetric distribution against symmetric alternatives. (to appear)

    Google Scholar 

  • Takeuchi, K., Meisner, M., Wanderling, J.: Asymptotic efficiencies of estimators of location. J. Statist. Comput. Simul. 2, 375–390 (1973)

    Article  Google Scholar 

  • Tukey, J.W.: The future of data analysis. Ann. Math. Statist. 33, 1–67 (1962)

    Article  MathSciNet  Google Scholar 

  • von Eden, C.: Efficiency-robust estimation of location. Ann. Math. Statist. 41, 172–181 (1970)

    Article  MathSciNet  Google Scholar 

  • Yhap, E.A.: Asymptotic optimally robust linear unbiased estimators of location for symmetric shapes. Ph.D. Dissertation, New York Unversity (1967)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Professor Emeritus, The University of Tokyo, Bunkyo-ku, Tokyo, Japan

    Kei Takeuchi

Authors
  1. Kei Takeuchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kei Takeuchi .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Japan KK, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Takeuchi, K. (2020). A Uniformly Asymptotically Efficient Estimator of a Location Parameter. In: Contributions on Theory of Mathematical Statistics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55239-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation