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On weakly equivariant estimators

  • M. ShamsEmail author
Regular Article
  • 13 Downloads

Abstract

In this paper, we shall generalize the concept of equivariance in statistics to “weak equivariance”. Then, we summarize the properties of weakly equivariant estimators and their applications in statistics. At first we characterize the class of all weakly equivariant estimators. Then, we shall consider the concept of cocycles and isovariance, and so we find their connection with weakly equivariant functions. It is natural to restrict attention to the class of weakly equivariant estimator to find minimum risk weakly equivariant estimators. If the group acts in two different ways, we shall find a relation between the minimum risk equivariant and minimum risk weakly equivariant estimator under the old and new group actions. Also we shall introduce a necessary and sufficient condition for the invariance of the loss function under the new action.

Keywords

Topological group Hausdorff space Locally compact group Orbit type Transitivity Homogeneous space Invariance Isovariance Weakly equivariance Cocycles 

Mathematics Subject Classification

Primary 62F10 Secondary 54H11 

Notes

References

  1. Berk RH (1967) A special group structure and equivariant estimation. Ann Math Stat 38(5):1436–1445MathSciNetCrossRefGoogle Scholar
  2. Bredon GH (1972) Introduction to compact transformation groups. Academic Press, New YorkzbMATHGoogle Scholar
  3. Brown LD (1968) Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. Ann Math Stat 39:29–48CrossRefGoogle Scholar
  4. Deitmar A, Echterhoff S (2009) Principles of harmonic analysis. Springer, New YorkzbMATHGoogle Scholar
  5. Eaton ML (1983) Multivariate statistics: a vector space approach. Wiley, New YorkzbMATHGoogle Scholar
  6. Eaton ML (1989) Group invariance applications in statistics. Institute of Mathematical Statistics and American Statistical Association, HaywardzbMATHGoogle Scholar
  7. Feres R, Katok A (2002) Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena), Handbook of dynamical systems, 1(A), 665–763. Elsevier, AmsterdamzbMATHGoogle Scholar
  8. Fisher RA (1973) Statistical methods and scientific inference. Hafner, New YorkzbMATHGoogle Scholar
  9. Folland GB (2015) A course in abstract harmonic analysis, 2nd edn. Chapman and Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  10. Fraser DAS (1961) The fiducial method and invariance. Biometrika 48:261–280MathSciNetCrossRefGoogle Scholar
  11. Fraser DAS (1968) The structure of inference. Wiley, New YorkzbMATHGoogle Scholar
  12. Garcia G, Oller JM (2001) Minimum Riemannian risk equivariant estimator for the univariate normal model. Stat Probab Lett 52:109–113MathSciNetCrossRefGoogle Scholar
  13. Hall WJ, Wijsman RAM (1965) Minimum Riemannian risk equivariant estimator for the univariate normal model. Staand Ghosh, J. KGoogle Scholar
  14. Hall WJ, Wijsman RAM (2001) The relationship between sufficiency and invariance with applications in sequential analysis. Ann Math Stat 36:575–614MathSciNetCrossRefGoogle Scholar
  15. Ilmonen P, Oja H, Serfling R (2012) On invariant coordinate system (ICS) functionals. Int Stat Rev 80:93–110MathSciNetCrossRefGoogle Scholar
  16. James W, Stein C (1960) Estimation with quadratic loss. In: Proc Fourth Berkeley Symp Math Stat Probab, vol. 1, pp. 361–380. University of Califomia PressGoogle Scholar
  17. Kiefer J (1957) Invariance, minimax sequential estimation and continuous time processes. Ann Math Stat 28:573–601MathSciNetCrossRefGoogle Scholar
  18. Konno Y (2007) Improving on the sample covariance matrix for a complex elliptically contoured distribution. J Stat Plan Inference 137:2475–2486MathSciNetCrossRefGoogle Scholar
  19. Kraft H (2016) Algebraic transformation groups: an introduction. Mathematisches Institut, Universitat BaselGoogle Scholar
  20. Kubokawa T, Konno Y (1990) Estimating the covariance matrix and the generalized variance under a symmetric loss. Ann Inst Stat Math 42:331–343MathSciNetCrossRefGoogle Scholar
  21. Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  22. Lehmann EL, Romano JP (2005) Testing statistical hypotheses, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  23. Olkin I, Selliah JB (1977) Estimating covariances in a multivariate normal distribution. In: Gupta S, Moore DS (eds) Statistical Decision theory and related topics II. Academic, New York, pp 313–326CrossRefGoogle Scholar
  24. Palais RS (1960) Classification of G-spaces, vol 36. Memoirs of the American Mathematical Society, ProvidencezbMATHGoogle Scholar
  25. Peisakoff M (1939) Transformation parameters, Thesis, Princeton University, Princeton, N.J. Pitman, E. J. GGoogle Scholar
  26. Peisakoff M (1950) The estimation of location and scale parameters of continuous population of any givan form. Biometrika 39:391–421Google Scholar
  27. Pitman EJG (1939) The estimation of location and scale parameters of continuous population of any givan form. Biometrika 39:391–421CrossRefGoogle Scholar
  28. Robert G (1971) A characterization of invariant loss functions. Ann Math Stat 42(4):1322–1327MathSciNetCrossRefGoogle Scholar
  29. Sanjar NF, Zakerzadeh H (2005) Estimation of a gamma scale parameter under asymmetric squared-log error loss. Commun Stat Theory Methods 34(5):1127–1135MathSciNetCrossRefGoogle Scholar
  30. Serfling R (2010) Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardization. J Nonparametr Stat 22:915–936MathSciNetCrossRefGoogle Scholar
  31. Serfling R (2015) On invariant within equivalence coordinate system (IWECS) transformations. In: Nordhausen K, Taskinen S (eds) Modern nonparametric, robust and multivariate methods. Springer, New York, pp 445–457Google Scholar
  32. Svensson L (2004) A useful identity for complex Wishart forms. Technical report, Department of Signals and Systems. Chalmers University of TechnologyGoogle Scholar
  33. Zhang S, Sha Q (1997) On the best equivariant estimator of covariance matrix of a multivariate normal population. Commun Stat Theory Methods 26(8):2021–2034MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of Mathematical SciencesUniversity of KashanKashanIran

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