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On weakly equivariant estimators

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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.

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References

  • Berk RH (1967) A special group structure and equivariant estimation. Ann Math Stat 38(5):1436–1445

    Article  MathSciNet  Google Scholar 

  • Bredon GH (1972) Introduction to compact transformation groups. Academic Press, New York

    MATH  Google Scholar 

  • Brown LD (1968) Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. Ann Math Stat 39:29–48

    Article  MathSciNet  Google Scholar 

  • Deitmar A, Echterhoff S (2009) Principles of harmonic analysis. Springer, New York

    MATH  Google Scholar 

  • Eaton ML (1983) Multivariate statistics: a vector space approach. Wiley, New York

    MATH  Google Scholar 

  • Eaton ML (1989) Group invariance applications in statistics. Institute of Mathematical Statistics and American Statistical Association, Hayward

    Book  Google Scholar 

  • 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, Amsterdam

  • Fisher RA (1973) Statistical methods and scientific inference. Hafner, New York

    MATH  Google Scholar 

  • Folland GB (2015) A course in abstract harmonic analysis, 2nd edn. Chapman and Hall/CRC Press, Boca Raton

    MATH  Google Scholar 

  • Fraser DAS (1961) The fiducial method and invariance. Biometrika 48:261–280

    Article  MathSciNet  Google Scholar 

  • Fraser DAS (1968) The structure of inference. Wiley, New York

    MATH  Google Scholar 

  • Garcia G, Oller JM (2001) Minimum Riemannian risk equivariant estimator for the univariate normal model. Stat Probab Lett 52:109–113

    Article  MathSciNet  Google Scholar 

  • Hall WJ, Wijsman RAM (1965) Minimum Riemannian risk equivariant estimator for the univariate normal model. Staand Ghosh, J. K

  • Hall WJ, Wijsman RAM (2001) The relationship between sufficiency and invariance with applications in sequential analysis. Ann Math Stat 36:575–614

    Article  MathSciNet  Google Scholar 

  • Ilmonen P, Oja H, Serfling R (2012) On invariant coordinate system (ICS) functionals. Int Stat Rev 80:93–110

    Article  MathSciNet  Google Scholar 

  • 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 Press

  • Kiefer J (1957) Invariance, minimax sequential estimation and continuous time processes. Ann Math Stat 28:573–601

    Article  MathSciNet  Google Scholar 

  • Konno Y (2007) Improving on the sample covariance matrix for a complex elliptically contoured distribution. J Stat Plan Inference 137:2475–2486

    Article  MathSciNet  Google Scholar 

  • Kraft H (2016) Algebraic transformation groups: an introduction. Mathematisches Institut, Universitat Basel

  • Kubokawa T, Konno Y (1990) Estimating the covariance matrix and the generalized variance under a symmetric loss. Ann Inst Stat Math 42:331–343

    Article  MathSciNet  Google Scholar 

  • Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Lehmann EL, Romano JP (2005) Testing statistical hypotheses, 3rd edn. Springer, New York

    MATH  Google Scholar 

  • 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 313–326

    Chapter  Google Scholar 

  • Palais RS (1960) Classification of G-spaces, vol 36. Memoirs of the American Mathematical Society, Providence

    MATH  Google Scholar 

  • Peisakoff M (1939) Transformation parameters, Thesis, Princeton University, Princeton, N.J. Pitman, E. J. G

  • Peisakoff M (1950) The estimation of location and scale parameters of continuous population of any givan form. Biometrika 39:391–421

    Google Scholar 

  • Pitman EJG (1939) The estimation of location and scale parameters of continuous population of any givan form. Biometrika 39:391–421

    Article  Google Scholar 

  • Robert G (1971) A characterization of invariant loss functions. Ann Math Stat 42(4):1322–1327

    Article  MathSciNet  Google Scholar 

  • Sanjar NF, Zakerzadeh H (2005) Estimation of a gamma scale parameter under asymmetric squared-log error loss. Commun Stat Theory Methods 34(5):1127–1135

    Article  MathSciNet  Google Scholar 

  • Serfling R (2010) Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardization. J Nonparametr Stat 22:915–936

    Article  MathSciNet  Google Scholar 

  • 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 445–457

    Google Scholar 

  • Svensson L (2004) A useful identity for complex Wishart forms. Technical report, Department of Signals and Systems. Chalmers University of Technology

  • Zhang S, Sha Q (1997) On the best equivariant estimator of covariance matrix of a multivariate normal population. Commun Stat Theory Methods 26(8):2021–2034

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Statistics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

    M. Shams

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  1. M. Shams
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Correspondence to M. Shams.

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Shams, M. On weakly equivariant estimators. Stat Papers 62, 1611–1650 (2021). https://doi.org/10.1007/s00362-019-01149-0

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  • DOI: https://doi.org/10.1007/s00362-019-01149-0

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