Skip to main content

Advertisement

SpringerLink
Log in

Matrix Euclidean norm Wielandt inequalities and their applications to statistics

  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

In this paper we shall establish a new matrix inequality which will fill the gap that there has not been any matrix Euclidean norm version of the Wielandt inequality in the literature yet. This inequality can be used to present an upper bound of a new measure of association which plays a very important role in statistics, especially in multivariate analysis. A new alternative based on Euclidean norm for relative gain of the covariance adjusted estimator of parameters is provided.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. Wiley, New York

    MATH  Google Scholar 

  • Baksalary JK, Puntanen S (1991) Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities. Aequationes Math 24: 182–187

    MathSciNet  Google Scholar 

  • Baksalary JK, Puntanen S, Styan GPH (1990) A property of the dispersion matrix of the best linear unbiased estimator in the general Gauss-Markov model. Sankhyā Ser A 52: 279–296

    MathSciNet  MATH  Google Scholar 

  • Bauer FL, Householder AS (1960) Some inequalities involving the Euclidean condition of a matrix. Numerische Mathematik 2: 308–311

    Article  MathSciNet  MATH  Google Scholar 

  • Bhatia R, Davis C (2000) A better bound on the variance. Am Math Mon 107: 353–357

    Article  MathSciNet  MATH  Google Scholar 

  • Bloomfield P, Watson GS (1975) The inefficiency of least squares. Biometrika 62: 121–128

    Article  MathSciNet  MATH  Google Scholar 

  • Drury SW, Liu SZ, Lu CY, Puntanen S, Styan GPH (2002) Some comments on several matrix inequalities with applications to canonical correlations: historical background and recent developments. Sankhyā Ser A 62: 453–507

    MathSciNet  Google Scholar 

  • Eaton ML (1976) A maximization problem and its applications to canonical correlation. J Multivar Anal 6: 422–425

    Article  MathSciNet  MATH  Google Scholar 

  • Groß J (2004) The general Gauss-Markov model with possibly singular dispersion matrix. Stat Pap 45: 311–336

    Article  MATH  Google Scholar 

  • Gustafson K (1999) The geometrical meaning of the Kantorovich-Wielandt inequalities. Linear Algebra Appl 296: 143–151

    Article  MathSciNet  MATH  Google Scholar 

  • Liu SZ (2000) Efficiency comparisons between the OLSE and the BLUE in a singular linear model. J Stat Plan Inference 84: 191–200

    Article  MATH  Google Scholar 

  • Liu SZ, Neudecker H (1999) A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities. Stat Pap 40: 55–73

    Article  MathSciNet  MATH  Google Scholar 

  • Liu SZ, Lu CY, Puntanen S (2009) Matrix trace Wielandt inequalities with statistical applications. J Stat Plan Inference 139: 2254–2260

    Article  MathSciNet  MATH  Google Scholar 

  • Lu CY (1999) A generalized matrix version of the Wielandt inequality with some applications. Research report, Department of Mathematics, Northeast Normal University, Changchun, China

  • Marshall AW, Olkin I (1990) Matrix version of the Cauchy and Kantorovich inequalities. Aequationes Math 40: 89–93

    Article  MathSciNet  MATH  Google Scholar 

  • Pec̆arić JE, Puntanen S, Styan GPH (1996) Some further matrix extensions of the Cauchy-Schwarz and Kantorovich inequalities, with some statistical applications. Linear Algebra Appl 237/238: 455–476

    Article  Google Scholar 

  • Poincaré H (1890) Sur les équations aux dérivées partielles de la physique mathématique. Am J Math 12: 211–294

    Article  Google Scholar 

  • Scott AJ, Styan GPH (1985) On a separation theorem for generalized eigenvalues and a problem in the analysis of sample surveys. Linear Algebra Appl 70: 209–224

    Article  MathSciNet  MATH  Google Scholar 

  • Wang SG, Ip WC (1999) A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl 296: 171–181

    Article  MathSciNet  MATH  Google Scholar 

  • Wang LT, Yang H (2009) Several matrix Euclidean norm inequalities involving Kantorovich inequality. J Inequal Appl. doi:10.1155/2009/291984

    Google Scholar 

  • Watson GS, Alpargu G, Styan GPH (1997) Some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor. Linear Algebra Appl 264: 13–54

    Article  MathSciNet  MATH  Google Scholar 

  • Yang H (1988) Extensions of the Kantorovich inequality and the error ratio efficiency of the mean square. Mathematica Applicata (in China) 4: 85–90

    Google Scholar 

  • Yang H, Wang LT (2009) An alternative form of the Watson efficiency. J Stat Plan Inference 139: 2767–2774

    Article  MATH  Google Scholar 

  • Zhang F (2001) Equivalence of the Wielandt inequality and the Kantorovich inequality. Linear Multilinear Algebra 48: 275–279

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, China

    Litong Wang

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, China

    Hu Yang

Authors
  1. Litong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Litong Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, L., Yang, H. Matrix Euclidean norm Wielandt inequalities and their applications to statistics. Stat Papers 53, 521–530 (2012). https://doi.org/10.1007/s00362-010-0357-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-010-0357-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation